Upper Limits on Gravitational Wave Bursts in LIGO’s Second Science RunLIGO-P040040-00-R

Circulation restricted to LIGO-I members

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

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A. Di Credico,27 M. Duaz,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

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H. Luck,31, 2 T. T. Lyons,12, t B. Machenschalk,1 M. MacInnis,13 M. Mageswaran,12 K. Mailand,12 W. Majid,12, b M. Malec,2, 31

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E. Messaritaki,39 C. Messenger,33 V. P. Mitrofanov,20 G. Mitselmakher,34 R. Mittleman,13 O. Miyakawa,12 S. Miyoki,12, w

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, x

D. Ouimette,12, r 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, y R. Riesen,15 K. Riles,36 B. Rivera,14 A. Rizzi,15, z

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S. Yoshida,25 K. D. Zaleski,28 M. Zanolin,13 I. Zawischa,31, ii L. Zhang,12 R. Zhu,1 N. Zotov,17 M. Zucker,15 and J. Zweizig12

(The LIGO Scientific Collaboration, http://www.ligo.org)

2

1Albert-Einstein-Institut, Max-Planck-Institut fur Gravitationsphysik, D-14476 Golm, Germany2Albert-Einstein-Institut, Max-Planck-Institut fur Gravitationsphysik, D-30167 Hannover, Germany

3Australian National University, Canberra, 0200, Australia4California Institute of Technology, Pasadena, CA 91125, USA

5California State University Dominguez Hills, Carson, CA 90747, USA6Caltech-CaRT, Pasadena, CA 91125, USA

7Cardiff University, Cardiff, CF2 3YB, United Kingdom8Carleton College, Northfield, MN 55057, USA

9Fermi National Accelerator Laboratory, Batavia, IL 60510,USA10Hobart and William Smith Colleges, Geneva, NY 14456, USA

11Inter-University Centre for Astronomy and Astrophysics, Pune - 411007, India12LIGO - California Institute of Technology, Pasadena, CA 91125, USA

13LIGO - Massachusetts Institute of Technology, Cambridge, MA 02139, USA14LIGO Hanford Observatory, Richland, WA 99352, USA

15LIGO Livingston Observatory, Livingston, LA 70754, USA16Louisiana State University, Baton Rouge, LA 70803, USA

17Louisiana Tech University, Ruston, LA 71272, USA18Loyola University, New Orleans, LA 70118, USA

19Max Planck Institut fur Quantenoptik, D-85748, Garching,Germany20Moscow State University, Moscow, 119992, Russia

21NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA22National Astronomical Observatory of Japan, Tokyo 181-8588, Japan

23Northwestern University, Evanston, IL 60208, USA24Salish Kootenai College, Pablo, MT 59855, USA

25Southeastern Louisiana University, Hammond, LA 70402, USA26Stanford University, Stanford, CA 94305, USA27Syracuse University, Syracuse, NY 13244, USA

28The Pennsylvania State University, University Park, PA 16802, USA29The University of Texas at Brownsville and Texas Southmost College, Brownsville, TX 78520, USA

30Trinity University, San Antonio, TX 78212, USA31Universitat Hannover, D-30167 Hannover, Germany

32Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain33University of Birmingham, Birmingham, B15 2TT, United Kingdom

34University of Florida, Gainesville, FL 32611, USA35University of Glasgow, Glasgow, G12 8QQ, United Kingdom

36University of Michigan, Ann Arbor, MI 48109, USA37University of Oregon, Eugene, OR 97403, USA

38University of Rochester, Rochester, NY 14627, USA39University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA

40Washington State University, Pullman, WA 99164, USA( RCS Revision: 1.3 ; compiled 1 October 2004)

We performed a search for gravitational wave bursts using data from the second science run of the LIGOdetectors. This search was sensitive to bursts of duration less than a second and with frequency content inthe 100-1100Hz range. It featured significant improvementsin the analysis pipeline respect to the previouslyreported one by LIGO [1]: issues in data pre-filtering were addressed and a waveform consistency test wasintroduced for the final event selection. The method we used in order to search for candidate signals is based ona wavelet time-frequency decomposition. Its sensitivity in terms of the burst amplituderoot-sum-square (rss)lays in the range ofhrss ∼ 10

−20 − 10−19

strain/√

Hz. This event selecting algorithm when combined withthe waveform consistency test allows exploring signals deeper into the instruments’ noise while maintaininga low false alarm rate, O(0.1)µHz. We interpret our search result in terms of an upper limit on the rate ofgravitational wave bursts which is at the level of 0.43 events per day. We then combine it with a measurement ofthe detection efficiency to given waveform morphologies in order to yield rate versus strength exclusion plots aswell as establish distance sensitivity to astrophysical systems that might emit them. Both the upper limit boundand its applicability to signal strengths improve our previously reported limits [1] and reflect the most sensitiveand stringent limit setting broad-band search for gravitational wave bursts so far.

PACS numbers: 04.80.Nn, 07.05.Kf, 95.30.Sf, 95.85.Sz

aCurrently at Stanford Linear Accelerator Center bCurrently at Jet Propulsion Laboratory

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I. INTRODUCTION

The Laser Interferometer Gravitational wave Observatory(LIGO) is a network of first generation interferometric de-tectors aiming to make the first direct observations of grav-itational waves. The construction of the LIGO detectors isessentially complete and over the last three years several data-taking runs interleaved with commissioning work took place.This effort was aimed to (a) bring the three interferometerstotheir final optical configuration (b) reduce the interferometers’noise floors and (c) pave the way to the first science observa-tions. A major milestone in the instruments’ commissioningand operation was the first science run (S1) that took place inthe summer of 2002. S1 represented the longest and most sen-sitive operation of broad-band interferometers in coincidenceup until 2002. Using the S1 data, an astrophysical search forthe four general categories of gravitational wave source types–inspiral, burst-like, stochastic and continuous– was pursuedby the LIGO Scientific Collaboration (LSC). The searches es-tablished the general methodology we expect to follow in theanalysis of the data from future running. We also interpretedthe S1 searches in terms of setting first upper limits usingLIGO science data [1–4]. In 2003 two additional science runsof the LIGO instruments collected data of improved sensitiv-ity with respect to S1, but still below the instruments’ design

cPermanent Address: HP LaboratoriesdCurrently at Rutherford Appleton LaboratoryeCurrently at University of California, Los AngelesfCurrently at Hofstra UniversitygPermanent Address: GReCO, Institut d’Astrophysique de Paris (CNRS)hCurrently at Keck Graduate InstituteiCurrently at National Science FoundationjCurrently at University of SheffieldkCurrently at Ball Aerospace CorporationlCurrently at European Gravitational ObservatorymCurrently at Intel Corp.nCurrently at University of Tours, FranceoCurrently at Lightconnect Inc.pCurrently at W.M. Keck ObservatoryqCurrently at ESA Science and Technology CenterrCurrently at Raytheon CorporationsCurrently at New Mexico Institute of Mining and Technology /MagdalenaRidge Observatory InterferometertCurrently at Mission Research CorporationuCurrently at Harvard UniversityvCurrently at Lockheed-Martin CorporationwPermanent Address: University of Tokyo, Institute for Cosmic Ray Re-searchxPermanent Address: University College DublinyCurrently at Research Electro-Optics Inc.zCurrently at Institute of Advanced Physics, Baton Rouge, LAaaCurrently at Thirty Meter Telescope Project at CaltechbbCurrently at European Commission, DG Research, Brussels, BelgiumccCurrently at University of ChicagoddCurrently at LightBit CorporationeePermanent Address: IBM Canada Ltd.ff Currently at University of DelawareggPermanent Address: Jet Propulsion LaboratoryhhCurrently at Shanghai Astronomical Observatoryii Currently at Laser Zentrum Hannover

goal. The second science run (S2) collected data in early 2003and the third science run (S3) at the end of the same year.

In this paper we report the results of a search for gravita-tional wave bursts using the LIGO S2 data. The astrophys-ical motivation for burst events is strong; it embraces catas-trophic phenomena in the universe with clear signatures in theelectromagnetic spectrum like supernovae explosions [5–7],the merging of compact binary stars as they form a singleblack hole [8–10] and the astrophysical engines that powerthe gamma ray bursts [11]. Perturbed or accreting blackholes, neutron star modes and instabilities as well as cosmicstring cusps may also account for potential burst sources. Theexpected rate, strength and waveform morphology for suchevents is not precisely known. For this reason, our assump-tions for the expected signals were minimal. The experimen-tal signatures on which our search focused can be describedas burst signals of short duration (<< 1s) and with enoughsignal strength in the LIGO sensitive band (100-1100Hz) tobe detected in coincidence in all three LIGO instruments.

The general methodology in pursuing this search followsthe one we presented in the analysis of the S1 data [1] withsome significant improvements. In the S1 analysis the ring-ing of the pre-filters limited our ability to perform tight time-coincidence between the triggers coming from the three LIGOinstruments. This was addressed by the use of a new searchmethod that is not dependent on any strong pre-filtering. Thisnew method also provided an improved event parameter esti-mation, including timing resolution. Finally, a waveform con-sistency test was introduced for events that passed the timeand frequency coincidence requirements in the three LIGOdetectors.

This search has yielded one event in coincidence amongthe three LIGO detectors during S2. Subsequent examinationof this event has established its origin to be of environmentalcharacter in the two Hanford detectors. Using the Feldman-Cousins approach [12], we set an upper limit on the numberof burst events at our detectors at the level of 0.43 per day atthe 90% confidence level. We have usedad hocwaveforms(Gaussians and sine-Gaussians) to establish the sensitivity ofthe S2 search pipeline to signal strengths and thus interpretour upper limit in a rate versus strength plot. The search sen-sitivity in terms of the burst amplituderoot-sum-square (rss)lays in the rangehrss ∼ 10−20 − 10−19 strain/

√Hz. Both

the upper limit (rate) and its applicability to signal strengths(sensitivity) reflect significant improvements respect to our S1result [1]. In addition, in this search we introduced modelsforrealistic waveforms for stellar core collapse [5–7] and binaryblack hole merger waves [8, 9] in order to establish distancesensitivity of our search to these systems.

In the following sections we present the LIGO instrumentsand the S2 run in more detail (section II) as well as anoverview of the search pipeline (section III). The proce-dure for selecting the data that we analyzed (section IV) isfollowed by a discussion on the role of the vetoes in thissearch (section V). We then present the search algorithm andthe waveform consistency test for doing the event selection(section VI), the description of the final event analysis (sec-tion VII), and the procedure for establishing the efficiencyof

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the search (section VIII). Our final results and discussion arepresented in sections IX and X.

II. THE LIGO SECOND SCIENCE RUN

LIGO comprises of three interferometers at two sites: 4kmlong interferometers at the LIGO Hanford and Livingston Ob-servatories (denoted H1 and L1 respectively) and a 2km in-terferometer sharing the same vacuum system at the HanfordObservatory (denoted H2). The LIGO interferometers areMichelson interferometers, with power recycling and resonantcavities in the two arms to increase the storage time (and con-sequently the phase shift) for the light returning to the beamsplitter due to motions of the end mirrors [13]. The mirrors aresuspended as pendulums from vibration-isolated platformstoprotect them from external noise sources. A detailed descrip-tion of the LIGO detectors as they were configured for the S1run may be found in ref. [14].

