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Search for G ravitationalW aves A ssociated w ith 39 G am m a-R ay B ursts

U sing D ata from the Second,T hird,and Fourth LIG O R uns

LIG O -P 060024-07-Z

B.Abbott,15 R.Abbott,15 R.Adhikari,15 J.Agresti,15 P.Ajith,2 B.Allen,2,54 R.Am in,19 S.B.Anderson,15

W .G .Anderson,54 M .Arain,41 M .Araya,15 H.Arm andula,15 M .Ashley,4 S.Aston,40 P.Aufm uth,38 C.Aulbert,1

S.Babak,1 S.Ballm er,15 H.Bantilan,9 B.C.Barish,15 C.Barker,16 D.Barker,16 B.Barr,42 P.Barriga,53

M .A.Barton,42 K .Bayer,18 K .Belczynski,26 S.J.Beruko�,1 J.Betzwieser,18 P.T.Beyersdorf,29 B.Bhawal,15

I.A.Bilenko,23 G .Billingsley,15 R.Biswas,54 E.Black,15 K .Blackburn,15 L.Blackburn,18 D.Blair,53 B.Bland,16

J.Bogenstahl,42 L.Bogue,17 R.Bork,15 V.Boschi,15 S.Bose,56 P.R.Brady,54 V.B.Braginsky,23 J.E.Brau,45

M .Brinkm ann,2 A.Brooks,39 D.A.Brown,15,7 A.Bullington,32 A.Bunkowski,2 A.Buonanno,43 O .Burm eister,2

D.Busby,15 W .E.Butler,46 R.L.Byer,32 L.Cadonati,18 G .Cagnoli,42 J.B.Cam p,24 J.Cannizzo,24 K .Cannon,54

C.A.Cantley,42 J.Cao,18 L.Cardenas,15 K .Carter,17 M .M .Casey,42 G .Castaldi,48 C.Cepeda,15 E.Chalkley,42

P.Charlton,10 S.Chatterji,15 S.Chelkowski,2 Y.Chen,1 F.Chiadini,47 D.Chin,44 E.Chin,53 J.Chow,4

N.Christensen,9 J.Clark,42 P.Cochrane,2 T.Cokelaer,8 C.N.Colacino,40 R.Coldwell,41 M .Coles,17 R.Conte,47

D.Cook,16 T.Corbitt,18 D.Coward,53 D.Coyne,15 J.D.E.Creighton,54 T.D.Creighton,15 R.P.Croce,48

D.R.M .Crooks,42 A.M .Cruise,40 P.Csatorday,18 A.Cum m ing,42 J.Dalrym ple,33 E.D’Am brosio,15

K .Danzm ann,38,2 G .Davies,8 E.Daw,49 D.DeBra,32 J.Degallaix,53 M .Degree,32 T.Delker,41 T.Dem m a,48

V.Dergachev,44 S.Desai,34 R.DeSalvo,15 S.Dhurandhar,14 M .D��az,35 J.Dickson,4 A.DiCredico,33

G .Diederichs,38 A.Dietz,8 H.Ding,15 E.E.Doom es,31 R.W .P.Drever,5 J.-C.Dum as,53 R.J.Dupuis,15

J.G .Dwyer,11 P.Ehrens,15 E.Espinoza,15 T.Etzel,15 M .Evans,15 T.Evans,17 S.Fairhurst,8,15 Y.Fan,53 D.Fazi,15

M .M .Fejer,32 L.S.Finn,34 V.Fium ara,47 N.Fotopoulos,54 A.Franzen,38 K .Y.Franzen,41 A.Freise,40 R.Frey,45

T.Fricke,46 P.Fritschel,18 V.V.Frolov,17 M .Fy�e,17 V.G aldi,48 K .S.G anezer,6 J.G arofoli,16 I.G holam i,1

J.A.G iaim e,17,19 S.G iam panis,46 K .D.G iardina,17 K .G oda,18 E.G oetz,44 L.G oggin,15 G .G onz�alez,19

S.G ossler,4 A.G rant,42 S.G ras,53 C.G ray,16 M .G ray,4 J.G reenhalgh,28 A.M .G retarsson,12 R.G rosso,35

H.G rote,2 S.G runewald,1 M .G uenther,16 R.G ustafson,44 B.Hage,38 D.Ham m er,54 C.Hanna,19 J.Hanson,17

J.Harm s,2 G .Harry,18 E.Harstad,45 T.Hayler,28 J.Heefner,15 G .Heinzel,2 I.S.Heng,42 A.Heptonstall,42

M .Heurs,2 M .Hewitson,2 S.Hild,38 E.Hirose,33 D.Hoak,17 D.Hosken,39 J.Hough,42 E.Howell,53 D.Hoyland,40

