Search for gravitational waves associated with the gamma ray burst GRB030329
using the LIGO detectors
B. Abbott,
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34,r
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15
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28
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1
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11
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PHYSICAL REVIEW D
72,
042002 (2005)
1550-7998
=
2005
=
72(4)
=
042002(17)$23.00
042002-1
2005 The American Physical Society
G. Woan,
34
R. Wooley,
14
J. Worden,
13
W. Wu ,
33
I. Yakushin,
14
H. Yamamoto,
11
S. Yoshida,
24
K. D. Zaleski,
27
M. Zanolin,
12
I. Zawischa,
30,
L. Zhang,
11
R. Zhu,
1
N. Zotov,
16
M. Zucker,
14
and J. Zweizig
11
(LIGO Scientific Collaboration)
1
Albert-Einstein-Institut, Max-Planck-Institut fu
̈
r Gravitationsphysik, D-14476 Golm, Germany
2
Albert-Einstein-Institut, Max-Planck-Institut fu
̈
r Gravitationsphysik, D-30167 Hannover, Germany
3
Australian National University, Canberra, 0200, Australia
4
California Institute of Technology, Pasadena, California 91125, USA
5
California State University Dominguez Hills, Carson, California 90747, USA
6
Caltech-CaRT, Pasadena, California 91125, USA
7
Cardiff University, Cardiff, CF2 3YB, United Kingdom
8
Carleton College, Northfield, Minnesota 55057, USA
9
Hobart and William Smith Colleges, Geneva, New York 14456, USA
10
Inter-University Centre for Astronomy and Astrophysics, Pune – 411007, India
11
LIGO –California Institute of Technology, Pasadena, California 91125, USA
12
LIGO –Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
13
LIGO Hanford Observatory, Richland, Washington 99352, USA
14
LIGO Livingston Observatory, Livingston, Louisiana 70754, USA
15
Louisiana State University, Baton Rouge, Louisiana 70803, USA
16
Louisiana Tech University, Ruston, Louisiana 71272, USA
17
Loyola University, New Orleans, Louisiana 70118, USA
18
Max Planck Institut fu
̈
r Quantenoptik, D-85748, Garching, Germany
19
Moscow State University, Moscow, 119992, Russia
20
NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
21
National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
22
Northwestern University, Evanston, Illinois 60208, USA
23
Salish Kootenai College, Pablo, Montana 59855, USA
24
Southeastern Louisiana University, Hammond, Louisiana 70402, USA
25
Stanford University, Stanford, California 94305, USA
26
Syracuse University, Syracuse, New York 13244, USA
27
The Pennsylvania State University, University Park, Pennsylvania 16802, USA
28
The University of Texas at Brownsville and Texas Southmost College, Brownsville, Texas 78520, USA
29
Trinity University, San Antonio, Texas 78212, USA
30
Universita
̈
t Hannover, D-30167 Hannover, Germany
31
Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
32
University of Birmingham, Birmingham, B15 2TT, United Kingdom
33
University of Florida, Gainesville, Florida 32611, USA
34
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
35
University of Michigan, Ann Arbor, Michigan 48109, USA
36
University of Oregon, Eugene, Oregon 97403, USA
37
University of Rochester, Rochester, New York 14627, USA
38
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA
39
Washington State University, Pullman, Washington 99164, USA
(Received 21 February 2005; published 12 August 2005)
We have performed a search for bursts of gravitational waves associated with the very bright gamma ray
burst GRB030329, using the two detectors at the LIGO Hanford Observatory. Our search covered the most
sensitive frequency range of the LIGO detectors (approximately
80
–
2048 Hz
), and we specifically
targeted signals shorter than
’
150 ms
. Our search algorithm looks for excess correlated power between
the two interferometers and thus makes minimal assumptions about the gravitational waveform. We
observed no candidates with gravitational-wave signal strength larger than a predetermined threshold. We
report frequency-dependent upper limits on the strength of the gravitational waves associated with
GRB030329. Near the most sensitive frequency region, around
’
250 Hz
, our root-sum-square (RSS)
gravitational-wave strain sensitivity for optimally polarized bursts was better than
h
RSS
’
6
10
21
Hz
1
=
2
. Our result is comparable to the best published results searching for association between
gravitational waves and gamma ray bursts.
DOI:
10.1103/PhysRevD.72.042002
PACS numbers: 04.80.Nn, 07.05.Kf, 95.85.Sz, 97.60.Bw
B. ABBOTT
et al.