A. Improvements to the LIGO detectors for S2

The LIGO interferometers [14] are still undergoing com-missioning and have not yet reached their final operating con-figuration and sensitivity. Between S1 and S2 a number ofchanges were made which resulted in improved sensitivity aswell as overall instrument stability and stationarity. Themostimportant of these are summarized below.

The analog suspension controller electronics on the H2 andL1 interferometers were replaced with digital suspension con-trollers of the type installed on H1 during the S1 run. Theaddition of a separate DC bias supply for alignment relievedthe range requirement of the coil driver. This, combined withflexibility of a digital system capable of coordinated switch-ing of analog and digital filters, enabled the new coil drivertooperate with much lower electronics output noise. In partic-ular, the system had two separate modes of operation: acqui-sition mode with higher range and noise, and run mode withreduced range and noise. Input noise to the coil driver due tothe discrete steps in the digital to analog converter (DAC) atthe output of the digital suspension computer was reduced bya filter pair surrounding the DAC: a digital filter to whiten thedata going into the DAC and an analog filter to restore it toits full dynamic range. Better filtering, better diagonalizationof the drive to the coils to eliminate length-to-angle couplingsand more flexible control/sequencing features also contributedto an overall performance improvement.

The noise from the optical lever servo that damps the angu-lar excitations of the interferometer optics was reduced. Themechanical support elements for the optical transmitter andreceiver were stiffened to reduce drift and low frequency vi-brational excitations. The reduction in the low frequency ex-citations enabled the filtering of the servo to be tailored tore-duce noise in the LIGO gravitational wave band. Input noiseto the servo due to the discrete steps in the analog to digitalconverter (ADC) was reduced by a filter pair surrounding the

ADC: an analog filter to whiten the data going into the ADCand a digital filter to restore it to its full dynamic range.

Further progress was made on commissioning the wave-front sensing (WFS) system for alignment control of the H1interferometer. This system uses the main laser beam to sensethe proper alignment for the suspended optics. During S1, allinterferometers had 2 degrees of freedom for the main inter-ferometer (plus four degrees of freedom for the mode cleaner)controlled by their WFS. For S2, the H1 interferometer had 8out of 10 alignment degrees-of-freedom for the main interfer-ometer under WFS control. As a result, it maintained a muchmore uniform operating point over the run than the other twointerferometers, which continued to have only 2 degrees offreedom under WFS control.

The high frequency sensitivity was increased by operatingthe interferometers with higher effective power. Two mainfactors enabled this power increase. Improved alignment tech-niques and better alignment stability (due to the wavefrontsensor and optical lever improvements above) reduced theamount of spurious light at the anti-symmetric port, whichthreatened to saturate the photodiode if the effective powerhad been increased in S1. Also, a new servo system to cancelthe “I-phase” photocurrent in the anti-symmetric photodiodewas added. This phase of the signal at the anti-symmetric portis nominally zero for a perfectly aligned and matched inter-ferometer, but various imperfections in the interferometer canlead to large low frequency signals. The new servo preventsthese signals from causing saturations in the photodiode andits RF preamplifier. During S2, the interferometers operatedwith about 1.5W incident on the mode cleaner and about 40Wincident on the beam splitter.

These changes led to a significant improvement in detec-tor sensitivity. Figure 1 shows typical spectra achieved bythe LIGO interferometers during the S2 run compared withLIGO’s design sensitivity. The differences among the threeLIGO spectra reflect differences in the operating parametersand hardware implementations of the three instruments whichare in various stages of reaching the final design configuration.The significant improvements in the noise of the interferome-ters over the S1 levels was a primary motivation for S2.

B. Description of S2

The data analyzed in this paper were taken during LIGO’ssecond science run (S2), which spanned approximately 59days from February 14 to April 14, 2003. During this time,operators and scientific monitors attempted to maintain con-tinuous low noise operation of the LIGO instruments. Theduty cycle for the interferometers, defined as the fraction ofthe total run time when the interferometer was locked and inits low noise configuration,were approximately 37% for L1,74% for H1 and 58% for H2. The longest continuous lockedstretch for any interferometer during S2 was 66 hours for H1.The main sources of lost time were high microseismic mo-

tion at both sites due to storms, and anthropogenic noise at theLivingston observatory.

Improved monitoring and automated alarms instituted after

5

102

103

10−23

10−22

10−21

10−20

10−19

10−18

frequency(Hertz)

stra

in/ √

Hz

S1: L1 (9 Sept ’02)S2: L1 (1 March ’03)S2: H1 (8 April ’04)S2: H2 (11 April ’04)

FIG. 1: Typical LIGO sensitivities instrain/√

Hz during the S2Run. The solid line denotes the design goal for the 4km instruments.

S1 gave the operators and scientific monitors better warningsof out-of-nominal operating conditions for the interferome-ters. As a result, the fraction of time lost to high noise orto missing calibration lines (both major sources of unanalyz-able data during the S1 run) was greatly reduced. Thus, eventhough the S2 run was less than a factor of four longer thanthe S1 run and the duty cycle for triple interferometer coinci-dence was in fact marginally lower (23.4% for S1 vs. 22.0%for S2),the total amount of analyzable triple coincidence datawas 303 hours (compared with 34 hours from S1).

As during S1, the strain signal for each interferometers(t) = [Lx(t) − Ly(t)]/L was derived from the error signalof the feedback loop used to control the differential motionof the interferometer arms. To calibrate the error signal, theeffect of the feedback loop gain was measured and dividedout. Although more stable than during S1, the response func-tions varied over the course of the S2 run due to drifts in thealignment of the optical elements; it was tracked by injectingfixed-amplitude sinusoidal signals (calibration lines) into thedifferential arm control loop, and monitoring the amplitudesof these signals at the measurement (error) point [15]. Cali-bration errors were< 14% for L1. For H1, the injected lineswere initially weaker than desired leading to errors up to 20%for the first part of the run. After the strength of the lines wasincreased (about midway through the run),the calibration er-rors were reduced to< 11%.

The S2 run also involved coincident running with theTAMA interferometer [16]. TAMA had comparable or bet-ter sensitivity to LIGO above about 1000Hz and a duty cy-cle of 81%, but poorer sensitivity at lower frequencies wherethe LIGO detectors had their best sensitivity. In addition,thelocation and orientation of the TAMA detector differs sub-stantially from the LIGO detectors, which further reduced thechance of a coincident detection at low frequencies. For thesereasons, the joint analysis of LIGO and TAMA data focusedon gravitational wave frequencies from 700-2000 Hz, and we

have performed a LIGO-only search from 100-1100Hz. It isthe latter search that is described in this paper, while the for-mer will be the subject of a future paper. The overlap be-tween these two searches (700-1100 Hz) is intended to ensurethat possible sources with frequency content spanning the twosearches will not be missed. Unlike S1, the GEO600 inter-ferometer [17] was undergoing commissioning and was notoperated in coincidence with the LIGO interferometers dur-ing S2.

III. SEARCH PIPELINE OVERVIEW

The overall burst search pipeline used in the S2 analysisfollows the one we introduced in our S1 search [1]. First, dataselection criteria are applied in order to identify periodswhenthe instruments are well behaved and the recorded data can beused for science searches (section IV). When candidate burstevents are identified, they can be compared with veto infor-mation from the thousands of auxiliary read-back channels ofthe servo control systems as well as of the physical environ-ment that are being recorded in LIGO (section V). Lastly, theprimary tool for reducing accidental coincidences and provid-ing confidence in detection is the requirement of coincidencebetween the three LIGO instruments.

A wavelet-based algorithm that we call WaveBurst (sec-tion VI) was used for the identification of candidate burstevents. Rather than forming single-interferometer triggersfirst, WaveBurst analyzes simultaneously the time series com-ing from two different LIGO interferometers and incorpo-rates strength thresholding as well as time-, and frequency-coincidence in a single module for selection of events. The fi-nal step in the coincidence analysis involves a waveform con-sistency test, ther-statistic (section VI). This test is basedon forming the normalized linear coefficient of the raw timeseries coming from the LIGO instruments and reflects ourexpectation that candidate events in our predominantly co-aligned LIGO interferometers should result into correlatedtime series. This is the final cut events have to pass. Theuse of Waveburst and ther-statistic are the major changes inthe S2 pipeline respect to the one invoked in S1 [1].

The measurement of the background in this search wasmeasured by artificially shifting in time the raw time seriesofone of the LIGO instruments, L1, and repeating the analysisas for the un-shifted data. We will describe the backgroundestimation in more detail in section VII. We have relied onhardware and software injections in order to establish the ef-ficiency of the pipeline. A set of signal morphologies [? ]were injected into the raw data at the beginning of our analy-sis pipeline and were used to establish the fraction of detectedover injected events as a function of their strength VIII. Inall cases, i.e., for raw, time-shifted and simulated (injection)data, exactly the same analysis pipeline was invoked for theiranalysis.

The search for bursts in the LIGO S2 data used roughly10%of the triple coincidence data set in order to perform allthe tunings and establish the end-to-end event selection cri-teria. This data set was chosen uniformly across the acquisi-

6

tion time and constituted the so-called “playground” for thesearch. Once the analysis pipeline was fixed using the play-ground data, it was applied to the remaining 90% of the datafor performing the search for bursts. All our reported searchresults in this paper reflect only the non-playground data set.

IV. DATA SELECTION

The selection of data to be analyzed was a key first step inthis search. We expect the presence of a candidate signal inall three LIGO instruments to be of primary importance in thedetection of gravitational wave bursts. In the case of a gen-uine astrophysical burst event this requirement will not onlyincrease our detection confidence but it will also allow us toextract in the best possible way the signal and source param-eters. Therefore, for this search we have confined ourselvesto periods of time when all three LIGO interferometers weresimultaneously locked in low noise mode with nominal oper-ating parameters (nominal servo loop gains, nominal filter set-tings, etc.), marked by a manually set bit (“science mode”) inthe data stream. This produced a total of 318 hours of poten-tial data for analysis. This total was reduced by the followingdata selection cuts:

• A minimum duration of 300 seconds was required fora triple coincidence segement to be analyzed. This cuteliminated 2.85 hours or∼ 0.9% of the initial data set.

• Post-run re-examination of the interferometer configu-ration and status channels included in the data streamidentified a small amount of time when the interferom-eter configuration deviated from nominal. In additionwe identified short periods of time when the timing sys-tem for the data acquisition had lost synchronization.These cuts eliminated 0.7 hours (∼ 0.2%) of the triplecoincidence data.

• It was discovered that large low frequency excitationsof the interferometer could cause the photodiode at theanti-symmetric port to saturate. This caused bursts ofexcess noise due to nonlinear up-conversion. These pe-riods of time were identified and eliminated from theanalysis.

