arXiv:gr-qc/0501068v3 24 Jun 2005
A Search for Gravitational Waves Associated with the Gamma Ray Burst GRB030329 Using the
LIGO Detectors
LIGO-P040007-08-D
B. Abbott,12 R. Abbott,15 R. Adhikari,13 A. Ageev,20, 27 B. Allen,39 R. Amin,34 S. B. Anderson,12 W. G. Anderson,29
M. Araya,12 H. Armandula,12 M. Ashley,28 F. Asiri,12, a P. Aufmuth,31 C. Aulbert,1 S. Babak,7 R. Balasubramanian,7
S. Ballmer,13 B. C. Barish,12 C. Barker,14 D. Barker,14 M. Barnes,12, b B. Barr,35 M. A. Barton,12 K. Bayer,13 R. Beausoleil,26, c
K. Belczynski,23 R. Bennett,35, d S. J. Berukoff,1, e J. Betzwieser,13 B. Bhawal,12 I. A. Bilenko,20 G. Billingsley,12
E. Black,12 K. Blackburn,12 L. Blackburn,13 B. Bland,14 B. Bochner,13, f L. Bogue,12 R. Bork,12 S. Bose,40 P. R. Brady,39
V. B. Braginsky,20 J. E. Brau,37 D. A. Brown,39 A. Bullington,26 A. Bunkowski,2, 31 A. Buonanno,6, g R. Burgess,13
D. Busby,12 W. E. Butler,38 R. L. Byer,26 L. Cadonati,13 G. Cagnoli,35 J. B. Camp,21 J. K. Cannizzo,21 C. A. Cantley,35
L. Cardenas,12 K. Carter,15 M. M. Casey,35 J. Castiglione,34 A. Chandler,12 J. Chapsky,12, b P. Charlton,12, h S. Chatterji,13
S. Chelkowski,2, 31 Y. Chen,6 V. Chickarmane,16, i D. Chin,36 N. Christensen,8 D. Churches,7 T. Cokelaer,7 C. Colacino,33
R. Coldwell,34 M. Coles,15, j D. Cook,14 T. Corbitt,13 D. Coyne,12 J. D. E. Creighton,39 T. D. Creighton,12 D. R. M. Crooks,35
P. Csatorday,13 B. J. Cusack,3 C. Cutler,1 E. D’Ambrosio,12 K. Danzmann,31, 2 E. Daw,16, k D. DeBra,26 T. Delker,34, l
V. Dergachev,36 R. DeSalvo,12 S. Dhurandhar,11 A. Di Credico,27 M. Diaz,29 H. Ding,12 R. W. P. Drever,4 R. J. Dupuis,35
J. A. Edlund,12, b P. Ehrens,12 E. J. Elliffe,35 T. Etzel,12 M. Evans,12 T. Evans,15 S. Fairhurst,39 C. Fallnich,31 D. Farnham,12
M. M. Fejer,26 T. Findley,25 M. Fine,12 L. S. Finn,28 K. Y. Franzen,34 A. Freise,2, m R. Frey,37 P. Fritschel,13 V. V. Frolov,15
M. Fyffe,15 K. S. Ganezer,5 J. Garofoli,14 J. A. Giaime,16 A. Gillespie,12, n K. Goda,13 G. González,16 S. Goßler,31
P. Grandclément,23, o A. Grant,35 C. Gray,14 A. M. Gretarsson,15 D. Grimmett,12 H. Grote,2 S. Grunewald,1 M. Guenther,14
E. Gustafson,26, p R. Gustafson,36 W. O. Hamilton,16 M. Hammond,15 J. Hanson,15 C. Hardham,26 J. Harms,19 G. Harry,13
A. Hartunian,12 J. Heefner,12 Y. Hefetz,13 G. Heinzel,2 I. S. Heng,31 M. Hennessy,26 N. Hepler,28 A. Heptonstall,35 M. Heurs,31
M. Hewitson,2 S. Hild,2 N. Hindman,14 P. Hoang,12 J. Hough,35 M. Hrynevych,12, q W. Hua,26 M. Ito,37 Y. Itoh,1 A. Ivanov,12
O. Jennrich,35, r B. Johnson,14 W. W. Johnson,16 W. R. Johnston,29 D. I. Jones,28 L. Jones,12 D. Jungwirth,12, s V. Kalogera,23
E. Katsavounidis,13 K. Kawabe,14 S. Kawamura,22 W. Kells,12 J. Kern,15, t A. Khan,15 S. Killbourn,35 C. J. Killow,35
C. Kim,23 C. King,12 P. King,12 S. Klimenko,34 S. Koranda,39 K. Kötter,31 J. Kovalik,15, b D. Kozak,12 B. Krishnan,1
M. Landry,14 J. Langdale,15 B. Lantz,26 R. Lawrence,13 A. Lazzarini,12 M. Lei,12 I. Leonor,37 K. Libbrecht,12 A. Libson,8
P. Lindquist,12 S. Liu,12 J. Logan,12, u M. Lormand,15 M. Lubinski,14 H. Lück,31, 2 T. T. Lyons,12, u B. Machenschalk,1
M. MacInnis,13 M. Mageswaran,12 K. Mailand,12 W. Majid,12, b M. Malec,2, 31 F. Mann,12 A. Marin,13, v S. Márka,12, w
E. Maros,12 J. Mason,12, x K. Mason,13 O. Matherny,14 L. Matone,14 N. Mavalvala,13 R. McCarthy,14 D. E. McClelland,3
M. McHugh,18 J. W. C. McNabb,28 G. Mendell,14 R. A. Mercer,33 S. Meshkov,12 E. Messaritaki,39 C. Messenger,33
V. P. Mitrofanov,20 G. Mitselmakher,34 R. Mittleman,13 O. Miyakawa,12 S. Miyoki,12, y S. Mohanty,29 G. Moreno,14
K. Mossavi,2 G. Mueller,34 S. Mukherjee,29 P. Murray,35 J. Myers,14 S. Nagano,2 T. Nash,12 R. Nayak,11 G. Newton,35
F. Nocera,12 J. S. Noel,40 P. Nutzman,23 T. Olson,24 B. O’Reilly,15 D. J. Ottaway,13 A. Ottewill,39, z D. Ouimette,12, s
H. Overmier,15 B. J. Owen,28 Y. Pan,6 M. A. Papa,1 V. Parameshwaraiah,14 C. Parameswariah,15 M. Pedraza,12 S. Penn,10
M. Pitkin,35 M. Plissi,35 R. Prix,1 V. Quetschke,34 F. Raab,14 H. Radkins,14 R. Rahkola,37 M. Rakhmanov,34 S. R. Rao,12
K. Rawlins,13 S. Ray-Majumder,39 V. Re,33 D. Redding,12, b M. W. Regehr,12, b T. Regimbau,7 S. Reid,35 K. T. Reilly,12
K. Reithmaier,12 D. H. Reitze,34 S. Richman,13, aa R. Riesen,15 K. Riles,36 B. Rivera,14 A. Rizzi,15, bb D. I. Robertson,35
N. A. Robertson,26, 35 L. Robison,12 S. Roddy,15 J. Rollins,13 J. D. Romano,7 J. Romie,12 H. Rong,34, n D. Rose,12 E. Rotthoff,28