S.H.Huttner,42 D.Ingram ,16 E.Innerhofer,18 M .Ito,45 Y.Itoh,54 A.Ivanov,15 D.Jackrel,32 O .Jennrich,42

B.Johnson,16 W .W .Johnson,19 W .R.Johnston,35 D.I.Jones,50 G .Jones,8 R.Jones,42 L.Ju,53 P.K alm us,11

V.K alogera,26 D.K asprzyk,40 E.K atsavounidis,18 K .K awabe,16 S.K awam ura,25 F.K awazoe,25 W .K ells,15

D.G .K eppel,15 F.Ya.K halili,23 C.J.K illow,42 C.K im ,26 P.K ing,15 J.S.K issel,19 S.K lim enko,41 K .K okeyam a,25

V.K ondrashov,15 R.K .K opparapu,19 D.K ozak,15 B.K rishnan,1 P.K wee,38 P.K .Lam ,4 M .Landry,16

B.Lantz,32 A.Lazzarini,15 B.Lee,53 M .Lei,15 J.Leiner,56 V.Leonhardt,25 I.Leonor,45 K .Libbrecht,15

A.Libson,9 P.Lindquist,15 N.A.Lockerbie,51 J.Logan,15 M .Longo,47 M .Lorm and,17 M .Lubinski,16 H.L�uck,38,2

B.M achenschalk,1 M .M acInnis,18 M .M ageswaran,15 K .M ailand,15 M .M alec,38 V.M andic,15 S.M arano,47

S.M �arka,11 J.M arkowitz,18 E.M aros,15 I.M artin,42 J.N.M arx,15 K .M ason,18 L.M atone,11 V.M atta,47

N.M avalvala,18 R.M cCarthy,16 D.E.M cClelland,4 S.C.M cG uire,31 M .M cHugh,21 K .M cK enzie,4

J.W .C.M cNabb,34 S.M cW illiam s,24 T.M eier,38 A.M elissinos,46 G .M endell,16 R.A.M ercer,41

S.M eshkov,15 E.M essaritaki,15 C.J.M essenger,42 D.M eyers,15 E.M ikhailov,18 S.M itra,14 V.P.M itrofanov,23

G .M itselm akher,41 R.M ittlem an,18 O .M iyakawa,15 S.M ohanty,35 G .M oreno,16 K .M ossavi,2 C.M owLowry,4

A.M oylan,4 D.M udge,39 G .M ueller,41 S.M ukherjee,35 H.M �uller-Ebhardt,2 J.M unch,39 P.M urray,42 E.M yers,16

J.M yers,16 S.Nagano,2 T.Nash,15 G .Newton,42 A.Nishizawa,25 F.Nocera,15 K .Num ata,24 P.Nutzm an,26

B.O ’Reilly,17 R.O ’Shaughnessy,26 D.J.O ttaway,18 H.O verm ier,17 B.J.O wen,34 Y.Pan,43 M .A.Papa,1,54

V.Param eshwaraiah,16 C.Param eswariah,17 P.Patel,15 M .Pedraza,15 S.Penn,13 V.Pierro,48 I.M .Pinto,48

M .Pitkin,42 H.Pletsch,2 M .V.Plissi,42 F.Postiglione,47 R.Prix,1 V.Q uetschke,41 F.Raab,16 D.Rabeling,4

H.Radkins,16 R.Rahkola,45 N.Rainer,2 M .Rakhm anov,34 M .Ram sunder,34 K .Rawlins,18 S.Ray-M ajum der,54

V.Re,40 T.Regim bau,8 H.Rehbein,2 S.Reid,42 D.H.Reitze,41 L.Ribichini,2 S.Richm an,18 R.Riesen,17

K .Riles,44 B.Rivera,16 N.A.Robertson,15,42 C.Robinson,8 E.L.Robinson,40 S.Roddy,17 A.Rodriguez,19

A.M .Rogan,56 J.Rollins,11 J.D.Rom ano,8 J.Rom ie,17 H.Rong,41 R.Route,32 S.Rowan,42 A.R�udiger,2

L.Ruet,18 P.Russell,15 K .Ryan,16 S.Sakata,25 M .Sam idi,15 L.Sancho de la Jordana,37 V.Sandberg,16

G .H.Sanders,15 V.Sannibale,15 S.Saraf,27 P.Sarin,18 B.S.Sathyaprakash,8 S.Sato,25 P.R.Saulson,33

R.Savage,16 P.Savov,7 A.Sazonov,41 S.Schediwy,53 R.Schilling,2 R.Schnabel,2 R.Scho�eld,45 B.F.Schutz,1