• There were occasional periods of time when the cali-bration lines (needed to correct the error signal for theeffects of the servo loop and obtain an accurate straincalibration) were either omitted or remained extremelyweak. These periods (approximately 2% of the totaltriple coincidence time) were identified and eliminatedfrom the analysis.

• The H1 interferometer had a known problem with amarginally stable servoloop, which lead to higher thannormal noise in the error signal for the differential armlength (the channel used in this search for gravitationalwaves). A data cut was imposed which eliminated anyperiods of time when the rms noise in the 200-400Hzband of this channel exceeded a threshold value for 5

consecutive minutes. The requirement for 5 consecutiveminutes was imposed to prevent a short burst of gravita-tional waves (the object of this search) from triggeringthis cut.

These data quality cuts eliminated a total of 13 hours fromthe original 318 hours of triple coincidence data. The fractionof data surviving these quality cuts (96%) is a significant im-provement over the experience in S1 when only 37% of thedata passed all the quality cuts.

V. VETOES

We performed several studies in order to establish any cor-relation of the triggers provided by the Event Trigger Gen-erators (ETGs) with environmental and instrumental glitches.LIGO records thousands of auxiliary read-back channels ofthe servo control systems employed in the instruments’ inter-ferometric operation as well as auxiliary channels monitoringthe instruments’ physical environment. These channels canprovide ways for establishing transients of non-astrophysicalorigin, i.e., glitches attributed to the instruments themselvesand their environment. Assuming that the coupling of thesechannels to a genuine gravitational wave burst is null (or be-low threshold within the context of a given analysis), suchglitches appearing in these auxiliary channels may be used toveto the events that appear in simultaneity in the gravitationalwave channel.

Given the number of auxiliary channels and the parameterspace that we need to explore for their analysis, an exhaustivea priori examination of all of them is a formidable task. Thestudy was limited to the S2 playground data set and to fewtens of channels which the instrument experts had identifiedasthe most interesting ones to pursue. Several different choicesof filter and threshold parameters were tested in running theglitch finding algorithms. For each of these configurations,theefficiency of the auxiliary channel in vetoing the event triggersas well as the dead-time introduced by using that auxiliarychannel as a veto were computed.

Efficiency and dead-time were then compared in order toestablish the effectiveness of the veto. None of the channelsand parameters we exercised resulted to an obviously goodveto (e.g., one with an efficiency of 50% or greater and a dead-time of the order of few percent) to be used in this search.Among the most interesting channels was the one in the L1instrument that recorded the DC level of the light out of the an-tisymmetric port of the interferometer, referred to as ASDC.That showed to correlate through a non-linear coupling be-tween interferometer alignment fluctuations with the gravita-tional wave channel. A candidate veto based on the ASDCchannel resulted in rejecting 15% of the events from an initialalgorithm in the L1 instrument with a 5% dead-time. We de-cided not to apply any vetoes in this search as their effect inthe results of this analysis would have been insignificant.

An inportant consideration in a veto analysis is to verifythe absence of coupling between a real gravitational waveburst and these auxiliary channels, or “safety”; in other words,

7

to verify that a real burst will not cause itself to be ve-toed. This was measured using hardware injections to thegravitational-wave channel and applying the same end-to-endanalysis pipeline including the veto candidates. For our choiceof candidate veto channels, no such coupling was found whichwould have impaired the safety of the vetoes. Although the fi-nal choice in the S2 veto analysis was based more on a heuris-tic approach, both the coupling analysis and the dead-time ver-sus efficiency study of the auxiliary channels recorded by theLIGO instruments are of paramount importance in the even-tuality of any event surviving the analysis pipeline As we willsee later on in this paper, one such event did survive all thecuts of the analysis but subsequent examination of auxiliarychannels established its origin to be of environmental charac-ter in the two Hanford detectors.

VI. METHODS FOR EVENT TRIGGER SELECTION

An accurate knowledge of gravitational wave burst wave-forms would allow the use ofmatched filtering[? ] alongthe lines of the search for binary inspirals [2, 18]. However,many different astrophysical systems may give rise to gravi-tational wave bursts, and the physics of these systems is oftenvery complicated. Even when numerical relativistic calcula-tions have been carried out, as in the case of core collapsesupernovae, they generally yield roughly representative wave-forms rather than exact predictions. Therefore, our searchesfor gravitational wave bursts use general algorithms whicharesensitive to a wide range of potential signals.

The first LIGO burst search used two Event Trigger Gen-erator (ETG) algorithms: a time-domain method designed todetect a large “slope” (time derivative) in the data stream af-ter suitable filtering, and a method calledTFCLUSTERS[19]which is based on identifying clusters of excess power in time-frequency spectrograms [1]. Several other burst-search meth-ods have been developed by members of the LIGO ScientificCollaboration. For this paper, we have chosen to focus ona single ETG called WaveBurst which identifies clusters ofexcess power once the signal is decomposed in the waveletdomain, as described below. Other methods which were ap-plied to the S2 data includeTFCLUSTERS; the excess powerstatistic [20]; and the “Block-Normal” time-domain algorithmwhich detects changes in the statistical properties of the datastream [21]. These methods were found to have somewhatpoorer sensitivities for the target waveforms described inSec-tion VIII than did WaveBurst, typically by a factor of 1.2 to3.

An integral part of our S2 search and the final event trig-ger selection is the waveform consistency test of the wave-form of the candidate events that are selected by the ETG.This is done using ther-statistic [22], a time-domain cross-correlation method sensitive to the coherent part of the candi-date signals.

A. WaveBurst

WaveBurst is an ETG that searches for gravitational wavebursts in the wavelet time-frequency domain. It is describedin greater detail in [23]. The method uses wavelet transforma-tions in order to obtain the time-frequency representationofthe data. The bursts are identified by searching for regions inthe wavelet time-frequency domain with an excess of powerthat is inconsistent with stationary detector noise.

The ETG processes gravitational wave data from two inter-ferometers at a time (two channels). As shown in figure 2 theanalysis is performed over three LIGO detectors resulting inthe production of triggers for three detector pairs. For eachdetector pair the analysis proceeds in the following steps:1)wavelet transformation, 2) selection of wavelet amplitudes, 3)coincidence between the channels, 4) generation of burst trig-gers, and 5) selection of burst triggers. During steps 1), 2)and 4) the data processing is independent for each channel.During steps 3) and 5) data from both channels are used.

FIG. 2: Block diagram of the WaveBurst analysis pipeline forthreeLIGO detectors, H1, H2 and L1 as applied in the S2 data.

The input data to the WaveBurst ETG are time series withduration of 120 seconds and sampling rate of16384 Hz. Be-fore the wavelet transformation is applied the data are down-sampled by factor of two. Using the orthogonal wavelet trans-formation –based on a SYMLET wavelet with the filter lengthof 60– the time series are converted into wavelet seriesWij ,wherei is the time index andj is the wavelet layer index. Eachwavelet layer can be associated with a certain frequency bandof the initial time series. Therefore, the wavelet seriesWij

can be displayed as a time-frequency scalogram consisting of64 wavelet layers withn = 15360 pixels (data samples) each.The time-frequency resolution of the WaveBurst scalogramsis the same for all the wavelet layers (1/128 sec× 64 Hz).This tiling is different from the one in the conventional dyadicwavelet decompositionwhere the time resolution adjusts tothescale [23–25]. The constant time-frequency resolution makesthe WaveBurst scalograms similar to spectrograms producedwith windowed Fourier transformations.

For each layer we first select a fixed fractionP of pixelswith the largest absolute amplitudes. These are calledblackpixels. The number of selected black pixels isnP . All otherwavelet pixels are calledwhite pixels. Then we calculate rankstatistics for the black pixels. The rankRij is an integer num-ber from 1 tonP , with the rank 1 assigned to the pixel withthe largest absolute amplitude in the layer. Given the rank of

8

wavelet amplitudesRij , the following non-parametric pixelstatistic is computed

yij = − ln

(

Rij

nP

)

. (6.1)

For white pixels the value ofyij is set to zero. The statisticyij has a meaning of the pixel’slogarithmic significance. As-suming Gaussian detector noise, the logarithmic significancecan be also calculated as

yij = gP (wij) = ln(P ) − ln

(

√

2/π

∫ ∞

wij

e−x2/2 dx

)

,

(6.2)wherewij is the absolute value of the pixel amplitude in unitsof the noise standard deviation. In practice, the LIGO detec-tor noise is not Gaussian and its probability distribution func-tion is not well known. Therefore, we use the non-parametricstatisticyij , which is a more robust measure of the pixel sig-nificance thanyij . Using the inverse function ofgP with yij

as an argument, we introduce thenon-parametric amplitude

wij = g−1P (yij), (6.3)

and theexcess power ratio

ρij = w2ij − 1, (6.4)

which characterizes the pixel excess power above the averagedetector noise.

After the black pixels are selected, we require their time-frequency coincidence in the two channels. Given a blackpixel of significanceyij in the first channel, this is acceptedif the significance of neighboring pixels in the second channel(y′ij) satisfies

y′(i−1)j + y′ij + y′(i+1)j > η, (6.5)

whereη is thecoincidence threshold. Otherwise, the pixel isrejected. This procedure is repeated for all the black pixels inthe first channel. The same coincidence algorithm is appliedto pixels in the second channel. As a result, a considerablenumber of black pixels in both channels produced by fluctua-tions of the detector noise is rejected. At the same time, blackpixels produced by coincident bursts have a much higher ac-ceptance probability because of the coherent excess of powerin two detectors.

After the coincidence procedure is applied to both channelsa clustering algorithm is applied jointly to the two channelpixel maps. As a first step, we merge the black pixels fromboth channels into one time-frequency plane. We then de-fine the black pixel neighbors, which could be both black andwhite. If a neighboring pixel is white, we call it ahalo pixeland distinguish it from the other white pixels. We define acluster as an isolated island on the time-frequency plane filledwith black and halo pixels. After the cluster reconstruction,we go back to the original time-frequency planes and calculatethe cluster parameters separately for each channel. Therefore,there are always two clusters, one per channel, which form aWaveBurst trigger.

The cluster parameters are calculated using black pixelsonly. For example, the cluster sizek is defined as the num-ber of black pixels. Other parameters, which characterize thecluster strength, are the cluster excess energy ratioρ and theclusterlogarithmic likelihoodY . Given a clusterC(k), theseare estimated as

ρ =∑

i,j∈C(k)

ρij and Y =∑

i,j∈C(k)

yij . (6.6)

As we will see in a while, the cluster size, likelihood and theexcess power ratio can be used for the further selection of trig-gers.

There are two main WaveBurst parameters: the black pixelfractionP and the coincidence thresholdη. The purpose ofthese parameters is to control the average black pixel occu-pancyO(P, η) of the scalograms used for the cluster recon-struction. To ensure robust cluster reconstruction, the occu-pancy should not be greater than1%. For white Gaussiandetector noise the functional dependenceO(P, η) can be cal-culated analytically. We require thatO(P, η) = 0.7%, whichsets a constraint onP andη.