S. Rowan,35 A. Rüdiger,2 P. Russell,12 K. Ryan,14 I. Salzman,12 V. Sandberg,14 G. H. Sanders,12, cc V. Sannibale,12
B. Sathyaprakash,7 P. R. Saulson,27 R. Savage,14 A. Sazonov,34 R. Schilling,2 K. Schlaufman,28 V. Schmidt,12, dd R. Schnabel,19
R. Schofield,37 B. F. Schutz,1, 7 P. Schwinberg,14 S. M. Scott,3 S. E. Seader,40 A. C. Searle,3 B. Sears,12 S. Seel,12 F. Seifert,19
A. S. Sengupta,11 C. A. Shapiro,28, ee P. Shawhan,12 D. H. Shoemaker,13 Q. Z. Shu,34, ff A. Sibley,15 X. Siemens,39
L. Sievers,12, b D. Sigg,14 A. M. Sintes,1, 32 J. R. Smith,2 M. Smith,13 M. R. Smith,12 P. H. Sneddon,35 R. Spero,12, b
G. Stapfer,15 D. Steussy,8 K. A. Strain,35 D. Strom,37 A. Stuver,28 T. Summerscales,28 M. C. Sumner,12 P. J. Sutton,12
J. Sylvestre,12, gg A. Takamori,12 D. B. Tanner,34 H. Tariq,12 I. Taylor,7 R. Taylor,35 R. Taylor,12 K. A. Thorne,28 K. S. Thorne,6
M. Tibbits,28 S. Tilav,12, hh M. Tinto,4, b K. V. Tokmakov,20 C. Torres,29 C. Torrie,12 G. Traylor,15 W. Tyler,12 D. Ugolini,30
C. Ungarelli,33 M. Vallisneri,6, ii M. van Putten,13 S. Vass,12 A. Vecchio,33 J. Veitch,35 C. Vorvick,14 S. P. Vyachanin,20
L. Wallace,12 H. Walther,19 H. Ward,35 B. Ware,12, b K. Watts,15 D. Webber,12 A. Weidner,19 U. Weiland,31 A. Weinstein,12
R. Weiss,13 H. Welling,31 L. Wen,12 S. Wen,16 J. T. Whelan,18 S. E. Whitcomb,12 B. F. Whiting,34 S. Wiley,5 C. Wilkinson,14
P. A. Willems,12 P. R. Williams,1, jj R. Williams,4 B. Willke,31 A. Wilson,12 B. J. Winjum,28, e W. Winkler,2 S. Wise,34
A. G. Wiseman,39 G. Woan,35 R. Wooley,15 J. Worden,14 W. Wu,34 I. Yakushin,15 H. Yamamoto,12 S. Yoshida,25
K. D. Zaleski,28 M. Zanolin,13 I. Zawischa,31, kk L. Zhang,12 R. Zhu,1 N. Zotov,17 M. Zucker,15 and J. Zweizig12
2
(The LIGO Scientific Collaboration, http://www.ligo.org)
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
Australian National University, Canberra, 0200, Australia
4
California Institute of Technology, Pasadena, CA 91125, USA
5
California State University Dominguez Hills, Carson, CA 90747, USA
6
Caltech-CaRT, Pasadena, CA 91125, USA
7
Cardiff University, Cardiff, CF2 3YB, United Kingdom
8
Carleton College, Northfield, MN 55057, USA
9
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
10
Hobart and William Smith Colleges, Geneva, NY 14456, USA
11
Inter-University Centre for Astronomy and Astrophysics, Pune - 411007, India
12
LIGO - California Institute of Technology, Pasadena, CA 91125, USA
13
LIGO - Massachusetts Institute of Technology, Cambridge, MA 02139, USA
14
LIGO Hanford Observatory, Richland, WA 99352, USA
15
LIGO Livingston Observatory, Livingston, LA 70754, USA
16
Louisiana State University, Baton Rouge, LA 70803, USA
17
Louisiana Tech University, Ruston, LA 71272, USA
18
Loyola University, New Orleans, LA 70118, USA
19
Max Planck Institut für Quantenoptik, D-85748, Garching, Germany
20
Moscow State University, Moscow, 119992, Russia
21
NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
22
National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
23
Northwestern University, Evanston, IL 60208, USA
24
Salish Kootenai College, Pablo, MT 59855, USA
25
Southeastern Louisiana University, Hammond, LA 70402, USA
26
Stanford University, Stanford, CA 94305, USA
27
Syracuse University, Syracuse, NY 13244, USA
28
The Pennsylvania State University, University Park, PA 16802, USA
29
The University of Texas at Brownsville and Texas Southmost College, Brownsville, TX 78520, USA
30
Trinity University, San Antonio, TX 78212, USA
31
Universität Hannover, D-30167 Hannover, Germany
32
Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
33
University of Birmingham, Birmingham, B15 2TT, United Kingdom
34
University of Florida, Gainesville, FL 32611, USA
35
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
36
University of Michigan, Ann Arbor, MI 48109, USA
37
University of Oregon, Eugene, OR 97403, USA
38
University of Rochester, Rochester, NY 14627, USA
39
University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
40
Washington State University, Pullman, WA 99164, USA
( RCS ; compiled 6 February 2008)
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 pre-determined 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 hRSS ≃6×10−21 Hz−1/2 . Our result is comparable to the best published results searching
for association between gravitational waves and GRBs.