2

P.Schwinberg,16 S.M .Scott,4 A.C.Searle,4 B.Sears,15 F.Seifert,2 D.Sellers,17 A.S.Sengupta,8 P.Shawhan,43

D.H.Shoem aker,18 A.Sibley,17 J.A.Sidles,52 X.Siem ens,15,7 D.Sigg,16 S.Sinha,32 A.M .Sintes,37,1

B.J.J.Slagm olen,4 J.Slutsky,19 J.R.Sm ith,2 M .R.Sm ith,15 K .Som iya,2,1 K .A.Strain,42 N.E.Strand,34

D.M .Strom ,45 A.Stuver,34 T.Z.Sum m erscales,3 K .-X.Sun,32 M .Sung,19 P.J.Sutton,15 J.Sylvestre,15

H.Takahashi,1 A.Takam ori,15 D.B.Tanner,41 M .Tarallo,15 R.Taylor,15 R.Taylor,42 J.Thacker,17 K .A.Thorne,34

K .S.Thorne,7 A.Th�uring,38 M .Tinto,5 K .V.Tokm akov,42 C.Torres,35 C.Torrie,42 G .Traylor,17 M .Trias,37

W .Tyler,15 D.Ugolini,36 C.Ungarelli,40 K .Urbanek,32 H.Vahlbruch,38 M .Vallisneri,7 C.Van Den Broeck,8

M .van Putten,18 M .Varvella,15 S.Vass,15 A.Vecchio,40 J.Veitch,42 P.Veitch,39 A.Villar,15 C.Vorvick,16

S.P.Vyachanin,23 S.J.W aldm an,15 L.W allace,15 H.W ard,42 R.W ard,15 K .W atts,17 D.W ebber,15 A.W eidner,2

M .W einert,2 A.W einstein,15 R.W eiss,18 L.W en,1 S.W en,19 K .W ette,4 J.T.W helan,1 D.M .W hitbeck,34

S.E.W hitcom b,15 B.F.W hiting,41 S.W iley,6 C.W ilkinson,16 P.A.W illem s,15 L.W illiam s,41 B.W illke,38,2

I.W ilm ut,28 W .W inkler,2 C.C.W ipf,18 S.W ise,41 A.G .W isem an,54 G .W oan,42 D.W oods,54 R.W ooley,17

J.W orden,16 W .W u,41 I.Yakushin,17 H.Yam am oto,15 Z.Yan,53 S.Yoshida,30 N.Yunes,34 K .D.Zaleski,34

M .Zanolin,18 J.Zhang,44 L.Zhang,15 C.Zhao,53 N.Zotov,20 M .Zucker,18 H.zurM �uhlen,38 and J.Zweizig15

(The LIG O Scienti�c Collaboration,http://www.ligo.org)1Albert-Einstein-Institut, M ax-Planck-Institut f�ur G ravitationsphysik, D-14476 G olm ,G erm any

2Albert-Einstein-Institut, M ax-Planck-Institut f�ur G ravitationsphysik, D-30167 Hannover,G erm any

3Andrews University, Berrien Springs, M I 49104 USA

4Australian NationalUniversity, Canberra, 0200, Australia

5California Institute of Technology, Pasadena, CA 91125, USA

6California State University Dom inguez Hills, Carson, CA 90747, USA

7Caltech-CaRT, Pasadena, CA 91125, USA

8Cardi� University, Cardi�, CF24 3AA, United K ingdom

9Carleton College, North�eld, M N 55057, USA

10Charles Sturt University, W agga W agga, NSW 2678, Australia

11Colum bia University, New York, NY 10027, USA

12Em bry-Riddle AeronauticalUniversity, Prescott, AZ 86301 USA13Hobart and W illiam Sm ith Colleges, G eneva, NY 14456, USA

14Inter-University Centre for Astronom y and Astrophysics, Pune - 411007, India15LIG O - California Institute of Technology, Pasadena, CA 91125, USA

16LIG O Hanford O bservatory, Richland, W A 99352, USA

17LIG O Livingston O bservatory, Livingston, LA 70754, USA

18LIG O - M assachusetts Institute ofTechnology, Cam bridge, M A 02139, USA

19Louisiana State University, Baton Rouge, LA 70803, USA20Louisiana Tech University, Ruston, LA 71272, USA21Loyola University, New O rleans, LA 70118, USA

22M ax Planck Institut f�ur Q uantenoptik, D-85748, G arching, G erm any

23M oscow State University, M oscow, 119992, Russia

24NASA/G oddard Space Flight Center, G reenbelt, M D 20771, USA

25NationalAstronom icalO bservatory of Japan, Tokyo 181-8588, Japan

26Northwestern University, Evanston, IL 60208, USA27Rochester Institute of Technology, Rochester, NY 14623, USA