PHYSICAL REVIEW D
72,
042002 (2005)
042002-2
I. INTRODUCTION
Gamma ray bursts (GRBs) are short but very energetic
pulses of gamma rays from astrophysical sources, with
duration ranging between 10 ms and 100 s. GRBs are
historically divided into two classes [1,2] based on their
duration: ‘‘short’’ (
<
2s
) and ‘‘long’’ (
>
2s
). Both
classes are isotropically distributed and their detection
rate can be as large as one event per day. The present
consensus is that long GRBs [2] are the result of the core
collapse of massive stars resulting in black hole formation.
The violent formation of black holes has long been pro-
posed as a potential source of gravitational waves. There-
fore, we have reason to expect strong association between
GRBs and gravitational waves [3–5]. In this paper, we
report on a search for a possible short burst of gravitational
waves associated with GRB030329 using data collected by
the Laser Interferometer Gravitational-Wave Observatory
(LIGO).
On March 29, 2003, instrumentation aboard the HETE-2
satellite [6] detected a very bright GRB, designated
GRB030329. The GRB was followed by a bright and
well-measured afterglow from which a redshift [7] of
z
0
:
1685
(distance
’
800 Mpc
[8]) was determined. After
approximately 10 days, the afterglow faded to reveal an
underlying supernova (SN) spectrum, SN2003dh [9]. This
GRB is the best studied to date, and confirms the link
between long GRBs and supernovae.
At the time of GRB030329, LIGO was engaged in a 2-
month long data run. The LIGO detector array consists of
three interferometers, two at the Hanford, WA site and one
at the Livingston, LA site. Unfortunately, the Livingston
interferometer was not operating at the time of the GRB;
therefore, the results presented here are based on the data
from only the two Hanford interferometers. The LIGO
detectors are still undergoing commissioning, but at the
time of GRB030329, their sensitivity over the frequency
band 80 to 2048 Hz exceeded that of any previous
gravitational-wave search, with the lowest strain noise of
’
6
10
22
Hz
1
=
2
around 250 Hz.
A number of long GRBs have been associated with
x-ray, radio and/or optical afterglows, and the cosmologi-
cal origin of the host galaxies of their afterglows has been
unambiguously established by their observed redshifts,
which are of order unity [2]. The smallest observed redshift
of an optical afterglow associated with a detected GRB
(GRB980425 [10 –12]) is
z
0
:
0085
(
’
35 Mpc
). GRB
emissions are very likely strongly beamed [13,14], a factor
that affects estimates of the energy released in gamma rays
(a few times
10
50
erg
), and their local true event rate (about
1 per year within a distance of 100 Mpc).
In this search, we have chosen to look for a burst of
gravitational waves in a model independent way. Core-
collapse [4], black hole formation [5,15] and black hole
ringdown [16,17] may each produce gravitational-wave
emissions, but there are no accurate or comprehensive
predictions describing the gravitational-wave signals that
might be associated with GRB type sources. Thus, a tradi-
tional matched filtering approach [18,19] is not possible in
this case. To circumvent the uncertainties in the wave-
forms, our algorithm does not presume any detailed knowl-
edge of the gravitational waveform and we only apply
general bounds on the waveform parameters. Based on
current theoretical considerations, we anticipate the signals
in our detectors to be weak, comparable to or less than the
detector’s noise [20 – 22].
This paper is organized as follows: Section II
summarizes the currently favored theories of GRBs and
their consequences for gravitational-wave detection.
Section III provides observational details pertinent to
r
Present address: ESA Science and Technology Center.
s
Present address: Raytheon Corporation.
t
Present address: NM Institute of Mining and Technology /
Magdalena Ridge Observatory Interferometer.
u
Present address: Mission Research Corporation.
v
Present address: Harvard University.
w
Permanent address: Columbia University.
x
Present address: Lockheed-Martin Corporation.
y
Permanent address: University of Tokyo, Institute for Cosmic
Ray Research.
z
Permanent address: University College Dublin.
aa
Present address: Research Electro-Optics Inc.
ab
Present address: Institute of Advanced Physics, Baton Rouge,
LA.
ac
Present address: Thirty Meter Telescope Project at Caltech.
ad
Present address: European Commission, DG Research,
Brussels, Belgium.
ae
Present address: University of Chicago.
af
Present address: LightBit Corporation.
al
Electronic address: http://www.ligo.org
d
Present address: Rutherford Appleton Laboratory.
c
Permanent address: HP Laboratories.
b
Present address: Jet Propulsion Laboratory.
ag
Permanent address: IBM Canada Ltd.
ah
Present address: University of DE.