PACS numbers: 04.80.Nn, 07.05.Kf, 95.85.Sz 97.60.Bw
a Currently
b Currently
at Stanford Linear Accelerator Center
at Jet Propulsion Laboratory
c Permanent
Address: HP Laboratories
3
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” (< 2 s) and “long” (> 2 s). 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 proposed as a potential source of gravitational
waves. Therefore, we have reason to expect strong association
between GRBs and gravitational waves [3, 4, 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 HETE2 satellite [6] detected a very bright GRB, designated
GRB030329. The GRB was followed by a bright and wellmeasured 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
d Currently
e Currently
at Rutherford Appleton Laboratory
at University of California, Los Angeles
f Currently at Hofstra University
g Permanent Address: GReCO, Institut d’Astrophysique de Paris (CNRS)
h Currently at La Trobe University, Bundoora VIC, Australia
i Currently at Keck Graduate Institute
j Currently at National Science Foundation
k Currently at University of Sheffield
l Currently at Ball Aerospace Corporation
m Currently at European Gravitational Observatory
n Currently at Intel Corp.
o Currently at University of Tours, France
p Currently at Lightconnect Inc.
q Currently at W.M. Keck Observatory
r Currently at ESA Science and Technology Center
s Currently at Raytheon Corporation
t Currently at New Mexico Institute of Mining and Technology / Magdalena
Ridge Observatory Interferometer
u Currently at Mission Research Corporation
v Currently at Harvard University
w Permanent Address: Columbia University
x Currently at Lockheed-Martin Corporation
y Permanent Address: University of Tokyo, Institute for Cosmic Ray Research
z Permanent Address: University College Dublin
aa Currently at Research Electro-Optics Inc.
bb Currently at Institute of Advanced Physics, Baton Rouge, LA
cc Currently at Thirty Meter Telescope Project at Caltech
dd Currently at European Commission, DG Research, Brussels, Belgium
ee Currently at University of Chicago
ff Currently at LightBit Corporation
gg Permanent Address: IBM Canada Ltd.
hh Currently at University of Delaware
ii Permanent Address: Jet Propulsion Laboratory
jj Currently at Shanghai Astronomical Observatory
kk Currently at Laser Zentrum Hannover
and supernovae.
At the time of GRB030329, LIGO was engaged in a 2month 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 cosmological 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, 11,
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 1050 erg), and
their local true event rate (about 1 per year within a distance
of 100Mpc).
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 traditional matched
filtering approach [18, 19] is not possible in this case. To
circumvent the uncertainties in the waveforms, our algorithm
does not presume any detailed knowledge 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, 21, 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 GRB030329. Section IV briefly
describes the LIGO detectors 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 efficiencies. 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,
4
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 (section 3 below) gives further support to
this association. This observation is consistent with the theory that the GRB itself is produced by an ultra-relativistic jet
associated with a central black hole. Stellar mass black holes
in supernovae must come from more massive stars. Ref. [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 fall-back accretion.
The observed pulsar kick velocities of ≃500 km/s hint at
a strong asymmetry around the time of maximum compression, 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 gravitational 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 physics to include
both 2D and 3D calculations, utilizing realistic pre-collapse
core models and a detailed, complex equation 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 gravitational wave emission in
these studies are different [20, 21, 26, 27, 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 endpoints are generally neutron stars,
rather than black holes. Another recent model involves accretion disks around Kerr black holes [29], subject 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 models, 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 almost 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.
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.
The timeliness of searching for a gravitational wave signal
associated with GRBs is keen in light of the recent work by
[21] and [20]. [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 largescale, 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 triggered the FREGATE instrument on board the HETE-2 satellite [6, 31, 32, 33]. The GRB had an effective duration of
≃50 s, and a fluence of 1.08×10−4 erg/cm2 in the 30-400 keV
band [33]. The KONUS detector on board the Wind satellite also detected it [34], triggering about 15 seconds after
HETE-2. KONUS observed the GRB for about 35 seconds,
and measured a fluence of 1.6×10−4 erg / cm2 in the 155000 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
minutes 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
4h51m after the HETE trigger, making this one of the brightest X-ray afterglows detected by RXTE [38]. The National
5
Radio Astronomy Observatory (NRAO) observed [39] the
radio afterglow, which was the brightest radio afterglow detected to date [40]. Spectroscopic measurements of the bright
optical afterglow [41] revealed emission and absorption lines,
and the inferred redshift (z = 0.1685, luminosity distance
DL ∼
= 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, 11, 12], and was
located in a spiral arm of the barred spiral galaxy ESO 184G82 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 ≃2×1052 ergs. This energy
release is about an order of magnitude larger than the energies
associated with typical supernovae.
to a value larger than Θj . 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 hours after the
initial HETE-2 detection.
Frail et al. [13] give a parametric relation between beaming
angle Θj , break time tj , and Eiso as:
3/8
−3/8
tj
1+z
Θj ≈ 0.057
×
24hours
2
−1/8
1/8
ηγ 1/8
n
Eiso
×
. (3.2)
53
−3
10 ergs
0.2
0.1cm
where Θj is measured in radians. It was argued that the fireball converts the energy in the ejecta into gamma rays efficiently [50] (ηγ ≈0.2), and that the mean circumburst density is n≈0.1 cm−3 [51]. Evaluating equation 3.2 for the
parameters of GRB030329 (tj ≈ 10 hours, z=0.1685, and
Eiso =2×1052 erg) gives Θj ≈0.07 rad.
Therefore the beaming factor that relates the actual energy
released in gamma rays (Eγ ) to the isotropic equivalent energy is Θ2j /2≈1/400, so that Eγ ≈5×1049 erg. Comparing
Eiso and Eγ with the histograms in Fig. 2 of Frail et al. [13]
, GRB030329 resides at the lower end of the energy distributions. The calculated isotropic energy from GRB980425, the
GRB associated with SN1998bw, is also low (≃1048 erg).
IV. OVERVIEW OF THE LIGO DETECTORS
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:
2
Eiso = 4π(BC)DL
f /(1 + z) ≈ 2 × 1052 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, β = −2.5, gives that the GRB’s total energy
integrated from 1 keV 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 Γ−1 , where Γ is the bulk Lorentz factor of the flow,
increases from a value smaller than the beaming angle Θj ,
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 identically
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 detector (L1). The two sites
are separated by ≃3000 km, representing 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
6
B. The second science run
FIG. 2: Typical LIGO Hanford sensitivity curves during the S2 Run
[strain Hz−1/2 ] (black and grey lines). The LIGO design sensitivity
goals (SRD) are also indicated (dotted and dashed lines).
excitations into the end test mass control systems, and monitoring the amplitude of these signals at the measurement (error) point. Calibration uncertainties at the Hanford detectors
were estimated to be < 11%.