The selection of black pixels effectively sets a threshold onthe wavelet amplitudeswij > g−1

P (0). The larger the value ofP the lower the threshold. [XXX doesn’t it get way too tech-nical here without remaining clear enough? XXX] However,the black pixel fraction should not be greater than 31.7%, oth-erwise pixels with negative values ofρij would be taken intothe analysis. From the other side, withP set too small (lessthan a few percent), noise outliers due to instrumental glitchesmay consume the entire time-frequency volume available forblack pixels and thus mask gravitational waves. To avoid satu-ration from the instrumental glitches, we run the analysis withP equal to 10%. This value ofP together with the occupancyconstraint onO(P, η) = 0.7% sets the coincidence thresholdη to 1.5.

All the tuning of the WaveBurst method was performed onthe S2 playground data set (section III). For the selected val-ues ofP andη, the average trigger rate per LIGO instrumentpair was approximately6 Hz. Within a factor of two thiswas consistent with the false alarm rate expected for the whiteGaussian detector noise. The trigger rate was further reducedby imposing cuts on the cluster sizek and the excess powerratioρ. For clusters of sizek > 1 we requiredρ > 6.25 whilefor single pixel clusters (k = 1) we used a more restrictivecut onρ > 9. This selection on the event parameters fur-ther reduced the counting rates per LIGO instrument pair to∼ 1 Hz. WaveBurst events passing these criteria were writtenonto disk. This allowed the further processing and selectionof these events without the need to re-analyze the full datastream, a process which is generally time and cpu intensive.

B. Triple coincidence

The further selection of WaveBurst events proceeds byforming their triple coincidences. The output of the Wave-Burst ETG is a set of coincident triggers for a selected inter-ferometer pair(X,Y ). Each WaveBurst trigger consists of

9

two clusters, one inX and one inY . For the three LIGO in-terferometers there are three possible pairs: (L1,H1),(H1,H2)and (H2,L1). In order to establish triple coincidence events,we simply require a time-frequency coincidence of the Wave-Burst triggers generated for these three pairs. Given the timeti of individual pixels, the cluster center time is calculatedas

T =∑

i,j∈C(k)

ti w2ij /

∑

i,j∈C(k)

w2ij . (6.7)

In applying the time coincidence we first constructTXY =(TX + TY )/2, i.e., the combined central time of theX andY clusters in each interferometer pair. Three such combinedcentral times are thus constructedTL1H1, TH1H2 andTH2L1.We then require that all possible differences of these combinedcentral times fall within a time windowTw = 20 ms. We alsoapply a loose requirement on the frequency coincidence of theWaveBurst triggers. First, we calculate the minimun (fmin)and maximum (fmax) frequency for each interferometer pair(X,Y )

fmin = min(fXlow, f

Ylow), fmax = max(fX

up, fYup), (6.8)

whereflow andfup are the lower and upper frequency bound-aries of theX andY clusters. Then, the trigger frequencybands are calculated asfmax − fmin for all pairs. For thefrequency coincidence, the bands of all three WaveBurst trig-gers are required to overlap. The final step in the coincidenceanalysis of the WaveBurst events involves the constructionof a single measure of their combined significance. As wedescribed already, triple coincidence events consist of threeWaveBurst triggers involving a total of six clusters. Each clus-ter has its parameters calculated on a per-interferometer basis.For the final selection of the burst triggers we use the proba-bilistic measure of thecluster significance

Z = Yk − ln

(

k−1∑

m=0

Y mk

m!

)

(6.9)

derived from the logarithic likelihoodY [23]. Given the sig-nificance of six clusters, we compute thecombined signifi-canceof the triple coincidence event as

ZG =(

ZL1L1H1 Z

H1L1H1 Z

H2H2L1 Z

L1H2L1 Z

H1H1H2 ZH2

H1H2

)1/6,

(6.10)whereZX

XY (ZYXY ) is the significance ofX (Y ) cluster for the

(X,Y) interferometer pair.Using 46 time-lagged instances of the S2 playground data

we have set the threshold onZG for this search in order toyield a targeted false alarm rate of O(10)µHz. Without com-promising much the pipeline sensitivity, this threshold wasselected to be 5.47, orln(ZG) > 1.7. For events fallingin the XX-1100Hz frequency band, the resulting false alarmrate in the S2 playground analysis was approximately15 µHz.The events the WaveBurst selection described so far are thenchecked for their waveform consistency using ther-statistic.

L1Entries 2845Mean 2.615RMS 1.926

significance1 10

even

ts

-210

-110

1

10

210 L1Entries 2845Mean 2.615RMS 1.926

H1Entries 2845Mean 2.603RMS 1.7

significance1 10

even

ts

-210

-110

1

10

210 H1Entries 2845Mean 2.603RMS 1.7

H2Entries 2845Mean 2.572RMS 1.826

significance1 10

even

ts

-210

-110

1

10

210 H2Entries 2845Mean 2.572RMS 1.826

L1xH1xH2Entries 2845Mean 2.235RMS 0.9299

combined significance1 10

even

ts

-210

-110

1

10

210 L1xH1xH2Entries 2845Mean 2.235RMS 0.9299

FIG. 3: The significance distribution of the WaveBurst events forindividual detectors (L1, H1, H2) and the combined significance oftheir triple coincidences (L1xH1xH2) for the S2 data set. Solid his-tograms reflect the zero-lag events, while the points with error barsrepresent background events as produced with unphysical time lagsbetween the Livingston and Hanford detectors (and normalized to S2live time).

C. r-statistic

Ther-statistic is a waveform consistency test [22, 26] thatwas applied as the final cut in the analysis of the LIGO data forthe identification of burst events. This cut was applied to allthe events that passed the WaveBurst ETG pipeline by goingback to the raw (uncalibrated) interferometric data aroundthetime of the reported coincident events in order to re-analyzethem and apply this criterion.

The fundamental building block in perfoming this wave-form consistency test is ther-statistic, or the normalized lin-ear correlation coefficient of two sequences –in this case thetwo gravitational wave signal time series–{xi} and{yi}:

r =

∑

i(xi − x)(yi − y)√∑

i(xi − x)2√∑

i(yi − y)2, (6.11)

wherex andy are their respective mean values. This quantityassumes values between -1 for fully anti-correlated sequencesand +1 for fully correlated sequences; the mid-value of 0 re-sults for uncorrelated ones. More generally, for uncorrelatedwhite noise, we expect ther-statistic values obtained for arbi-

10

trary sets of points of lengthN to follow a normal distributionwith zero mean andσ = 1/

√N . Any coherent component

in the two sequences will causer to deviate from the abovenormal distribution. As a normalized quantity, ther-statisticis not sensitive to the relative amplitude of the two sequences.At the same time though, it offers the advantage of being ro-bust against fluctuations of detector response and noise floor.

The number of pointsN considered in calculating thestatistic in 6.11 –or equivalently theintegration windowτ–is the most important parameter in the construction of ther-statistic. Its optimal value depends in general on the dura-tion of the signal being considered for detection. Ifτ is toowide, the candidate signal is “washed out” by the noise whencomputingr. On the other hand, if it is too short, statisticalfluctuations in the values ofr reduce its power as a waveformconsistency discriminant. Simulation studies have shown thatmost of the short-lived signals of interest to the LIGO burstsearch can be retrieved successfully using the three integra-tion time windows of 20, 50 and 100ms length.

Within its LIGO implementation, ther-statistic analysisfirst perfomed some necessary data conditioning. That aimedto restricting the frequency content of the signals within theLIGO’s sensitive band as well as suppressing any coherentlines and instrumental artifacts. This eventually led to aswhitedata as possible. For this purpose, we first band-passed thedata with an 8th order Butterworth filter and then downsam-pled them to a 2048Hz rate. The band-pass had a low cornerfrequency of 100Hz, in order to suppress the contribution ofseismic noise, and a high corner at 1572Hz in order to havea 20dB suppression at 2048Hz and thus avoid aliasing. Theband-passed data were then effectively whitened with a linearerror predictor filter of order XXXXX trained on a 10s periodbefore the event start time. The filter removed predictable con-tent, including lines that were stationary over a 10s time scale.It also had the effect of subtracting from the data the spectralshape, thus emphasizing transients [25, 26].

The next step in ther-statistic analysis involved the con-struction of all the possibler coefficients given the number ofinterferometer pairs involved in the trigger, their possible rel-ative time-delays due to their geographic separation as well asthe various integration windows being considered. Althoughther coefficients generally assume values in[−1, 1], we havechosen to use only the absolute value of this statistic, i.e., |r|.This is mostly because our emphasis has been to test by howmuch the correlation coefficients deviate from 0 (the uncor-related case). For a given integration windowp, pair of in-trumentsl, m(l 6= m) and relative time-delayk, values of{|rk

plm|} were obtained by the sliding (with 50% overlap) ofthe integration window on the data. A Kolmogorov-Smirnovtest with 5% significance was used to compare the distributionof these values of{|rk

plm|} to the null hypothesis expectation

of a normal distribution with zero mean andσ = 1/√

Np,whereNp is the number of data samples in thep-th integrationwindow. If the two were inconsistent, a one-sided significanceand its associated logarithmic confidence were constructed??.The maximum of this quantity among all relative time-delays(±10ms) and averaged over all possible instrument (ordered)pairs defined the correlation confidence for a given integration

window and a given piece of the data. Finally,β, the multi-interferometer combined correlation confidence was assignedto an event by taking thesupremumamong the ones corre-sponding to all the possible integration windows and time se-ries intervals that were being considered. Events with a valueof β above a given threshold were finally selected.

A similar method based on this type of time-domain cross-correlation and with a more detailed (and computationally in-tense) scan of integration windows, has been implemented inthe externally triggered search in LIGO [27, 28] and else-where [29].

The r-statistic threshold onβ was tuned primarily in or-der to yield a targeted final coincidence rate of events. Otherchoices entering in ther-statistic implementation (like the fil-ter parameters, the Kolmogorov-Smirnov’s test significanceor the integration windows) were madea priori. We followedthe criterion of much less than one false alarm expected inthe whole S2, which translated into anO(0.1)µHz false alarmrate. Using the WaveBurst trigger production on the S2 play-ground (which as we mentioned in the previous section re-sulted to aO(15)µHz rate) the threshold on ther-statistic’sβwas chosen so that to ensure at least 99% of the ETG back-ground events were suppressed by the test.

TABLE I: Percentage of S2 background events rejected by ther-statistic for two different values ofβ.

Event Production β > 3 β > 4

200 ms white gaussian noise ... ± ...% ... ± ...%

200 ms real noise (random)99.72 ± 0.013% 99.996 ± 0.002%

WaveBurst 95.1 ± 0.8% 99.5 ± 0.3%

Table I shows the rejection efficiency of ther-statistic fortwo thresholds onβ when the test is applied to white gaus-sian noise (200 ms segments), to randomly selected interfer-ometer noise (200 ms segments) or to the background eventsproduced by the WaveBurst ETG coincidence analysis in theplayground. On the basis of these considerations, we choseto operate ther-statistic waveform consistency test with thethreshold onβ set at 4. As we will discuss in the efficiencystudies section VIII that follows, ther-statistic waveform con-sistency test withβ = 4 presents, for the waveforms we con-sidered, a sensitivity that is equal to or better than that oftheWaveBurst ETG. As a result of this, the false dismissal prob-ability of the test does not impair the efficiency of the wholepipeline.