ai
Permanent address: Jet Propulsion Laboratory.
ak
Present address: Laser Zentrum Hannover.
aj
Present address: Shanghai Astronomical Observatory.
a
Present address: Stanford Linear Accelerator Center.
e
Present address: University of CA, Los Angeles.
f
Present address: Hofstra University.
g
Permanent address: GReCO, Institut d’Astrophysique de
Paris (CNRS).
h
Present address: La Trobe University, Bundoora VIC,
Australia.
i
Present address: Keck Graduate Institute.
j
Present address: National Science Foundation.
k
Present address: University of Sheffield.
l
Present address: Ball Aerospace Corporation.
m
Present address: European Gravitational Observatory.
n
Present address: Intel Corp.
o
Present address: University of Tours, France.
p
Present address: Lightconnect Inc.
q
Present address: W.M. Keck Observatory.
SEARCH FOR GRAVITATIONAL WAVES ASSOCIATED
...
PHYSICAL REVIEW D
72,
042002 (2005)
042002-3
GRB030329. Section IV briefly describes the LIGO detec-
tors and their data. Section V discusses the method of
analysis of the LIGO data. In Section VI we compare the
events in the signal region with expectations and we use
simulated signal waveforms to determine detection effi-
ciencies. We also present and interpret the results in this
section. Section VII offers a comparison with previous
analyses, a conclusion, and an outlook for future searches
of this type.
II. PRODUCTION OF GRAVITATIONAL WAVES IN
MASSIVE CORE COLLAPSES
The apparent spatial association of GRB afterglows with
spiral arms, and by implication star formation regions in
remote galaxies, has lead to the current ‘‘collapsar’’ or
‘‘hypernova’’ scenario [23,24] in which the collapse of a
rotating, massive star to a Kerr black hole can lead to
relativistic ejecta emitted along a rotation axis and the
associated production of a GRB jet. The identification of
GRB030329 with the supernova SN2003dh (Sec. III be-
low) gives further support to this association. This obser-
vation is consistent with the theory that the GRB itself is
produced by an ultrarelativistic jet associated with a central
black hole. Stellar mass black holes in supernovae must
come from more massive stars. Reference [24] presents
‘‘maps’’ in the metallicity-progenitor mass plane of the end
states of stellar evolution and shows that progenitors with
25
M
can produce black holes by fallback accretion.
The observed pulsar kick velocities of
’
500 km
=
s
hint
at a strong asymmetry around the time of maximum com-
pression, which may indicate deviations from spherical
symmetry in the progenitor. The resulting back reaction
on the core from the neutrino heating provides yet another
potential physical mechanism for generating a gravita-
tional-wave signal. In the model of [25] it imparts a kick
of
400
–
600 km
=
s
and an induced gravitational-wave strain
roughly an order of magnitude larger than in [20] and an
order of magnitude smaller than [26].
Theoretical work on gravitational-wave (GW) signals in
the process of core collapse in massive stars has advanced
much in recent years, but still does not provide detailed
waveforms. Current models take advantage of the increase
in computational power and more sophisticated input phys-
ics to include both 2D and 3D calculations, utilizing real-
istic precollapse core models and a detailed, complex equa-
tion of state of supernovae that produce neutron stars. The
most recent studies by independent groups give predictions
for the strain amplitude within a similar range, despite the
fact that the dominant physical mechanisms for gravita-
tional-wave emission in these studies are different
[20,21,26 – 28]. The calculations of [20] are qualitatively
different from previous core-collapse simulations in that
the dominant contribution to the gravitational-wave signal
is neutrino-driven convection, about 20 times larger than
the axisymmetric core-bounce gravitational-wave signal.
The applicability of the above models to GRBs is not
clear, since the model end points are generally neutron
stars, rather than black holes. Another recent model in-
volves accretion disks around Kerr black holes [29], sub-
ject to nonaxisymmetric Papaloizou-Pringle instabilities
[30] in which an acoustic wave propagates toroidally
within the fallback material. They are very interesting
since they predict much higher amplitudes for the
gravitational-wave emission.
For our search, the main conclusion to draw is that in
spite of the dramatic improvement in the theoretical mod-
els, there are no gravitational waveforms that could be
reliably used as templates for a matched filter search, and
that any search for gravitational waves should ideally be as
waveform independent as practical. Conversely, detection
of gravitational waves associated with a GRB would al-
most certainly provide crucial new input for GRB/SN
astrophysics. It is also clear that the predictions of
gravitational-wave amplitudes are uncertain by several
orders of magnitude, making it difficult to predict the
probability to observe the gravitational-wave signature of
distant GRBs.