Significant improvements were made to the LIGO detectors
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 suspension
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 wavefront sensing system for the H1 interferometer
was used to control 8 of 10 alignment degrees 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 detector sensitivity. Figure 2 shows typical spectra achieved by
the LIGO interferometers during the S2 run. The differences
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.
The data analyzed in this paper were taken during LIGO’s
second Science Run (S2), which spanned approximately 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 interferometers were locked and taking science mode data. For this
analysis we relied on the single, ≃4.5 hours long coincident
lock stretch, which started ≃3.5 hours 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 section V). 60 seconds 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 non-stationary behavior of the interferometric detectors.
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 adhoc and model-based waveforms (Section VI B) and the false
alarm rate of the detection algorithm respectively.
The analysis takes advantage of the information provided
by the astrophysical trigger. The trigger time determined
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 predictions and timing
uncertainties. The source direction was needed to calculate
the attenuation due to the LIGO detector’s antenna pattern for
the astrophysical interpretation.
The two co-located and co-aligned 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
7
fore the GRB trigger time is sufficient; roughly ten 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 gravitational 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 second region after the GRB trigger time, is sufficient to cover these predictions and also contains allowance
for up to 30 seconds 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.
FIG. 3: The schematic of the analysis pipeline
B. Search algorithm
ratio test [53, 54].
The schematic of the full analysis pipeline is shown in Figure 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 second overlap between consecutive segments (both ends), providing a tiling of the data with
300 second 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) transients well outside of the 300 second
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
second long window (see Figure 1), starting 120 seconds be-
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.
The raw data streams have narrowband lines, associated
with the power line harmonics at multiples of 60Hz, 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 addition, the broadband
noise shows significant variations over timescales of hours
and smaller variations over timescales of minutes and seconds
due to alignment drift and fluctuations. The background data
must portray a representative sample of the detector behavior
around the time of the trigger. Broadband non-stationarity 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 Hz 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 non-stationary lines and enhance the efficiency of the follow up adaptive filtering stage.
8
The conditioned data has a consistent white spectrum without
major lines and sufficient stationarity, from segment to segment, throughout the background and signal regions.
The end result of the pre-processing is a data segment with
a flat power spectral density (white noise), between ≃80 Hz
and ≃2048 Hz. The data conditioning was applied consistently after the signal injections. This ensures that any change
in detection efficiency due to the pre-processing is properly
taken into account.
2.
Gravitational Wave Search Algorithm
The test statistics for a pair of data streams are constructed
as follows. We take pairs of short segments, one from each
stretch, and compute their cross-correlation function. The ace m,n ) is identical to
tual form of the cross-correlation used (C
k,p,j
the common Euclidean inner product:
e m,n =
C
k,p,j
j
X
Hm [k + i]Hn [k + p + i] ,
(5.1)
i=−j
where the pre-conditioned time series from detector “x” is
Hx = {Hx [0], Hx [1], . . .} 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 (H1 ) or 2 (H2 ). There
are therefore three free parameters to scan when searching for
coherent segments of data between a pair of interferometers
(m,n): 1. the center time of the segment from the first detector
(k); 2. the relative time lag between the segments from the
two detectors (p); and 3. the common duration of segments
(2j+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 computed from segment pairs with start times and lengths varying over values,
which should, respectively, cover the expected arrival times
(signal region) and consider durations of the gravitational
wave burst signals [20, 21, 25, 26, 27, 28] (∼O(1-128ms)).
Hence we apply a search algorithm [54] that processes the
data in the following way.
(1) A three dimensional quantity (Ck,j [p]) is constructed:
2
2 1/2
e1,2
e 2,1
Ck,j [p] = C
+
,
(5.2)
C
k,p,j
k,−p,j
scanning the range of segment center times (k), integration
lengths (2j+1) and relative time shifts (p = 0, ±1, ±2, . . .). A
coherent and coincident signal is expected to leave its localized signature within this three dimensional quantity.
We use a fine rectangular grid in relative time shift (p) and
integration length (2j+1) space. The spacing between grid
points is ≃1 ms 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 ms to ≃128 ms.
Introducing small, non-physical 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 non-stationarity in the interferometer
outputs. Accordingly, Ck,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 the relative phase
uncertainty of the datastreams. 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 Ck,j [p] to estimate the mean (b
µk,j ) and variance (b
σk,j ) of the local noise
distribution, which is also useful in countering the effects of
non-stationarity.
(2) The three dimensional quantity is reduced to a two dimensional image (see Fig. 4), called a corrgram, as follows.
The values of Ck,j [p] in the core region are standardized by
subtracting µ
bk,j and then dividing by σ
bk,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 characterized 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 background 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% CL Poisson errors, based on the the number of
9
FIG. 4: Examples of corrgram images. The horizontal axes are time
(linearly scaled) and the vertical axes are integration length (logarithmically scaled). The color axis, an indicator of the excess correlation, is independently auto-scaled 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.
events within the given bin. We used this distribution to fix
the event strength threshold used in the subsequent analysis.
We chose an event strength threshold with an associated
false alarm rate of less than ≃5×10−4 Hz, equivalent to less
than ≃9% chance for a false alarm within the 180 second long
signal region.
B.
Efficiency determination
The detection sensitivity of the analysis was determined
by simultaneously adding simulated signals of various amplitudes and waveforms to both data streams in the background
region and evaluating the efficiency of their detection as a
function of the injected amplitude and waveform type.
The waveforms we considered include Sine-Gaussians to
emulate short narrow-band bursts, Gaussians to emulate short
FIG. 5: False alarm rate as a function of the event strength threshold
as determined from background data. The error bars reflect 90% CL
Poisson errors, based on the the number of events within the given
bin. The pointer indicates the event strength threshold used for the
analysis, which corresponds to an interpolated 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.
broad-band signals, and Dimmelmeier-Font-Müller numerical
waveforms [26], as examples of astrophysically motivated signals.
Calibration of the waveforms from strain to ADC counts
was performed in the frequency domain, and was done separately for each interferometer. Calibration procedures of the
LIGO detectors are described in Ref. [52]. The transformed
signals, now in units of counts of raw interferometer noise,
were then simply added to the raw data stream.
The amplitudes and the times of the injections were randomly varied. In this way we ensured that each amplitude
region sampled the full, representative range of noise variations and that we had no systematic effects, for example, due
to a regular spacing in time.
To a reasonable approximation the sensitivity of our analysis pipeline can be expressed in terms of the frequency content, the duration and the strength of the gravitational wave
signal. Therefore, it is sufficient to estimate the sensitivity of
our search for a representative set of broad and narrow band
waveforms, which span the range of frequencies, bandwidth,
and duration we wish to search.