VII. SIGNAL AND BACKGROUND RATES

In the preceding section we described the methods that weused for the selection of burst events. These were applied tothe S2 triple coincidence data set (excluding the playground)for a total of8.6 × 105 seconds of observation time. This istheS2 data set of this analysis and every aspect of the analysisdiscussed from this point on will refer only to it.

11

A. Event analysis

The WaveBurst analysis applied on the S2 data yielded 16coincidence events (at zero-lag). These are shown in table II.The application of ther-statistic cut rejected 15 of them leav-ing us with a single event that passed all the analysis criteria.

The background in this search is generally attributed to ran-dom coincidences of otherwise uncorrelated instruments. Wehave measured this background by artificially shifting the rawtime series of the L1 instrument. As in our S1 search, we havechosen not to time-shift relative to each other the two Hanfordinstruments (H1, H2). Although we had no evidence of H1-H2 correlations in the S1 burst search, indications for suchcorrelations in other LIGO searches exist [3]. A total of 46artificial lags of the raw time series of the L1 instrument wereused in order to make a measurement of the accidental rateof coincidences, i.e, the background. The background eventsgenerated in this way were also processed by ther-statistic inan identical way with the one used for the zero-lag events.The total simulated live-time for the purpose of measuringthe background in this search was thus33.8 × 106 seconds.A proper correction of each time-shift experiment’s live-timedue to the reduction of common observation time betweenthe LIGO instruments was taken into account when calculat-ing background rates. This correction was necessary becausethe observation time was non-contiguous. The time-lags werechosen in the [-115,115] seconds range and with a 5s step.This step was larger than the duration of any signal that wesearched for and well above the time-scale within which anyautocorrelation might have been present in the data. This canbe seen in figure 4 where the time between consecutive eventshistogram is shown for the double and their resulting triplecoincidence WaveBurst events before any geometric signifi-cance orr-statistic cut is applied. These distributions followthe expected exponential form, indicating a quasi-stationaryPoisson process.

A scatter plot of the background rates established in eachof the 46 time-lag experiments as a function of the lag time isshown in figure 5. These rates reflect the measurementbeforethe application of ther-statistic. A Poisson fit is also shownin the adjacent panel; the fit describes reasonably well the re-spective distribution of rates confirming the hypothesis thatthe triple coincidence events are Poisson distributed.

All the events that resulted from the zero-lag coincidenceanalysis as well as the 46 time-lag experiments were subjectedto the r-statistic analysis. In figure 6 we show theβ val-ues, i.e., the multi-interferometer combined correlationcon-fidence. The plot displays the cummulative rate of events (y-axis) above a givenβ value (x-axis) both for the zero-lag coin-cidence and background events. For the choice ofβ thresholdthat we have made for this analysis (β > 4), only a singleevent in the zero-lag analysis and a single background eventin all the 46 time-lags of the WaveBurst analysis survived thefinal waveform consistency cut (r-statistic).

In table III we summarize the number of zero-lag coin-cidence and background events that the WaveBurst pipelineyielded. The events before and after the application of ther-statistic are also shown. The measurement for the background

H1H2Entries 346262

/ ndf 2χ 508.8 / 118Constant 0.01± 10.06 Slope 0.0007± -0.2486

time between consecutive events, seconds0 10 20 30 40 50 60

even

ts

1

10

210

310

410

510 H1H2Entries 346262

/ ndf 2χ 508.8 / 118Constant 0.01± 10.06 Slope 0.0007± -0.2486

H1L1Entries 355456

/ ndf 2χ 308.7 / 112Constant 0.01± 10.16 Slope 0.001± -0.275

time between consecutive events, seconds0 10 20 30 40 50 60

even

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310

410

510 H1L1Entries 355456

/ ndf 2χ 308.7 / 112Constant 0.01± 10.16 Slope 0.001± -0.275

L1H1H2Entries 2141

/ ndf 2χ 27.17 / 19Constant 0.032± 6.457

Slope 0.000091± -0.003761

time between consecutive events, seconds0 200 400 600 800 1000 1200 1400 1600 1800 2000

even

ts

1

10

210

310 L1H1H2Entries 2141

/ ndf 2χ 27.17 / 19Constant 0.032± 6.457

Slope 0.000091± -0.003761

FIG. 4: Time between consecutive WaveBurst events (prior tothe ap-plication of ther-statistic test). The triple coincidence events are rea-sonably well described by a Poisson process of constant mean. Theexponential fits are performed for time delays greater than [XXX].

lag [s]-100 -50 0 50 100

Hz]

µra

te [

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 910

Entries 46

Mean 14.4

RMS 3.5

/ ndf 2χ 7.7 / 13

Prob 0.9

A 6.6± 41.1

µ 0.7± 14.0

FIG. 5: WaveBurst event rate (prior to ther-statistic test, inµHz)versus lag time (ins) of the L1 interferometer respect to the H1 andH2. The zero-lag measurement, i.e., the only coincidence measure-ment that is physical, is also shown. Due to the fragmentation ofthe data set each time-lag has slightly different live-time; for thisreason, rates rather than event numbers are shown. A projection ofthe event rates to an one-dimensional histogram with a Poisson fit isalso shown in the adjacent panel. These background rates reflect theWaveBurst analysisprior to the application of the final event selec-tion, ther-statistic.

reported in this table is normalized to the same live-time as

12

TABLE II: Reconstructed parameters of the 16 events that passed the WaveBurst analysis. Only one of them (event #9) passes ther-statisticcut of aβ > 4.

Event hrss frequency (Hz) Geometricr-statistic

L1 H1 H2 L1 H1 H2 Significance β

1 5.8e-21 3.9e-21 1.4e-20 471.5 412.3 455.8 1.704 2.10

2 2.3e-21 3.2e-21 3.1e-21 416.0 416.0 416.0 1.727 1.21

3 2.7e-20 2.2e-20 1.7e-20 97.5 126.0 129.1 1.913 1.76

4 6.7e-21 4.7e-21 1.2e-20 629.6 482.2 618.5 1.774 2.32

5 3.2e-20 4.7e-21 5.9e-20 1376.4 454.3 1447.1 1.892 1.74

6 4.7e-21 4.4e-21 6.7e-21 530.7 531.7 548.4 1.766 0.89

7 3.3e-20 5.3e-21 4.1e-20 1167.2 961.2 950.1 2.433 1.20

8 2.0e-20 2.6e-21 2.2e-20 1371.2 438.7 1272.1 1.739 1.64

9 2.7e-21 6.4e-20 1.4e-19 139.8 136.1 128.7 1.722 4.10

10 1.3e-20 5.1e-21 4.5e-21 984.5 1027. 992.0 1.908 2.03

11 7.8e-21 2.4e-21 1.1e-20 599.1 480.0 519.8 1.785 1.16

12 5.8e-20 4.4e-21 6.5e-20 1337.2 329.4 1307.7 2.081 1.00

13 9.0e-21 2.9e-21 7.2e-21 492.4 480.0 510.9 1.785 1.47

14 1.2e-20 6.0e-21 2.0e-20 525.3 485.4 550.9 1.773 1.59

15 6.0e-21 2.9e-21 7.4e-21 440.1 446.0 428.1 1.755 2.73

16 7.5e-20 3.5e-21 7.0e-20 1227.9 253.7 1170.3 1.749 1.06

βr-statistic threshold 0 1 2 3 4 5 6 7

βr-statistic threshold 0 1 2 3 4 5 6 7

Hz]

µra

te [

10-2

10-1

1

10Coincidences

Background

FIG. 6: Cumulative event rates versusr-statisticβ threshold forWaveBurst ETG zero-lag (in red) and background (in black) anal-yses.

the one for the zero-lag coincidence measurement.Table III shows that a single event in the triple coincidence

data set survived all previously described analysis cuts. Giventhe expected background of 0.05 events, the probability of anevent surviving by chance is only [XXX]. This candidate grav-itational wave detection triggered investigations of the instru-ment’s auxiliary interferometric and environmental monitor-ing channels with the purpose of determining if any of thethree coincident triggers could have been produced by an in-terferometer malfunction or an event in the environment of theinstrument.

The investigation indicated that two of the three triggerscould have been produced by an acoustic event at Han-

TABLE III: Event statistics and rate bounds for the S2 burst search.The coincidence and background number of events reflect the samelive-time. The upper limits (UL) at the three confidence levels (CL)are derived according to the Feldman-Cousins [12] formulation.

WaveBurst events in 862186 sec rate

pre-r test

Coincidences 16 18.6 ± 4.6 µHz

Background 12.3 14.3 ± 0.7 µHz

post-r test

Coincidences 1 1.2 ± 1.2 µHz

Background 0.05 0.06 ± 0.04 µHz

90% CL UL 4.3 0.43/day

95% CL UL 5.1 0.51/day

99% CL UL 6.9 0.69/day

ford, where two of the three interferometers are co-located.The couplings of environmental signals to the interferometerswere measured during S2 by generating acoustic and otherenvironmental disturbances and comparing the resulting sig-nals on the gravitational and environmental monitoring chan-nels. The S2 acoustic coupling measurements indicated thatthe amplitude as well as the frequency of signals on both theH1 and H2 gravitational wave channels during the triple co-incidence event could be accounted for by the simultaneousacoustic event in the recorded microphone signals. On thisbasis, the single surviving event of this analysis was discardedas a plausible gravitational wave burst.

The source of the acoustic event appears to have been anaircraft. The sequence of Doppler shifts in the microphonesignals from the five Hanford stations were consistent with the

13

overflight of an airplane on a typical approach, roughly paral-leling the X arm of the interfermeters, to the Pasco, Washing-ton airport. Similar signatures at other times have been vi-sually confirmed as over-flying airplanes. The signals on thegravitational wave channel shifted in frequency along withthemicrophone signals at known acoustic coupling sites.

Because of the sensitivity of the interferometers to theacoustic environment during S2, a program to reduce acousticcoupling was undertaken prior to S3. The acoustic sensitiv-ities of the interferometers were reduced by 2 to 3 orders ofmagnitude by addressing the coupling mechanisms on opticstables located outside of the vacuum system, and by acousti-cally isolating the main coupling sites.

B. Error analysis

The statistical error in the estimated rate of backgroundevents (0.05) is small due to the use of 46 statistically inde-pendent time-lags of the data sample. The standard error onthe Poisson mean of this measurement is 0.03 events. Sourcesof systematic errors may arise in the choices we have made onhow to perform the time-lag experiments, namely the choiceof step and window size as well as the time-lag method by it-self. We have performed time-lag experiments using differenttime steps, all of which yielded statistically consistent resultsThe 1-σ systematic errors due to this is estimated to be lessthan 0.04 events.