The timeliness of searching for a gravitational-wave
signal associated with GRBs is keen in light of the recent
work by [20,21]. Reference [20] finds that the signal due to
neutrino convection exceeds that due to the core bounce
and therefore a chaotic signal would be expected. Studies
with simplified or no neutrino transport (e.g., [21,22]) find
the core bounce to be the dominant contributor to the GW
signal. The large-scale, coherent mass motions involved in
the core bounce leads to a predicted gravitational-wave
signal resembling a damped sinusoid.
III. GRB030329 RELATED OBSERVATIONAL
RESULTS
A. Discovery of GRB 030329 and its afterglow
On March 29, 2003 at 11:37:14.67 UTC, a GRB trig-
gered the FREGATE instrument on board the HETE-2
satellite [6,31– 33]. The GRB had an effective duration of
’
50 s
, and a fluence of
1
:
08
10
4
erg
=
cm
2
in the 30 –
400 keV band [33]. The KONUS detector on board the
Wind satellite also detected it [34], triggering about 15 sec
after HETE-2. KONUS observed the GRB for about 35 sec,
and measured a fluence of
1
:
6
10
4
erg
=
cm
2
in the 15–
5000 keV band. The measured gamma ray fluences place
this burst among the brightest GRBs. Figure 1 shows the
HETE-2 light curve for GRB030329 [35].
The rapid localization of the GRB by HETE ground
analysis gave an accurate position which was distributed
about 73 min after the original trigger. A few hours later, an
optical afterglow [7,36] was discovered with magnitude
R
12
:
4
, making it the brightest optical counterpart to any
GRB detected to date. The RXTE [37] satellite measured a
x-ray flux of
1
:
4
10
10
erg s
1
cm
2
in the 2 –10 keV
band about 4 h 51 m after the HETE trigger, making this
B. ABBOTT
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one of the brightest x-ray afterglows detected by RXTE
[38]. The National Radio Astronomy Observatory (NRAO)
observed [39] the radio afterglow, which was the brightest
radio afterglow detected to date [40]. Spectroscopic mea-
surements of the bright optical afterglow [41] revealed
emission and absorption lines, and the inferred redshift
(
z
0
:
1685
, luminosity distance
D
L
800 Mpc
) made
this the second nearest GRB with a measured distance.
To date, no host galaxy has been identified. It is likely that
numerous other GRBs have been closer than GRB030329,
but the lack of identified optical counterparts has left their
distances undetermined.
Spectroscopic measurements [8,42,43], about a week
after the GRB trigger, revealed evidence of a supernova
spectrum emerging from the light of the bright optical
afterglow, which was designated SN2003dh. The emerging
supernova spectrum was similar to the spectrum of
SN1998bw a week before its brightness maximum [44,45].
SN1998bw was a supernova that has been spatially and
temporally associated with GRB980425 [10 –12], and was
located in a spiral arm of the barred spiral galaxy ESO 184-
G82 at a redshift of
z
0
:
0085
(
’
35 Mpc
), making it the
nearest GRB with a measured distance. The observed
spectra of SN2003dh and SN1998bw, with their lack of
hydrogen and helium features, place them in the Type Ic
supernova class. These observations, together with the
observations linking GRB980425 (which had a duration
of
’
23 s
) to SN1998bw, make the case that collapsars are
progenitors for long GRBs more convincing. In the case of
SN1998bw, Woosley
et al.
[46] and Iwamoto
et al.
[11]
found that its observed optical properties can be well
modeled by the core collapse of a
C
O
core of mass
6
M
(main sequence mass of
25
M
) with a kinetic energy
of
’
2
10
52
ergs
. This energy release is about an order of
magnitude larger than the energies associated with typical
supernovae.
B. GRB030329 energetics
A widely used albeit naive quantity to describe the
energy emitted by GRBs is the total isotropic equivalent
energy in gamma rays:
E
iso
4
BC
D
2
L
f=
1
z
2
10
52
erg
(3.1)
where
f
is the measured fluence in the HETE-2 waveband
and BC is the approximate bolometric correction for
HETE-2 for long GRBs. Using a ‘‘band spectrum’’ [47]
with a single power law to model the gamma ray spectrum,
and using a spectral index,
2
:
5
, gives that the GRB’s
total energy integrated from 1 to 5 GeV is greater than that
present in the band 30 – 400 keV by a factor 2.2.