We characterize the strength of an arbitrary waveform by its
root-sum-square amplitude (hRSS ), which is defined as [56]:
sZ
∞
hRSS =
−∞
| h(t) |2 dt .
(6.1)
The above definition of hRSS includes all frequencies, while
the gravitational wave detectors and search algorithms are
only sensitive in a restricted frequency band. In principle,
one can analogously define a “band-limited” hRSS , in which
10
FIG. 6: Efficiency of the detection algorithm for a sample waveform as a function of signal strength (hRSS ); in this case a SineGaussian of f0 = 250 Hz and Q = 8.9. To extract this curve numerous simulated waveforms were embedded in a representative fraction
of the background data at random times with randomly varying signal strength. The plot shows the fraction of signals detected as the
function of amplitude and a sigmoid function fit. The reconstructed
signal onset times were required to fall within ±60 ms of the true
onsets, which also explains why the low hRSS end of the curve falls
near zero. This is a typical plot and in general, the agreement between the measured values, and the fit is better than ≃5%. We relied
on the fit to extract our upper limits for an optimally oriented and
polarized source. Section VI E below describes the corrections due
to non-optimal source direction and polarization.
only the sensitive frequency band of the analysis is taken into
account. Within this paper we choose to adopt the Eq. 6.1
definition of hRSS for historical reasons.
The extracted sensitivities (see the examples in Figures 6
and 7) can be used to generalize our measurements and estimate the pipeline’s sensitivity for other similar band-limited
waveforms.
To assess the sensitivity for relatively narrow band waveforms, we used Sine-Gaussian injections of the form:
2
h(t) = h◦ sin(ω◦ t)e−t
/2σ2
,
(6.2)
with a central angular frequency of ω◦ = 2πf◦ , and Q =
ω◦ σ = 2πf◦ σ. The relation between h◦ and hRSS is given
as:
r√
πσ
hSG
1 − e−Q2 ≈
RSS = h◦
2
s
Q
1 Q≫1
√
≈ h◦
.
×
0.8 Q ≃ 1
4 π f◦
(6.3)
The injected signals covered the frequency range between
100 and 1850 Hz with 13 values of f◦ . To test the dependence of the sensitivity on signal duration, we used three values of Q (4.5, 8.9 and 18) for each frequency (see Table I).
FIG. 7: Detected event strength versus hRSS of the injected SineGaussian waveform with f0 = 250 Hz and Q ≈ 8.9. The dots indicate the scatter of the distribution of raw measurements. The gray
band shows the quadratic polynomial fit, which allows us to convert
the strength of an observed event into the equivalent hRSS value and
determine the associated 90% CL error bars. The markers with error bars represent the 90% CL regions for subsets of the data. For
each marker, 90% of the measurements used were within the horizontal error bars and 90% of the detected event strengths values fell
within the vertical error bars. The vertical dash-dot line represents
the 50% detection efficiency associated with the waveform type and
the chosen detection threshold (horizontal dotted line). As expected,
the crossing of the threshold and the 50% efficiency lines agree well
with the fit and the center of the corresponding marker. The vertical dashed line represents the boundary of the region where we have
better than 90% detection efficiency. The “corner” defined by the
event strength threshold and the 90% detection efficiency boundary
(dashed lines) agrees well with the curve outlined by the lower end
of the vertical error bars of the markers. All events in the upper right
corner of the plot (above and beyond the dashed lines) are detectable
with high confidence. This plot is typical for different waveforms
considered in the analysis.
Near the most sensitive frequency region, around ≃250 Hz,
our gravitational wave strain sensitivity for optimally polarized bursts was better than hRSS ≃5×10−21 Hz−1/2 . Figure 8
shows the sensitivity for these narrow band waveforms. The
symbols mark the simulated event strength (hRSS ) necessary
to achieve 90% detection efficiency for each waveform. We
quote the gravitational wave signal strength associated with
the 90% detection efficiency, as this can be related to the upper limits on the gravitational wave strength associated with
the source. Figure 8 also illustrates the insensitivity of the
detection efficiency to the Q of the Sine-Gaussian waveforms
with the same central frequency, as these reach their 90% efficiency levels at similar gravitational wave strengths, even
though their Q differ by a factor of ≃4; for a given hRSS ,
a longer signal (higher Q) would of course, have a smaller
hP EAK . This strength is frequency dependent, naturally following the frequency dependence of the detector sensitivities,
which are also indicated in Figure 8.
Table II shows a similar set of efficiencies estimated using
broad-band simulated signals. We used two types of broadband waveforms, Sine-Gaussians with unity quality factor and
11
FIG. 8: Sensitivity of the detection algorithm for detecting SineGaussian waveforms versus characteristic frequency. The plot shows
the strength necessary for 90% detection efficiency. The grey spectra
illustrate the sensitivity of the 2K and 4K Hanford detectors during
the time surrounding the GRB030329 trigger. The error bars reflect
a total 15% error.
Gaussians. Both are short bursts, however, the Gaussians are
even functions while the Sine-Gaussians are odd, leading to
different peak amplitudes with the same hRSS value. Gaussians were parametrized as:
2
h(t) = h◦ e−t
/2σ2
.
q
√
πσ.
(6.5)
The estimated sensitivities indicate that the 90% detection
efficiency limits for short bursts are similar to those obtained
for the narrow band waveforms when one takes into account
that only part of the power of the broad-band waveforms is
confined to the analysis frequency band. Longer Gaussian
bursts are more difficult to detect, as their spectrum has a significant low frequency component, outside the sensitive band
of our analysis.
We have also estimated our efficiency for a set of astrophysically motivated burst waveforms [26] (see Table III). These
simulated waveforms are not expected to be necessarily associated with GRBs, rather these results are presented here to
further illustrate the waveform independence of the analysis.
C.
placed an upper limit on the detected strength of gravitational
waves associated to GRB030329. Our fixed false alarm rate
permitted the results of simulations to be used directly in setting upper limits.
The upper limits on hRSS for narrow band waveforms are
given in Table I. Tables II and III show the upper limits for the
broadband simulations and astrophysically motivated waveforms, respectively.
(6.4)
The relationship between h◦ and hRSS for a Gaussian is:
hGA
RSS = h◦
FIG. 9: Number of events versus event strength in the signal region (circle). The diamonds show the expected distribution based
on the background region. The squares mark the expected distribution based on non-physical time shifts (ranging from 2 to 9 seconds)
between the H1 and H2 datastreams in the background region. The
error bars reflect 90% CL Poisson errors. The position of the symbols correspond to the center of the bins.