C. Rate bound

We can now use our measurement of zero-lag coincidenceand background events in this search for bursts in LIGO’s S2data in order to set bounds on the rate of such events in ourdetectors. Although the single event remaining in the zero-lag analysis was clearly identified as non-astrophysical, wehave chosen to include it in the upper limit statistics. Thiswasmostly due to the subtleties that are involved in introducingformally apost factoveto that was going to remain unbaised.The upper limits thus derived are weaker (more conservative)by roughly a factor of 3 from what they would had been other-wise. As in our S1 analysis [1] we establish these bounds us-ing the Feldman-Cousins [12] approach. In table III we reportthe upper rate bound on the signal interval set by the Feldman-Cousins method at the 90%, 95% and 99% confidence level(CL). Due to the single zero-lag event that we retained for theupper limit statistics, the lower bound on this interval is notconsistent with the zero rate of signal events at the 90% and95% CL. At the 90% confidence level the S2 upper rate boundstands at 0.43 events per day and it reflects an improvement ofa factor of 4 respect to our S1 result [1].

VIII. EFFICIENCY OF THE SEARCH

A. Target waveforms and signal generation

In order to estimate the sensitivity of the burst analysispipeline we studied its response to simulated signals of var-ious waveform morphologies and strengths. The simulatedsignals were injected in software (and off-line) into the grav-itational wave data stream of the three LIGO detectors andover the same S2 triple coincidence data set that was searchedfor bursts [30]. Severalad hocand astrophysically motivatedwaveforms were selected for injections:

• Gaussian waveforms of the form [1]h(t + t0) =h0 exp(−t2/τ2) and withτ equal to 0.1, 0.5, 1.0, 2.5,4.0 and 6.0ms;

• sine-Gaussian waveforms of the form [1]h(t + t0) =

h0 sin(2πf0t) exp(−t2/τ2), whereτ = Q√2πf0

, Q=9,

andf0 assumed the value of 100, 153, 235, 361, 554,and 849 Hz;

• waveforms resulting from numerical simulations ofcore-collapse supernovae as they are available in the lit-erature [31–33];

• binary black hole merger waveforms as described in [9]and for black hole masses of 10, 30, 50, 70 and 90 solarmasses.

The sine-Gaussian and Gaussian waveforms were chosen torepresent the two general classes of short-lived gravitationalwave bursts of broad-band and narrow-band character respec-tively. The supernovae and binary black hole merger wave-forms were adopted as a more realistic model for a gravita-tional wave burst.

In order to make the simulation as realistic as possible wetook into account the exact geometry of the individual LIGOdetectors respect to the impinging gravitational burst wave-front. A gravitational wave burst is generally comprised oftwo waveformsh+(t) andh×(t) which represent its two po-larizations. The signal produced on the output of a LIGO de-tectorhdet(t) is a linear combination of these two waveforms,

hdet(t) = F+h+(t) + F×h×(t), (8.1)

whereF+ andF× are the antenna pattern functions [13, 34].The antenna pattern functions depend on the source locationon the sky (spherical polar anglesθ andφ) and the wave’spolarization angleψ. In order to generate the antenna pat-tern functions for the two Hanford detectors –H1 and H2–we explicitly constructed the rotational transformation fromthe source coordinate frame to the Handford frame. In or-der to obtain the L1 one we applied a second transformation–rotation from the Hanford to the Livingston frame. The timesof the injections were chosen randomly according to a Poissonmean rate of one per minute. The source coordinates –θ andφ – were also chosen randomly so that they would appear uni-formly distributed on the sky. For every source direction therelative time delay between the Hanford and Livingston sites

14

was also acounted for. For the twoad hocwaveform fami-lies (Gaussians, sine-Gaussians) as well as for the supernovaeones, a linearly polarized wave was assumed with a polariza-tion angle that was taking values randomly between 0 and2π.The binary black hole merger waveforms come with two po-larizations [9] andthey were simulated appropriately.Finally,the amplitudes of the injected signals were varied so that theefficiency of the search pipeline could be mapped as a func-tion of the injection strength.

B. Software injection results

In order to add the aforementioned waveforms to the rawdetector data, their signals were first digitized at the LIGOsampling frequency of 16384Hz. Their amplitudes definedin strain dimensionless units were converted in units of ADCcounts using the inverse calibration function. The resultingtime series of ADC(ti) were then added to the raw detectordata and were made available to the analysis pipeline whichprocessed them in a way indentical to the analysis of the raw,real interferometric data. For every combination of waveformand detector a total ofapproximately 3000signals were in-jected uniformly over (a) the S2 data set that was used forsetting the rate bound and (b) the signal strengths (logarithmi-cally, for a total of 20different values). As in our S1 signalinjection analysis, we have used theroot-sum-square (rss)am-plitude spectral density defined by

hrss ≡√

∫

|h(t)|2dt (8.2)

in order to quantify the strength of the injected signals. This isa measure of the square root of the signal “energy” and it canbe shown that, when divided by the detector spectral noise, itapproximates the signal-to-noise ratio (SNR) that is used toquantify the detectability of a signal in optimal filtering.Onemay notice thathrss has units ofstrain/

√Hz and it can thus

relate directly to the detector sensitivity curves as measuredby power spectral densities over long time-scale.The quan-tities (discriminators) used by the WaveBurst ETG for eventselection are found to be monotonically related to the signals’hrss.

The efficiency was defined as the ratio of the detected (atthe end of the pipeline) over injected events. The softwareinjections exercised a range of signal strengths that allowedus to measure –in most cases– the turn on of the efficiency allthe way from 0 to 1. The measurements were fitted with anasymmetric sigmoid of the form

efficiency(x) =ε

1 +(

xx0

)α(1+β tanh( xx0

), (8.3)

wherex is thehrss of the injected signals,ε corresponds tothe efficiency for strong signals (and should be close to 1),x0 is thehrss value corresponding toε2 efficiency,β is theparameter that describes the asymmetry of the sigmoid (and

takes values within±1) andα describes the slope. The an-alytic expression of the fits were then used to determine thesignal strengthhrss for which an efficiency of 50%, 90% and95% was reached.

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0

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1 100 Hz

153 Hz

235 Hz

361 Hz

554 Hz

850 Hz

100 Hz

153 Hz

235 Hz

361 Hz

554 Hz

850 Hz

100 Hz

153 Hz

235 Hz

361 Hz

554 Hz

850 Hz

100 Hz

153 Hz

235 Hz

361 Hz

554 Hz

850 Hz

100 Hz

153 Hz

235 Hz

361 Hz

554 Hz

850 Hz

100 Hz

153 Hz

235 Hz

361 Hz

554 Hz

850 Hz

FIG. 7: Detection efficiency of the analysis pipeline as a function ofthe signal strength for sine-Gaussian waveforms of Q=9 and centralfrequencies of 100, 153, 235, 361, 554 and 850Hz. The efficienciesplotted reflect the averages over the S2 data set, the sky positions andpolarization angles.

In analyzing the injection data,every aspectof the analy-sis pipeline that starts with single-interferometer time seriesADC(ti) and ends with a collection of event triggers was keptidentical to the one that was used in the analysis of the real,interferometric data. In accepting a signal injection as suc-cessfully detected the time interval associated with the Wave-Burst trigger had to be closer than 200 milliseconds from theknown peak time of the injection. In figure 7 we show theefficiency curves, i.e, the efficiency versus signal strength ofour end-to-end burst search pipeline for the case of the 6 dif-ferent sine-Gaussian waveforms we have introduced earlierinthis section. Given the way the signal injections were per-formed, we should point out that these curves reflect the av-eraging over theentiredata-taking run and over random (uni-form) source polarizations and directions. As expected giventhe instruments’ noise floor (see figure 1), the best sensitiv-ity is attained for sine-Gaussians with a central frequencyof235Hz; for this signal type, the required strength in order toreach 50% efficiency is athrss =1.5×10−20strain/

√Hz and

it stands roughly20times above the average noise floor of theLIGO instruments during S2. In figure 8 we show the samecurves for the Gaussian family of waveforms we considered.

C. Parameter estimation of signals

The software signal injections we just described provided agood way of not only measuring the efficiency of the searchbut also benchmarking WaveBurst’s ability in extracting thesignal parameters. An accurate estimation of the signal pa-rameters by a detection algorithm is essential for the suc-cessful implementation of the time and frequency coincidenceamong candidate triggers coming from the three LIGO detec-tors.

We have confronted the central time of a WaveBurst event

15

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0

0.2

0.4

0.6

0.8

1 = 0.1 msτ

= 0.5 msτ

= 1.0 msτ

= 2.5 msτ

= 4.0 msτ

= 6.0 msτ

= 0.1 msτ

= 0.5 msτ

= 1.0 msτ

= 2.5 msτ

= 4.0 msτ

= 6.0 msτ

= 0.1 msτ

= 0.5 msτ

= 1.0 msτ

= 2.5 msτ

= 4.0 msτ

= 6.0 msτ

= 0.1 msτ

= 0.5 msτ

= 1.0 msτ

= 2.5 msτ

= 4.0 msτ

= 6.0 msτ

= 0.1 msτ

= 0.5 msτ

= 1.0 msτ

= 2.5 msτ

= 4.0 msτ

= 6.0 msτ

= 0.1 msτ

= 0.5 msτ

= 1.0 msτ

= 2.5 msτ

= 4.0 msτ

= 6.0 msτ

FIG. 8: Same plot as in figure 7 but for Gaussian injections ofτ =0.1, 0.5, 1.0, 2.5, 4.0 and 6.0ms.

TABLE IV: Summary of the S2 pipelinehrss sensitivity toad hocwaveforms in units of10−20

strain/√

Hz (these are livetime, polar-ization and sky averaged results).

95% 90% 50%noise PSD

sine-Gaussianf0=100Hz 55 34 8.0

sine-Gaussianf0=153Hz 40 24 5.4

sine-Gaussianf0=235Hz 12 7.1 1.5

sine-Gaussianf0=361Hz 12 7.7 1.8

sine-Gaussianf0=554Hz 17 10 2.3

sine-Gaussianf0=850Hz 31 19 4.2

Gaussianτ=0.1ms 31 19 4.8

Gaussianτ=0.5ms 18 11 2.8

Gaussianτ=1.0ms 26 16 3.3

Gaussianτ=2.5ms 123 73 15

Gaussianτ=4.0ms 252 152 33

Gaussianτ=6.0ms 568 382 119

(section VI) with the known central time of the signal injec-tion. For each of the twoad hocwaveform families consideredso far as well as for each of the astrophysical waveforms wewill introduce later (section IX), WaveBurst was able to re-solve the time of the event on the average within better than3ms and with an equalσ. The apparent bias has a contributioncoming from the calibration phase error. Another contributioncomes from the fact that the detected central time is based ona finite time-frequency volume of the signal’s decompositionwhich is obtained after thresholding. It remains however wellwithin our needs for a tight time coincidence between interfer-ometers. In figure 9 we show a typical plot of the timing errorfor the case of all the sine-Gaussian injections we exercised inthe software simulations and for the three LIGO instrumentstogether. For the same type of signals, we show in figure 10the reconstructed versus injected central frequency of them.The measurements are consistent within the signal bandwidth.