However, it is generally believed that GRBs are strongly
beamed, and that the change in slope in the afterglow light
curve corresponds to the time when enough deceleration
has occurred so that relativistic beaming is diminished to
the point at which we ‘‘see’’ the edge of the jet. This occurs
during the time in which the relativistic ejecta associated
with the GRB plows through the interstellar medium, and
the beaming factor
1
, where
is the bulk Lorentz factor
of the flow, increases from a value smaller than the beam-
ing angle
j
, to a value larger than
j
. Effectively, prior to
this time the relativistic ejecta appears to be part of a
spherical expansion, the edges of which cannot be seen
because they are outside of the beam, while after this time
the observer perceives a jet of finite width.
This leads to a faster decline in the light curves. Zeh
et
al.
and Li
et al.
[48,49] show that the initial ‘‘break’’ or
strong steepness in the light curve occurs at about 10 h after
the initial HETE-2 detection.
Frail
et al.
[13] give a parametric relation between
beaming angle
j
, break time
t
j
, and
E
iso
as
j
0
:
057
t
j
24 h
3
=
8
1
z
2
3
=
8
E
iso
10
53
ergs
1
=
8
0
:
2
1
=
8
n
0
:
1cm
3
1
=
8
;
(3.2)
where
j
is measured in radians. It was argued that the
fireball converts the energy in the ejecta into gamma rays
efficiently [50] (
0
:
2
), and that the mean circumburst
density is
n
0
:
1cm
3
[51]. Evaluating Eq. (3.2) for the
parameters of GRB030329 (
t
j
10 h
,
z
0
:
1685
, and
E
iso
2
10
52
erg
)gives
j
0
:
07 rad
.
Therefore the beaming factor that relates the actual
energy released in gamma rays (
E
) to the isotropic
equivalent energy is
2
j
=
2
1
=
400
, so that
E
5
10
49
erg
. Comparing
E
iso
and
E
with the histograms
in Fig. 2 of Frail
et al.
[13], GRB030329 resides at the
lower end of the energy distributions. The calculated iso-
FIG. 1. The GRB030329 light curve as measured by the
HETE-2 FREGATE B instrument. The arrow indicates the
HETE trigger time. The signal region analyzed in this study is
indicated by the horizontal bar at the top. This figure is the
courtesy of the HETE Collaboration.
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tropic energy from GRB980425, the GRB associated with
SN1998bw, is also low (
’
10
48
erg
).
IV. OVERVIEW OF THE LIGO DETECTORS
The three LIGO detectors are orthogonal arm Michelson
laser interferometers, aiming to detect gravitational waves
by interferometrically monitoring the relative (differential)
separation of mirrors, which play the role of test masses.
The LIGO Hanford Observatory (LHO) operates two iden-
tically oriented interferometric detectors, which share a
common vacuum envelope: one having 4 km long arms
(H1), and one having 2 km long arms (H2). The LIGO
Livingston Observatory operates a single 4 km long detec-
tor (L1). The two sites are separated by
’
3000 km
, rep-
resenting a maximum arrival time difference of
’
10 ms
.
A complete description of the LIGO interferometers as
they were configured during LIGO’s first science run (S1)
can be found in Ref. [52].
A. Detector calibration and configuration
To calibrate the error signal, the response to a known
differential arm strain is measured, and the frequency-
dependent effect of the feedback loop gain is measured
and compensated for. During detector operation, changes
in calibration are tracked by injecting continuous, fixed-
amplitude sinusoidal excitations into the end test mass
control systems, and monitoring the amplitude of these
signals at the measurement (error) point. Calibration un-
certainties at the Hanford detectors were estimated to be
<
11%
.
Significant improvements were made to the LIGO de-
tectors following the S1 run, held in early fall of 2002:
(1) The analog suspension controllers on the H2 and L1
interferometers were replaced with digital suspen-
sion controllers of the type installed on H1 during
S1, resulting in lower electronics noise.
(2) The noise from the optical lever servo that damps
the angular excitations of the interferometer optics
was reduced.
(3) The wave front sensing system for the H1 interfer-
ometer was used to control 8 of 10 alignment de-
grees of freedom for the main interferometer. As a
result, it maintained a much more uniform operating
point over the run.
(4) The high frequency sensitivity was improved by
operating the interferometers with higher effective
power, about 1.5 W.
These changes led to a significant improvement in de-
tector sensitivity. Figure 2 shows typical spectra achieved
by the LIGO interferometers during the S2 run. The dif-
ferences among the three LIGO spectra reflect differences
in the operating parameters and hardware implementations
of the three instruments which are in various stages of
reaching the final design configuration.