Signal region
The analysis of the signal region (Fig. 9) yielded only
events well below the predetermined event strength threshold
(.60% of threshold). Since we had no candidate event, we
D. Errors
The analysis method, the procedures used to determine the
efficiencies, and the non-stationary nature of the data, all contribute to the uncertainty associated with the results.
The efficiency (versus hRSS ) values have an estimated
≃11% uncertainty due to our limited knowledge of the calibrated response of our detectors. This estimate also accounts
for the slight difference in calibrated response between the
signal region and background data used for the simulations.
An additional uncertainty arises from the non-stationarity
of the data. The results of the simulations exhibit a slight dependence on the choice of the actual data segments (“base”
data) used for the injections. This dependence was characterized via simulations using numerous different sub-segments
of the background data. We repeated the full efficiency estimation process several times for the same waveform, while
injecting into various base data stretches. The variation in the
measured upper limits indicated ≃10% uncertainty due to the
dependency of our upper limits on the base data. This uncertainty shall also account for the statistical error due to the
finite number of simulations used.
We characterized the detection efficiencies for each waveform considered via fits of sigmoid functions (see for example
Figure 6). The fits agree well with the data, but small differ-
12
ences are occasionally observed in the & 90% efficiency region. We estimate that using these fits can underestimate the
90% limits by . 5%.
The uncertainties listed above are taken into account by
specifying a total 15% uncertainty for each measurement in
Figure 8 and in all Tables.
The false alarm rate associated with the results was also
measured. The false alarm rate limit is based on the measurement with zero lag data plus the 90% confidence Poisson
error bars. We have checked the assumption of Poisson background statistics by examining the time intervals between consecutive triggers and the variance in trigger counts for varying
ES thresholds when the background sample is divided into
50 equal-length intervals. Good agreement with the Poisson
expectation is observed. This choice provides a conservative
estimate of our associated (≃5×10−4 Hz) false alarm rate.
E.
Astrophysical interpretation
GRB030329 has a well-determined redshift, therefore we
can relate our observed limits on strain to a measure of the
total gravitational wave energy emission. For a strain h(t) at
distance DL from a source of gravitational radiation, the associated power is proportional to ḣ2 (ḣ = dh/dt), though the
proportionality constant will depend on the (unknown) emission pattern of the source and the antenna pattern of the detector (for the known source position, but unknown polarization
angle).
In general, it is not possible to relate our upper limit on
the strain from a particular waveform to a limit on the energy
radiated by the source, without assuming a model. Sources
that radiate energy EGW might produce an arbitrarily small
signal h(t) in the detector, e.g., if the dynamics in the source
were purely axi-symmetric with the detector located on the
axis. Nevertheless, we can associate a strain h(t) in the detector with some minimum amount of gravitational-wave energy radiated by the source by choosing an “optimistic” emission pattern, thereby obtaining a measure of the minimum
amount of energy that would need to be radiated in order to
obtain a detectable signal. We will show that the progenitor
of GRB030329 is not expected to have produced a detectable
signal.
We are interested in a “plausible case scenario” of gravitational wave emission in order to obtain the minimum (plausible) amount of gravitational-wave energy radiated that could
be associated with a detector signal h(t). We do not expect the
gravitational waves to be strongly beamed, and we expect that
we are observing the GRB progenitor along some preferred
axis. We take a model best case scenario to be that of gravitational wave emission from a triaxial ellipsoid rotating about
the same axis as the GRB (i.e., the direction to the Earth). If
we assume quadrupolar gravitational wave emission, the plusand cross-polarization waveforms, emitted at a polar angle θ
from the axis of rotation to be:
1
(1 + cos2 θ) h+,0
2
= cos θ h×,0
h+ =
(6.6)
h×
(6.7)
where h+,0 and h×,0 are two orthogonal waveforms (e.g., a
Sine-Gaussian and a Cosine-Gaussian), each containing the
same amount of radiative power. That is, we assume that the
same amount of gravitational-wave energy is carried in the
two polarizations and that they are orthogonal:
Z ∞
Z ∞
Z ∞
ḣ+,0 ḣ×,0 dt = 0.
ḣ2×,0 dt and
ḣ2+,0 dt =
−∞
−∞
−∞
(6.8)
Thus, we would expect that the gravitational waves travelling
along the rotational axis (toward the Earth) would be circularly polarized, and that the detector would receive the signal
h = F+ h+,0 + F× h×,0
(6.9)
where F+ and F× represent the detector responses to the polarization components h+,0 and h×,0 [57], and depend on the
position of the source in the sky and on a polarization angle.
The radiated energy from such a system is calculated to be
Z
Z ∞
2 Z ∞
c3
c3 D L
2
2
EGW =
dA
(ḣ+ +ḣ× )dt =
ḣ2 dt
2
16πG
5G
η
−∞
−∞
(6.10)
where η 2 = F+2 + F×2 (which depends only on the position
of the source on the sky) and where we are integrating over a
spherical shell around the source with radius DL (the distance
to the Earth). Alternatively, using Parseval’s identity, we have
EGW
2
8π 2 c3 DL
=
5G η 2
Z
0
∞
|f h̃|2 df
(6.11)
where
h̃(f ) =
Z
∞
h(t)e−2πif t dt.
(6.12)
−∞
Whereas optimal orientation gives η = 1 for a source at
zenith, the position of GRB030329 was far from optimal. The
angle with respect to zenith was 68◦ and the azimuth with respect to the x-arm was 45◦ , which yields η = 0.37.
We now relate EGW to the strain upper limits using the
specific waveforms used in the analysis. For a Gaussian waveform [see Eq. (6.4)]:
√ 3 2 2
πc
D L h◦
(6.13)
EGW =
10G
σ
and for a sine-Gaussian waveform [see Eq. (6.2)]:
√ 3 2 2
2
D L h◦
πc
(1 + 2Q2 − e−Q )
EGW =
20G
σ
(6.14)
where Q = ω◦ σ = 2πf◦ σ. The relation between h◦ and
hRSS is given in Eqs. (6.5) and (6.3).