The accuracy of the WaveBurst algorithm in determiningthe signal strength had the highest fractional error among allthe estimated signal parameters, as can be seen in figure 11.The signal strength was on the average underestimated for

detected - injected time, ms-20 -15 -10 -5 0 5 10 15 20

Entries 8608Mean 1.223RMS 3.034

detected - injected time, ms-20 -15 -10 -5 0 5 10 15 20

even

ts

0

50

100

150

200

250 Entries 8608Mean 1.223RMS 3.034

FIG. 9: Timing error of the WaveBurst algorithm in the three LIGOinstruments during S2 when sine-Gaussian injections of varying fre-quency and strength were injected. For comparison, the timesepa-ration between the two LIGO sites is 10ms and the coincidencetimewindow we used in this analysis is 25ms.

injected frequency, Hz100 200 300 400 500 600 700 800 900

injected frequency, Hz100 200 300 400 500 600 700 800 900

det

ecte

d f

req

uen

cy, H

z

100

200

300

400

500

600

700

800

900

FIG. 10: Central frequency reconstruction for sine-Gaussian injec-tions ofQ = 9.

all signal morphologies. It was typically within a factor 3 ofthe injected signals’root-sum-square (rss)amplitude spectraldensity. Several factors contributed to this. WaveBurst limitsthe signalhrss integration within the detected time-frequencyvolume of an event and not over the entire theoretical supportof a signal (which is infinite). Moreover, errors in the deter-mination of the signal’s time-frequency volume due to thresh-olding may lead to systematic uncertainties in the determina-tion of its strength. Thehrss shown in figure 11 [XXX] alsoreflects the folding from the measurements coming from allthree LIGO instruments and thus calibration errors and noisefluctuations in either of them were immediately affecting it.Our simulation analysis has shown that the detected signal’s

16

hrss is the most sensitive quantity to detector noise and itsvariability; for this, it was not used in any step of the analysiseither as part of the coincidence analysis or for the final eventselection.

Hzstrain(injected hrss)*(beam pattern coefficient),

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ind

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ted

hrs

s,

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D. Hardware injection results

During the S2 data taking as well as shortly before and afterit, several run intervals were designated for hardware signalinjections. These injections were intended to address any in-strumental issues, including calibrations, and provide a robustend-to-end test of LIGO’s data analyses pipelines. An arbi-trary function generator connected to the actuators providedthe capability of exciting the mirrors according to a gravi-tational wave pattern. The waveforms injected through thishardware calibration included several of the ones described inthe target waveform section above. The injections among thethree LIGO instruments during S2 were not performed so thatto correspond to a single point source in the sky emitting grav-itational wave bursts that impinge on the network of our detec-tors. Thus their coincidence analysis using the same end-to-end pipeline invoked in the analysis of the real (and software-simulated) data was not fully appropriate. We have restrictedourselves in examining the performance of the LIGO instru-ments and of the analysis methods in detecting these eventsand reconstructing their signal parameters. In table V we sum-marize the accuracy in time and frequency reconstruction ofthe sine-Gaussian injections. Out of the 121 injected signals,111 were detected. Both the time and frequency reconstruc-tion by the WaveBurst algorithm on these hardware-injectedsignals is consistent with our software injections and withinour expectations. Although ther-statistic cut was not appliedto these events, we expect its effect on reducing the efficiencyto be insignificant.

E. Error analysis

The largest source of systematic error in the determinationof the efficiency of this search is by far the calibration un-certainty. We are estimating this to be at the level of 10%,mostly reflecting the amplitude calibration error [35]. No sig-nificant systematic error is attributed to the procedure we fol-lowed in order to perform the efficiency measurement. Theentire S2 data sample was indeed used in order to model thedetector noise and superimpose onto it various signal mor-phologies. The statistical error attributed to the finite num-ber of simulations that were used for the efficiency measure-ment is reflected in the goodness of the sigmoid-like fits andis estimated to less than 5% [XXX]. Throughout the S2 anal-ysis, the efficiency measurement was performed in multipleslightly-varying ways all of which yielded results within 1-sigma [XXX]. These variations included different samplingof the S2 data set, different versions of the calibration con-stants as well as different number and placing of the signalinjections. Combining all uncertainties, we estimate our effi-ciency to any given signal morphology to be accurate at the11% level or better.

IX. SEARCH RESULTS

A. Rate versus strength upper limit

As we have seen in section VII, using the zero-lag and back-ground rate measurements we set an upper bound on the rateof gravitational wave bursts at the instruments at the levelof0.43 events per day. We will now use the measurement ofthe efficiency of the search as was described in the previoussection in order to associate the above rate bound with thestrength of the searched for gravitational wave burst events.This is the rate versus strength analysis that we introducedinour previous search for bursts in LIGO using the S1 data??.The rate bound of our search to signals of strengthhrss isgenerally given by

R(hrss) =η

ε(hrss) × T(9.1)

where the numeratorη is determined by the Feldman-Cousinsstatistics given our event counting and the denominator is theproduct of the live-timeT of the search (8.6 × 105 seconds)by its efficiencyε to signals of strengthhrss. This rate ver-sus strength interpretation makes the same assumptions on thesignal morphology and origin as the ones that enter in the de-termination of the efficiency. In figure 12 we show the rateversus strength upper limit for the two classes ofad hocwave-forms we considered in this search, i.e., the Gaussians andsine-Gaussians. For a given signal strengthhrss these plotsgive the upper limit at the 90% confidence on the rate of burstevents at the instruments with strength equal or greaterhrss.In that sense, the part of the plot above these curves definesthe region of signal strength-rate excluded by this search ata given confidence level. As one would expect, for strong

17

TABLE V: Time and frequency reconstruction by the WaveBurstalgorithm for sine-Gaussian hardware injection performedin the three LIGOinstruments during S2.

central time (ms) frequency (Hz) Events Events

frequency (Hz) detected-injected detected-injected injected detected

L1 H1 H2 L1 H1 H2

100 5.4 4.9 5.0 -1.3 -3.2 -3.1 21 20

153 1.5 0.3 1.3 6.7 4.3 4.1 21 20

235 4.3 3.9 3.0 6.0 17.1 22.2 20 20

361 1.7 1.7 0.5 -11.6 -11.9 1.1 20 20

554 3.7 3.9 3.2 -6.2 0. -5.1 19 18

850 0.9 0.7 0.5 -29.8 -21.5 -10.2 20 13

enough signals the efficiency of the search is 1 for all the sig-nal morphologies: this part of the plot remains flat at a levelthat is set primarily by the observation time of this search.For weaker signals the efficiency reduces and the strength-rate plot curves up. Eventually, as the efficiency vanishes therate limit reaches asymptotically infinity. These plots forthevarious waveforms are not identical as the detailed turn offofthe efficiency is dependent on the waveform. The exclusionrate-strength plots obtained from the S2 analysis represent asignificant improvement respect to the S1 [1] result. As al-ready noted in section VII, the flat part of the plot determinedby the observation time is improved by a factor of 4 while thesensitivity-limited curved part of it reflects an imrpovement inthe efficiency of at least of factor of 18 (or better, dependingon the waveform morphology) [XXX check XXX].

B. Astrophysical interpretation

As mentioned in the introduction, potential sources of grav-itational wave bursts targeted in this search include core col-lapse supernovae, merging compact binaries (neutron starsand/or black holes) and gamma ray bursts. Unlike othergravitational wave searches pursued with the LIGO data, thissearch focuses on short duration bursts whose precise wave-form is not well predicted. In recent years there has beenmuch effort devoted to predicting gravitational wave burstwaveforms from astrophysical sources. Most of them rely ondetailed numerical and approximation methods. In particu-lar, we focus on simulations of the core collapse of rapidlyspinning massive stars [5–7], and of binary black hole merg-ers [8, 9].

The core collapse simulations employ detailed hydrody-namical models in two dimensions, enforcing axisymmetryof the rotating star throughout its evolution. The core collapseis initiated articifically (e.g., through a change in the adiabaticindex of the core material [5]). An accelerating quadrupolemoment is calculated in 2D from the distribution and flow ofmatter during the collapse, from which the gravitational wavesignal is derived. The rapid spinning of the progenitor startypically produces multiple bounces of the dense core, whichis reflected in the waveform of the emitted waves. Simplemodels of the differential rotation of material in the star also

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nts/

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0

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1.5

2

2.5

3 = 0.1 msτ

= 0.5 msτ

= 1.0 msτ

= 2.5 msτ

= 4.0 msτ

= 6.0 msτ

FIG. 12: Rate versushrss exclusion plots at the 90% confidencelevel derived from the LIGO burst search using the S2 data. Thetop plot corresponds to burst events modeled by sine-Gaussians ofQ = 9 and frequencies ranging from 100Hz to 850Hz as indicatedby the corresponding colours, while the bottom plot corresponds toones events modeled by Gaussians of theτ ’s shown.

lead to significant differences in the resulting waveforms.Rel-ativistic effects [6], if included, serve to effectively ”stiffen”the core, shifting the waves to higher frequencies and shorterdurations. The simulation is followed through the core col-lapse phase when most of the gravitational wave signal is pro-duced; it need not be continued through to the explosion ofthe outer layers (and indeed, these simulations may not pro-duce such explosions). The simulations attempt to sample thespace of important parameters (progenitor star angular mo-mentum, differential angular momentum versus radius, den-sity versus radius, adiabatic index of the core,etc.), resultingin collections of waveforms with widely varying morpholo-

18

gies; but of course the actual distributions of such parametersare poorly known. In ref. [7] the authors employ updated pro-genitor models and nuclear equation of state. For the stud-ies described here, we make use of 78 waveforms supplied inref. [5], 26 from ref. [6], and 72 from ref. [7]. We emphasizethat we are studying these waveforms only as a guide for oursearch; we do not endorse these waveforms as being particu-larly realistic or believable, or of providing a reliable samplingof the population of progenitor stars.

The binary black hole merger waveforms are calculated us-ing the Lazarus project approach [8, 9], applying numericalsimulation of the vacuum Einstein’s equations for the mostsignificantly nonlinear part of the interaction, together withclose-limit perturbation theory for the late-time dynamics.The authors in [8, 9] generate waveforms from simulationsof equal mass binary black holes with no intrinsic spin start-ing from near the innermost stable circular orbit followingabinary black hole inspiral. It should be kept in mind that thesewaveforms would naturally occur after an inspiral waveform,which is searched for using matched filtering techniques [2]and before the ringdown phase of the system [36]

In all of these models, the simulations and calculationspredict gravitational wave bursts with time durations rangingfrom a fraction of ams to 10’s or 100’s ofms and with asignificant fraction of their power in the LIGO most sensitivefrequency band (100-1100Hz). This observation motivates thechoice of parameters for the sine-Gaussian waveforms used tooptimize and evaluate the efficiency for our search pipeline,as discussed in section VIII. After tuning our pipeline algo-rithms using thead hocsine-Gaussian waveforms, we evalu-ate the efficiency for our search to detect the waveforms pre-dicted from the astrophysical simulations. The amplitudesofthese waveforms are predicted by the simulations, so that inaddition to evaluating the efficiency as a function ofhrss (atthe detectors), we can also present them as a function of dis-tance to the source. We have evaluated the efficiency versusactual distance, averaged over source directions and orienta-tions, assuming uniform distribution in source direction andorientation. This assumption becomes invalid for supernovaprogenitors in the Galactic disk, when the LIGO detectorsbecome sensitive to supernovae at distances greater than thedisk thickness (on the order of 150 pc). It is also invalid forextra-galactic binary black hole mergers, since the distributionof nearby galaxies is far from isotropic at the 1 Mpc scale.This evaluation of the efficiency as a function of distance isonly to “set the scale” for the current and future astrophysicalsearches.