B. The second science run
The data analyzed in this paper were taken during
LIGO’s second science run (S2), which spanned approxi-
mately 60 days from February 14 to April 14, 2003. During
this time, operators and scientific monitors attempted to
maintain continuous low noise operation. The duty cycle
for the interferometers, defined as the fraction of the total
run time when the interferometer was locked and in its low
noise configuration, was approximately 74% for H1 and
58% for H2. The longest continuous locked stretch for any
interferometer during S2 was 66 hours for H1.
At the time of the GRB030329 both Hanford inter-
ferometers were locked and taking science mode data.
For this analysis we relied on the single,
’
4
:
5h
long
coincident lock stretch, which started
’
3
:
5h
before the
trigger time. With the exception of the signal region, we
utilized
’
98%
of the data within this lock stretch as the
background
region (defined in Sec. V). 60 sec of data
before and after the signal region were not included
in the background region. Data from the beginning
and from the end of the lock stretch were not included
in the background region to avoid using possibly non-
stationary data, which might be associated with these
regions.
As described below, the false alarm rate estimate, based
on background data, must be applicable to the data within
the signal region. We made a conservative choice and
avoided using background data outside of the lock stretch
containing the GRB trigger time. This is important when
considering the present nonstationary behavior of the in-
terferometric detectors.
FIG. 2.
Typical LIGO Hanford sensitivity curves during the S2
Run [strain
Hz
1
=
2
] (black and gray lines). The LIGO design
sensitivity goal (SRD) is also indicated (dashed line).
B. ABBOTT
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V. ANALYSIS
The goal of the analysis is either to identify significant
events in the signal region or, in the absence of significant
events, to set a limit on the strength of the associated
gravitational-wave signal. Simulations and background
data were used to determine the detection efficiency for
various
ad hoc
and model-based waveforms (Sec. VI B)
and the false alarm rate of the detection algorithm,
respectively.
The analysis takes advantage of the information pro-
vided by the astrophysical trigger. The trigger time deter-
mined when to perform the analysis. As discussed below,
the time window to be analyzed around the trigger time
was chosen to accommodate most current theoretical pre-
dictions and timing uncertainties. The source direction was
needed to calculate the attenuation due to the LIGO de-
tector’s antenna pattern for the astrophysical interpretation.
The two colocated and coaligned Hanford detectors had
very similar frequency-dependent response functions at the
time of the trigger. Consequently, the detected arrival time
and recorded waveforms of a gravitational-wave signal
should be essentially the same in both detectors. It is
natural then to consider cross correlation of the two data
streams as the basis of a search algorithm. This conclusion
can also be reached via a more formal argument based on
the maximum log-likelihood ratio test [53,54].
The schematic of the full analysis pipeline is shown in
Fig. 3. The underlying analysis algorithm is described in
detail in Ref. [54]. The background data, the signal region
data and the simulations are all processed identically. The
background region consists of the data where we do not
expect to have a gravitational-wave signal associated with
the GRB. We scan the background to determine the false
alarm distribution and to set a threshold on the event
strength that will yield an acceptable false alarm rate.
This threshold is used when scanning the signal region
and simulations. In order to estimate our sensitivity to
gravitational waves, simulated signals of varying strength
are added to the detector data streams.
The signal region around the GRB trigger is scanned to
identify outstanding signals. If events were detected above
threshold, in this region, their properties would be tested
against those expected from gravitational waves. If no
events were found above threshold, we would use the
estimated sensitivity to set an upper limit on the
gravitational-wave strain at the detector.
The output from each interferometer is divided into
330 sec long segments with a 15 sec overlap between
consecutive segments (both ends), providing a tiling of
the data with 300 sec long segments. In order to avoid
edge effects, the 180 sec long signal region lies in the
middle of one such 300 sec long segment. This tiling
method also allows for adaptive data conditioning and
places the conditioning filter (see Sec. V B 1 below) tran-
sients well outside of the 300 sec long segment containing
the signal region.
A. Choice of signal region
Current models suggest [2] that the gravitational-wave
signature should appear close to the GRB trigger time. We
conservatively chose the duration and position of the signal
region to over-cover most predictions and to allow for the
expected uncertainties associated with the GRB trigger
timing. A 180 sec long window (see Fig. 1), starting
120 sec before the GRB trigger time is sufficient; roughly
10 times wider than the GRB light curve features, and wide
enough to include most astrophysical predictions. Most
models favor an ordering where the arrival of the gravita-
tional wave precedes the GRB trigger [2], but in a few
other cases the gravitational-wave arrival is predicted to be
contemporaneous [5,55] to the arrival and duration of the
gamma rays (i.e. after the GRB trigger). The 60 sec region
after the GRB trigger time, is sufficient to cover these
predictions and also contains allowance for up to 30 sec
uncertainty on trigger timing, which is a reasonable choice
in the context of the HETE light curve. Figure 1 shows a
signal rise time of order
10 s
, precursor signals separated
from the main peak, and significant structure within the
main signal itself. Effects due to the beaming dynamics of
the GRB and the instrumental definition of the trigger time
can also be significant contributors to the timing
uncertainty.