We can relate the observed limit on hRSS to an equivalent
mass MEQ which is converted to gravitational radiation with
100% efficiency, EGW = MEQ c2 , at a luminosity distance
DL ≈ 800 Mpc. For sine-Gaussian waveforms with f◦ =
13
250 Hz and Q = 8.9, MEQ = 1.9 × 104 η −2 M⊙ . For Gaussian waveforms with σ = 1 ms, MEQ = 3.1 × 104 η −2 M⊙ .
However, we would not expect that the gravitational-wave luminosity of the source could exceed ≃ c5 /G = 2 × 105M⊙ c2
per second [58], so we would not expect an energy in gravitational waves much more than ≃ 2 × 103 M⊙ c2 in the ≃ 10 ms
Sine-Gaussian waveform, or an energy of much more than
≃ 3 × 104 M⊙ c2 in the maximum duration (150 ms) of the
search; far below the limits on MEQ c2 that we find in this
analysis. Present theoretical expectations on the gravitational
wave energy emitted range from 10−6 M⊙ c2 - 10−4 M⊙ c2 to
10−1 M⊙ c2 - M⊙ c2 for some of the most optimistic models
[see e.g. [5, 55, 59, 60]]. Nonetheless, these scalings indicate how we can probe well below these energetic limits with
future analyses. For example, assuming similar detector performance for an optimally oriented trigger like GRB980425
(DL ≈35 Mpc) the limit on the equivalent mass would be
MEQ ≈60 M⊙ for the Gaussian waveforms mentioned above
with σ = 1 ms.
VII. SUMMARY
A.
Comparison with previous searches for gravitational waves
from GRBs
Our result is comparable to the best published results
searching for association between gravitational waves and
GRBs [61], however these studies differ in their most sensitive
frequency.
Tricarico et al. [62] used a single resonant mass detector,
AURIGA [63] , to look for an excess in coincidences between the arrival times of GRBs in the BATSE 4B catalog.
They used two different methods. They searched for events
identified above a certain threshold in the gravitational wave
data, and also attempted to establish a statistical association
between GRBs and gravitational waves. No significant excess
was found with the former method. The latter used a variant of the correlation based Finn-Mohanty-Romano (FMR)
method [64]. However, instead of using the cross-correlation
of two detectors, as proposed in the FMR method, only the
variance of the single detector output was used. A sample of
variances from times when there were no GRBs was compared with a corresponding sample from data that spanned
the arrival times of the GRBs. An upper limit on the sourceaveraged gravitational wave signal root mean square value of
1.5 × 10−18 was found using 120 GRBs. This limit applies at
the AURIGA resonant frequencies of 913 and 931 Hz, which
are very far from the most sensitive frequency of the LIGO detectors (≃250 Hz). This work [62] was later extended [65],
which led to an improved upper limit.
The data analysis method employed in Modestino & Moleti [66] is another variant of the FMR method. Instead of
constructing off-source samples from data segments that are
far removed from the GRB trigger, the off-source samples
are constructed by introducing non-zero time shifts between
the two detector data streams and computing their crosscorrelation. For narrowband resonant mass detectors, the di-
rectional information of a GRB cannot be exploited to discriminate against incorrect relative timing since the signal in
the output of the detector is spread out by the detector response over time scales much larger than the light travel time
between the detectors.
Astone et al. [67, 68] report on a search for a statistical
association between GRBs and gravitational waves using data
from the resonant mass detectors EXPLORER [69] and NAUTILUS [70]. They report a Bayesian upper limit on gravitational wave signal amplitudes of 1.2×10−18, at 95% probability, when the maximum delay between the GRB and gravitational wave is kept at 400 sec. The upper limit improves
to 6.5×10−19 when the maximum delay is reduced to 5 sec.
However, the absence of directional and/or distance information for most of these GRBs precluded accounting for source
variations; the gravitational wave signal amplitude was assumed to be the same for all of the GRBs.
Astone et al. [71] report on the operation of the resonant
mass detectors EXPLORER during the closest ever gamma
ray burst (GRB980425) with known redshift and direction. At
the time of the burst, EXPLORER was taking data with close
to optimal orientation. GRB980425 was ≃23 times closer to
Earth than GRB030329 giving a ≃520 increase in energy sensitivity. Based on their sensitivity and the loudest event within
±5 minutes of the GRB980425 trigger the authors quote a
limit of ≃1600 M⊙ for a simple model assuming isotropic
gravitational wave emission.
Recently, Astone et al. [61] executed a search aiming to
detect a statistical association between the GRBs detected by
the satellite experiments BATSE and BeppoSAX, and the EXPLORER and NAUTILUS gravitational wave detectors. No
association was uncovered. Their upper limit is the lowest
published result, which is based on bar-detector gravitational
wave data.
B. Conclusion
We have executed a cross-correlation-based search for possible gravitational wave signatures around the GRB030329
trigger, which occurred during the Second Science Run (S2)
of the LIGO detectors. We analyzed a 180 second signal region around the GRB and 4.5 hours of background data, surrounding the signal region, corresponding to a single coincident lock stretch. These data were sufficient to characterize
the background, scan the signal region and estimate our efficiency. We used the same procedure, based on cross correlation, for each of these studies. We evaluated the sensitivity of
the search to a large number of broad and narrow band waveforms.
We observed no candidates with gravitational wave signal
strength larger than a pre-determined threshold, therefore we
set upper limits on the associated gravitational wave strength
at the detectors. The present analysis covers the most sensitive
frequency range of the Hanford detectors, approximately from
80 Hz to 2048 Hz. The frequency dependent sensitivity of our
search was hRSS ≃O(6×10−21) Hz−1/2 .
The prospect for future searches is promising, as the sen-
14
sitivity of the instruments improves with further commissioning.
Once operating at target sensitivity, the detectors will be
more sensitive to strain than they were during S2 by factors of
10 - 100, depending on frequency (see Figure 2.). This implies
an improvement of a factor of ∼1000 in sensitivity to EGW ,
since EGW scales like ∼ h2RSS (see for example Eq. 6.13).
Detection of GRBs with measured redshifts significantly smaller than GRB030329’s is certainly possible.
GRB030329’s electromagnetic brightness was due to a favorable combination of distance and our position in its beam. One
year of observation will incorporate hundreds of GRBs with
LIGO data coverage and some of these GRBs, even though
fainter [72, 73, 74] than GRB030329, could be significantly
closer, as was 1998bw. We can also hope for sources with
more optimal direction and coincidence between three or four
observing interferometers.
VIII. ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support of the
United States National Science Foundation for the construc-
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16
TABLE I: hRSS [Hz −1/2 ] for 90% detection efficiency for SineGaussians (SG) waveforms at various frequencies (f◦ ) and Q (see
eq. 6.2). The quoted values are the results of simulations and are
subject to a total of ≃ 15% statistical and systematic errors, which
are taken into account when quoting the U L90%CL
hRSS values. Note that
at the low and at the high frequency end, the low Q waveforms have
significant power outside of the analysis frequency band.