In figure 13 we plot the detection efficiency of our pipelineversus distance for core collapse supernovae. We considerthe collections of waveforms from the three studies discussedabove [5–7] as separate populations. A total of 176 super-novae waveforms were considered of which XXX had theircentral frequency within the sensitive band of this search (100-1100Hz). The curves shown in figure 13 reflect the averagesover these in-band only waveforms. From this plot, it can beseen that we are sensitive to gravitational waves from core-collapse supernovae within a distance of∼ 100 pc. Suchclose-by supernovae are, of course, quite rare. In figure 14

we show the expectedhrss (at the detectors) and the centralfrequency for each of the 176 supernovae waveforms assum-ing they originate from a core collapse supernovae that is op-timally oriented and polarized and is located at 100pc fromthe detectors. Superimposed, we show the sensitivity of theLIGO instruments during S3. Extending this analysis to thedesign sensitivity of the LIGO-I and advanced LIGO detec-tors we show in figure 15 the same plot but for a supernovaevent located 10kpc from us. As we may see from this plotthe advanced LIGO detectors will provied coverage over ourentire galaxy for the vast majority of the in-band supernovaewaveforms of the models we have considered.

distance [pc]1 10 10

210

3

distance [pc]1 10 10

210

3ra

te [e

vent

s/da

y]0

0.5

1

1.5

2

2.5

3

ZM

DFM

OB

FIG. 13: Detection efficiency of this search versus distance(see text)to supernovae (in-band to LIGO’ search) models from three studies:black circles of top plot are for the XXX/78 waveforms from [5]; redcircles of top plot are for the XXX/26 waveforms from [6]; blackcircles of the bottom plots are for the XXX/72 waveforms from[7].In obtaining the distance sensitivities, an averaged is performed overthe waveforms of a given family that have their central frequencywithin the sensitive band of this search (100-1100Hz). [XXXtheplot is actually REAL distance averaged over all sky/polarizs XXX]

Frequency [Hz]10

210

3

Frequency [Hz]10

210

3

h(f)

[1/r

tHz]

10-24

10-23

10-22

10-21

10-20

10-19

10-18

10-17

10-16

LLO-4km

LHO-4km

LHO-2km

ZM 100 pc

DFM 100 pc

OB 100 pc

FIG. 14: Signal strength (inhrss at the detectors versus central fre-quency for the 176 supernovae waveforms from references [5–7].The supernova event was positioned in optimal orientation and po-larization at 100pc from the detectors. The S3 sensitivity curves arealso superimposed.

For the purpose of the binary black hole merger waveforms,we have considered systems of black hole mass in the range of10-100M� (per black hole). The [XXX] frequency of the re-

19

Frequency [Hz]10

210

3

Frequency [Hz]10

210

3

h(f)

[1/r

tHz]

10-24

10-23

10-22

10-21

10-20

10-19

10-18

10-17

10-16

SRD 4km

SRD 2km

Advanced LIGO

ZM 10 kpc

DFM 10 kpc

OB 10 kpc

FIG. 15: Same plot as in 14 but for the supernova event placed at10kpc from the detectors. The design sensitivity of the LIGO-I andadvanced LIGO detectors is also shown.

sulting waveform is inversely proportional [XXX] to the massof the system and thus five different masses of 10, 30, 50, 70and 90M� were chosen so that to span the nominal frequencycoverage of this search, i.e., the 100-1100Hz band. Moreover,the waveforms of these systems come with two polarizationsand they thus offered a realistic check of the robustness ofthe waveform consistency test, ther-statistic, against com-plex morphologies. In figure 16 we show the efficiency ofour search as a function of the signal strength for the 5 differ-ent mass binary black hole systems. The efficiency is calcu-lated over the whole sky and considering the two polarizationwaveforms. The best performing mass system corresponds to50M�: the [XXX] frequency of this system corresponds tothe best operating point of the LIGO instruments, i.e., closeto 250Hz. On the contrary, the two worst performing masssystems reflect frequencies at the two ends of the LIGO in-strument’s sensitivities relevant to this search, i.e., 100Hz and1100Hz.

[strain/rtHz]rssh10

-2110

-2010

-1910

-18

[strain/rtHz]rssh10

-2110

-2010

-1910

-18

Effi

cien

cy

0

0.2

0.4

0.6

0.8

1 10 Mo

30 Mo

50 Mo

70 Mo

90 Mo

10 Mo

30 Mo

50 Mo

70 Mo

90 Mo

10 Mo

30 Mo

50 Mo

70 Mo

90 Mo

10 Mo

30 Mo

50 Mo

70 Mo

90 Mo

10 Mo

30 Mo

50 Mo

70 Mo

90 Mo

FIG. 16: Efficiency versus strength for the 5 different solarmassbinary black hole systems we simulated according to the Lazarusproject [8, 9]. The best sensitivity is for the 50M� while the worstone for the 10M� case.

We should note that all four waveform families we haveconsidered for our simulations, eitherad hocor astrophysi-cally motivated, have frequency content that ranges over ourentire band of search. Within each of these families, signalstrengths in order to reach fix efficiency (e.g., 50%) range overapproximately an order of magnitude: this is primarily a man-ifestation of the different frequency content of each waveform

and the fact that the LIGO detectors’ frequency response isnot flat (and of about this ratio over the 100-1100Hz band aswe may see in figure 1). However, for all the waveform fami-lies, this signal strengthhrss range remains the same, namely10−20−10−19strain/

√Hz. This builds confidence in our ap-

proach to usehrss as a measure of the signal strength relevantto burst detection in LIGO as well as to use the performance ofthe search to sine-Gaussian injections to set the benchmarkfornarrow band, sort-lived transients independent of their precisewaveform structure. We are in the process of enriching ourwaveform library considered in the LIGO burst search [XXX]something that will give us further opportunities to test theabove findings.

X. SUMMARY AND DISCUSSION

We presented a search for gravitational wave bursts us-ing the data the three LIGO detectors collected during theirsecond science run in 2003. Short-lived transients withenough energy in LIGO’s sensitive band of 100-1100Hz weresearched for. A search for gravitational wave bursts with fre-quency content above the 1100Hz range was pursued in co-incidence with the TAMA [16] detector and a separate publi-cation on this is forthcoming [? ]. We saw no evidence forthe presence of astrophysical events in our data stream andusing the Feldman-Cousins approach we set an upper limiton their rate at 0.43 events per day at the 90% confidencelevel. This rate improves our previously published limit [1]by a factor of 4. The efficiency of this search was measuredusing various waveform morphologies. Besides the familiesof ad hocwaveforms we introduced in our previous search,we also measured the efficiency of our search to astrophysi-cally motivated waveforms resulting from numerical simula-tions of the core collapse supernovae and binary black holemergers [? ]. For these morphologies we found that thehrss

root-sum-square (rss)amplitude of the signal is a good mea-sure [XXX quantify XXX] of the signal property relevant fordetection independent of the detailed structure of the signal.The 50%hrss efficiencies lie for each of the waveform fam-ilies in the10−20 − 10−19strain/

√Hz range. The attained

performance of this search reflects an improvement respectto S1 by at least afactor of 18. This is generally frequencydependent and reflects first of all the instruments’ noise floorimprovement by afactor of 10. The rest is attributed to im-provements of the search algorithm and the use of the wave-form consistency test (r-statistic). The interpreted results ofthis search include the rate versus strength exclusion plots ona per waveform morphology basis. The improvements on therate and signal strength sensitivity are both reflected in signifi-cantly more stringent regions now allowed in these rate versusstrength plots.

In our S1 paper, we made a rough comparison with the re-sults of the most recent IGEC search for gravitational wavebursts [? ]. The IGEC search had somewhat better sensitivityfor bursts whose power was concentrated near the bars’ res-onant frequencies than did the LIGO S1 search. The IGECsearch also had the benefit of much longer observing time,

20

so was able to set rate limits far below what we were ableto do in S1. However, LIGO’s broad band response allowedus to set better limits on bursts whose power was mainly atfrequencies away from the bars’ resonances. In S2, our sen-sitivity is substantially better than in S1. For example, Wave-Burst had 50% efficiency to an ideally-oriented 846 Hz sine-Gaussian at a strength of1.0 × 10−20/

√Hz. By the rough

method of comparison that we used in our S1 paper, this cor-responds to a burst strength in bars’ natural units of approxi-mately3.2×10−22/Hz. [XXX check XXX] This is now sub-stantially to the left of the boundary of the excluded regioninthe rate-strength plot of Figure 13 in [? ]. However, althoughS2’s increased observing time allows a lower rate limit thancould be set in S1, it is still the case that the IGEC search al-lows substantially better rate limits to be set, for signalsstrongenough for its detectors to have seen.

Furthermore, we are interested in a comparison with the re-sults reported in the 2002 paper of the Rome group [37]. Thepresent LIGO search approached the sensitivity correspond-ing to bar events of strength 100 mK, suggested as an inter-esting level in that paper. To check the indications of possiblesignals, though, we would need to be able to see or set limits atthe level of about 0.5 per day [37, 38]. That is in fact roughlythe event rate bound we can set in the limit of strong signals.For signals nearer the threshold of our sensitivity, though, ourrate limits are poorer (see Figure 12.) Thus, our results inthis paper are not quite able to check the Rome results. Thesignificant improvements in sensitivity and longer observation

times that we expect in new LIGO searches in the near futurewill enable us to make a definitive comparison with the claimsin ref. [37, 38]

[XXX add a short paragraph here on for future of LIGOburst searches XXX]

Acknowledgments

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the construc-tion and operation of the LIGO Laboratory and the ParticlePhysics and Astronomy Research Council of the United King-dom, the Max-Planck-Society and the State of Niedersach-sen/Germany for support of the construction and operation ofthe GEO600 detector. The authors also gratefully acknowl-edge the support of the research by these agencies and by theAustralian Research Council, the Natural Sciences and Engi-neering Research Council of Canada, the Council of Scientificand Industrial Research of India, the Department of Scienceand Technology of India, the Spanish Ministerio de Cienciay Tecnologia, the John Simon Guggenheim Foundation, theDavid and Lucile Packard Foundation, the Research Corpora-tion, and the Alfred P. Sloan Foundation.

This document has been assigned LIGO Laboratory docu-ment number LIGO-P040040-00-R.

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