B. Search algorithm
1. Data conditioning
The data-conditioning step was designed to remove
instrumental artifacts from the data streams. We used an
identical data-conditioning procedure when processing the
background, the signal region and the simulations.
FIG. 3.
The schematic of the analysis pipeline.
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The raw data streams have narrowband lines, associated
with the power line harmonics at multiples of 60 Hz, the
violin modes of the mirror suspension wires and other
narrow band noise sources. The presence of lines has a
detrimental effect on our sensitivity because lines can
produce spurious correlations between detectors. In addi-
tion, the broadband noise shows significant variations over
time scales of hours and smaller variations over time scales
of minutes and seconds due to alignment drift and fluctua-
tions. The background data must portray a representative
sample of the detector behavior around the time of the
trigger. Broadband nonstationarity can limit the duration of
this useful background data and hence the reliability of our
estimated false alarm rate.
Our cross-correlation-based algorithm performs best on
white spectra without line features. We use notch filters to
remove the well-known lines, such as power line and violin
mode harmonics from both data streams. Strong lines of
unknown origin with stationary mean frequency are also
removed at this point. We also apply a small correction to
mitigate the difference between the phase and amplitude
response of the two Hanford detectors.
We bandpass filter and decimate the data to a sampling
rate of 4096 Hz to restrict the frequency content to the
’
80
to
’
2048 Hz
region, which was the most sensitive band
for both LIGO Hanford detectors during the S2 run.
In order to properly remove weaker stationary lines and
the small residuals of notched strong lines, correct for
small slow changes in the spectral sensitivity and whiten
the spectrum of the data, we use adaptive line removal and
whitening. As all strong lines are removed before the
adaptive whitening, we avoid potential problems due to
nonstationary lines and enhance the efficiency of the
follow-up adaptive filtering stage. The conditioned data
has a consistent white spectrum without major lines and
sufficient stationarity, from segment to segment, through-
out the background and signal regions.
The end result of the preprocessing is a data segment
with a flat power spectral density (white noise), between
’
80
and
’
2048 Hz
. The data conditioning was applied
consistently after the signal injections. This ensures that
any change in detection efficiency due to the preprocessing
is properly taken into account.
2. Gravitational wave search algorithm
The test statistics for a pair of data streams are con-
structed as follows. We take pairs of short segments, one
from each stretch, and compute their cross-correlation
function. The actual form of the cross correlation used
(
~
C
m;n
k;p;j
) is identical to the common Euclidean inner prod-
uct:
~
C
m;n
k;p;j
X
j
i
j
H
m
k
i
H
n
k
p
i
;
(5.1)
where the preconditioned time series from detector ‘‘
x
’’ is
H
x
f
H
x
0
;H
x
1
;
...
g
and
i;k;p
and
j
are all integers
indexing the data time series, with each datum being
1
=
4096
s
long. As we now only consider the two
Hanford detectors ‘‘
m
’’ and ‘‘
n
’’ can only assume values
of 1 (
H
1
)or2(
H
2
). There are therefore three free parame-
ters to scan when searching for coherent segments of data
between a pair of interferometers (
m;n
): (i) the center time
of the segment from the first detector (
k
); (ii) the relative
time lag between the segments from the two detectors (
p
);
and (iii) the common duration of segments (
2
j
1
) called
the integration length.
The optimum integration length to use for computing
the cross correlation depends on the duration of the signal
and its signal-to-noise ratio, neither of which is
a priori
known. Therefore the cross correlation should be com-
puted from segment pairs with start times and lengths
varying over values, which should, respectively, cover the
expected arrival times (signal region) and consider dura-
tions of the gravitational-wave burst signals [20,21,25– 28]
[
O
1
–
128 ms
].
Hence we apply a search algorithm [54] that processes
the data in the following way.
(1) A three dimensional quantity (
C
k;j
p
) is con-
structed:
C
k;j
p
~
C
1
;
2
k;p;j
2
~
C
2
;
1
k;
p;j
2
1
=
2
;
(5.2)
scanning the range of segment center times (
k
), integration
lengths
(
2
j
1
)
and
relative
time
shifts
(
p
0
;
1
;
2
;
...