−1/2
−1/2
Waveform σ [ms] Q f◦ [Hz] h90%
] U L90%CL
]
RSS [Hz
hRSS [Hz
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
7.2
2.9
2
1.6
1.3
1
0.84
0.72
0.53
0.49
0.46
0.42
0.39
14
5.7
3.9
3.1
2.6
2
1.7
1.4
1
0.97
0.91
0.83
0.77
29
11
7.9
6.3
5.2
4.1
3.4
2.9
2.1
2
1.8
1.7
1.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
8.9
8.9
8.9
8.9
8.9
8.9
8.9
8.9
8.9
8.9
8.9
8.9
8.9
18
18
18
18
18
18
18
18
18
18
18
18
18
100
250
361
458
554
702
850
1000
1361
1458
1554
1702
1850
100
250
361
458
554
702
850
1000
1361
1458
1554
1702
1850
100
250
361
458
554
702
850
1000
1361
1458
1554
1702
1850
17 × 10−21
4.8 × 10−21
5.8 × 10−21
7.0 × 10−21
7.9 × 10−21
10 × 10−21
12 × 10−21
15 × 10−21
27 × 10−21
30 × 10−21
37 × 10−21
43 × 10−21
50 × 10−21
18 × 10−21
4.6 × 10−21
6.0 × 10−21
7.1 × 10−21
7.3 × 10−21
8.9 × 10−21
10 × 10−21
13 × 10−21
20 × 10−21
23 × 10−21
26 × 10−21
32 × 10−21
38 × 10−21
23 × 10−21
5.0 × 10−21
6.4 × 10−21
7.9 × 10−21
7.7 × 10−21
9.8 × 10−21
10 × 10−21
12 × 10−21
19 × 10−21
21 × 10−21
22 × 10−21
29 × 10−21
34 × 10−21
20 × 10−21
5.6 × 10−21
6.7 × 10−21
8.0 × 10−21
9.1 × 10−21
11 × 10−21
14 × 10−21
17 × 10−21
31 × 10−21
34 × 10−21
43 × 10−21
50 × 10−21
58 × 10−21
21 × 10−21
5.3 × 10−21
6.9 × 10−21
8.1 × 10−21
8.4 × 10−21
10 × 10−21
12 × 10−21
15 × 10−21
23 × 10−21
27 × 10−21
30 × 10−21
37 × 10−21
44 × 10−21
26 × 10−21
5.7 × 10−21
7.4 × 10−21
9.1 × 10−21
8.9 × 10−21
11 × 10−21
12 × 10−21
14 × 10−21
21 × 10−21
24 × 10−21
25 × 10−21
33 × 10−21
39 × 10−21
17
TABLE II: As in Table I, hRSS [Hz −1/2 ] for 90% detection efficiency for Gaussian (GA) waveforms of various durations (σ) (see
eq. 6.4) and for Sine-Gaussians (SG) waveforms at various frequencies (f◦ ) and Q = 1 (see eq. 6.2). Note that these broadband waveforms have significant power outside of the analysis frequency band.
−1/2
−1/2
Waveform σ [ms] Q f◦ [Hz] h90%
] U L90%CL
]
RSS [Hz
hRSS [Hz
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
SG
GA
GA
GA
GA
GA
GA
GA
GA
GA
1.6
0.64
0.44
0.35
0.29
0.23
0.19
0.16
0.12
0.11
0.1
0.094
0.086
0.5
0.75
1
2
3
4
5.5
8
10
1
1
1
1
1
1
1
1
1
1
1
1
1
100
250
361
458
554
702
850
1000
1361
1458
1554
1702
1850
10 × 10−21
6.5 × 10−21
8.4 × 10−21
10 × 10−21
13 × 10−21
18 × 10−21
23 × 10−21
26 × 10−21
39 × 10−21
44 × 10−21
46 × 10−21
55 × 10−21
61 × 10−21
8.3 × 10−21
9.6 × 10−21
1.3 × 10−20
3.3 × 10−20
8.2 × 10−20
1.9 × 10−19
8.5 × 10−19
1.3 × 10−17
1.0 × 10−16
12 × 10−21
7.4 × 10−21
9.7 × 10−21
12 × 10−21
14 × 10−21
20 × 10−21
26 × 10−21
30 × 10−21
45 × 10−21
51 × 10−21
52 × 10−21
63 × 10−21
70 × 10−21
9.6 × 10−21
1.1 × 10−20
1.5 × 10−20
3.8 × 10−20
9.5 × 10−20
2.2 × 10−19
9.8 × 10−19
1.5 × 10−17
1.2 × 10−16
18
TABLE III: As in Table I, hRSS [Hz −1/2 ] for 90% detection efficiency for astrophysically motivated waveforms. These waveforms
are described in detail in Ref. [26]. Note that most of these waveforms have significant power outside of the analysis frequency band.
−1/2
−1/2
Simulation Waveform h90%
] U L90%CL
]
RSS [Hz
hRSS [Hz
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
DFM
A1B1G1
A1B2G1
A1B3G1
A1B3G2
A1B3G3
A1B3G5
A2B4G1
A3B1G1
A3B2G1
A3B2G2
A3B2G4
A3B3G1
A3B3G2
A3B3G3
A3B3G5
A3B4G2
A3B5G4
A4B1G1
A4B1G2
A4B2G2
A4B2G3
A4B4G4
A4B4G5
A4B5G4
A4B5G5
12 × 10−21
13 × 10−21
12 × 10−21
12 × 10−21
12 × 10−21
34 × 10−21
24 × 10−21
19 × 10−21
20 × 10−21
15 × 10−21
14 × 10−21
28 × 10−21
17 × 10−21
12 × 10−21
30 × 10−21
23 × 10−21
26 × 10−21
38 × 10−21
32 × 10−21
42 × 10−21
39 × 10−21
17 × 10−21
12 × 10−21
21 × 10−21
19 × 10−21
14 × 10−21
15 × 10−21
14 × 10−21
14 × 10−21
14 × 10−21
39 × 10−21
27 × 10−21
21 × 10−21
23 × 10−21
17 × 10−21
16 × 10−21
33 × 10−21
20 × 10−21
14 × 10−21
34 × 10−21
27 × 10−21
29 × 10−21
44 × 10−21
36 × 10−21
48 × 10−21
45 × 10−21
19 × 10−21
13 × 10−21
25 × 10−21
22 × 10−21