). A coherent and coincident signal is ex-
pected to leave its localized signature within this three
dimensional quantity.
We use a fine rectangular grid in relative time shift (
p
)
and integration length (
2
j
1
) space. The spacing be-
tween grid points is
’
1ms
for the segment center time
(
k
) and
1
=
4096
s
for the relative time shift. The spacing
of the integration lengths is approximately logarithmic.
Each consecutive integration length is
’
50%
longer than
the previous one, covering integration lengths from
’
1
to
’
128 ms
.
Introducing small, nonphysical relative time shifts
(much larger than the expected signal duration) between
the two data streams before computing the cross-
correlation matrix suppresses the average contribution
from a GW signal. This property can be used to estimate
the local noise properties, thereby mitigating the effects of
nonstationarity in the interferometer outputs. Accordingly,
C
k;j
p
contains the autocorrelation of the coherent signal
for relative time shifts at and near
p
0
(called ‘‘
core
’’),
while far away, in the ‘‘side
lobes
,’’ the contribution from
the signal autocorrelation is absent, sampling only the
random contributions to the cross correlation arising from
the noise. The optimal choice of the core size depends on
the expected signal duration (integration length), the
underlying detector noise and it cannot be smaller than
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the relative phase uncertainty of the data streams. The core
region can reach as far as 5 ms, as it increases with
increasing integration length. The size of each side lobe
is twice the size of the core region and the median time
shift associated with the side lobes can be as large as
325 ms as it is also increasing with increasing integration
length. We use the side lobes of
C
k;j
p
to estimate the
mean (
^
k;j
) and variance (
^
k;j
) of the local noise distribu-
tion, which is also useful in countering the effects of
nonstationarity.
(2) The three dimensional quantity is reduced to a two
dimensional image (see Fig. 4), called a
corrgram
,as
follows. The values of
C
k;j
p
in the core region are stand-
ardized by subtracting
^
k;j
and then dividing by
^
k;j
.
Positive
standardized values in the core region are summed
over
p
to determine the value of the corrgram pixel. Each
pixel is a measure of the excess cross correlation in the core
region when compared to the expected distribution char-
acterized by the side lobes for the given (
k;j
) combination.
(3) A list of events is found by recursively identifying
and characterizing significant regions (called ‘‘clusters’’)
in the corrgram image. Each event is described by its
arrival time, its optimal integration length and its strength
(ES). The event’s arrival time and its optimal integration
length correspond to the most significant pixel of the
cluster. The event strength is determined by averaging
the five most significant pixels of the cluster, as this is
helpful in discriminating against random fluctuations of the
background noise.
The strength of each event is then compared to a preset
detection threshold corresponding to the desired false
alarm rate. This detection threshold is determined via
extensive scans of the background region.
VI. RESULTS
A. False alarm rate measurements
In order to assess the significance of the cross-correlated
power of an event, we determined the false alarm rate
versus event strength distribution. We used the full back-
ground data stretch for this measurement.
Figure 5 shows the event rate as a function of the event
strength threshold for the background region. The error
bars reflect 90% C.L. Poisson errors, based on the number
of events within the given bin. We used this distribution to
FIG. 4 (color).
Examples of corrgram images. The horizontal
axes are time (linearly scaled) and the vertical axes are integra-
tion length (logarithmically scaled). The color axis, an indicator
of the excess correlation, is independently autoscaled for each
quadrant for better visibility, therefore the meaning of colors
differ from quadrant to quadrant. The time ticks also change
from quadrant to quadrant for better visibility. The rainbow type
color scale goes from blue to red, dark red marking the most
significant points within a quadrant. The upper two quadrants
show the corrgram image of injected sine Gaussians (250 Hz,
Q
8
:
9
). The bottom quadrants are examples of noise. The
maximum of the intensity scale is significantly higher for both
quadrants with injections, when compared to the noise examples.
The top left injection is strong enough to be significantly above
the preset detection threshold, while the top right injection is
weak enough to fall significantly below the detection threshold.
FIG. 5.
False alarm rate as a function of the event strength
threshold as determined from background data. The error bars
reflect 90% C.L. Poisson errors, based on the number of events
within the given bin. The pointer indicates the event strength
threshold used for the analysis, which corresponds to an inter-
polated false alarm rate of less than
5
10
4
Hz
. Note that the
signal region data is not included in this calculation. The position
of symbols correspond to the center of the bins.
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