28Rutherford Appleton Laboratory, Chilton,Didcot, O xon OX11 0Q X United K ingdom29San Jose State University, San Jose, CA 95192, USA

30Southeastern Louisiana University, Ham m ond, LA 70402, USA

31Southern University and A& M College, Baton Rouge, LA 70813, USA

32Stanford University, Stanford, CA 94305, USA

33Syracuse University, Syracuse, NY 13244, USA

34The Pennsylvania State University, University Park, PA 16802, USA

35The University ofTexas atBrownsville and Texas Southm ostCollege,Brownsville,TX 78520,USA

36Trinity University, San Antonio, TX 78212, USA

37Universitat de les Illes Balears, E-07122 Palm a de M allorca, Spain

38Universit�at Hannover, D-30167 Hannover, G erm any

39University of Adelaide, Adelaide, SA 5005, Australia40University of Birm ingham , Birm ingham , B15 2TT, United K ingdom

41University of Florida, G ainesville, FL 32611, USA42University of G lasgow, G lasgow, G 12 8Q Q , United K ingdom

43University of M aryland, College Park, M D 20742 USA44University of M ichigan, Ann Arbor, M I 48109, USA45University of O regon, Eugene, O R 97403, USA

3

46University of Rochester, Rochester, NY 14627, USA

47University of Salerno, 84084 Fisciano (Salerno), Italy

48University of Sannio at Benevento, I-82100 Benevento, Italy49University of She�eld, She�eld, S3 7RH, United K ingdom

50University of Southam pton, Southam pton, SO 17 1BJ, United K ingdom

51University of Strathclyde, G lasgow, G 1 1XQ , United K ingdom

52University of W ashington, Seattle, W A, 98195

53University of W estern Australia, Crawley, W A 6009, Australia

54University of W isconsin-M ilwaukee, M ilwaukee, W I 53201, USA55Vassar College, Poughkeepsie, NY 12604

56W ashington State University, Pullm an, W A 99164, USA

(D ated:April7,2013)

W epresenttheresultsofa search forshort-duration gravitational-wave burstsassociated with 39

gam m a-ray bursts(G RBs)detected by gam m a-ray satellite experim entsduring LIG O ’sS2,S3,and

S4 science runs. The search involves calculating the crosscorrelation between two interferom eter

data stream s surrounding the G RB trigger tim e. W e search for associated gravitationalradiation

from single G RBs,and also apply statisticalteststo search fora gravitational-wave signature asso-

ciated with the whole sam ple.Forthe sam ple exam ined,we �nd no evidence forthe association of

gravitationalradiation with G RBs,eitheron a single-G RB basisoron a statisticalbasis.Sim ulating

gravitational-waveburstswith sine-gaussian waveform s,wesetupperlim itson theroot-sum -square

ofthegravitational-wave strain am plitudeofsuch waveform satthetim esoftheG RB triggers.W e

also dem onstratehow a sam pleofseveralG RBscan beused collectively to setconstraintson popu-

lation m odels.The sm allnum berofG RBsand thesigni�cantchange in sensitivity ofthedetectors

overthethreeruns,however,lim itstheusefulnessofa population study fortheS2,S3,and S4 runs.

Finally,we discussprospects for the search sensitivity for the ongoing S5 run,and beyond for the

nextgeneration ofdetectors.

I. IN T R O D U C T IO N

Ithasbeen overthreedecadessincegam m a-ray bursts

(G RBs)were�rstdetected bytheVelasatellites[1].Dur-

ing the1990s,when theBurstand TransientSource Ex-

perim ent(BATSE)[2]and BeppoSAX [3]were in oper-

ation,im portantdiscoveriesand observationsrelating to

G RBs were m ade,such as the isotropic distribution of

G RBs [4];the bim odaldistribution ofburst durations,

suggesting long and shortclassesofG RBs[5];detections

ofthe �rst x-ray [6],optical[7],and radio [8]counter-

parts;the�rstredshiftm easurem ents[9,10,11];and the

�rsthintsoftheassociation oflong-duration G RBswith

core-collapse supernovae [12,13,14]. Today,im portant

questionsaboutG RB progenitors,em ission m echanism s

and geom etry linger,and observationsm ade by the cur-

rentgeneration ofgam m a-ray satelliteexperim entssuch

asSwift[15],HETE-2 [16],INTEG RAL [17],and others

continue to provide new and exciting inform ation which

help usanswerthesequestionsand betterunderstand the

origin and physicsofthese astrophysicalobjects.

Currentlyfavored m odelsofG RB progenitorsarecore-

collapse supernovae for long-duration G RBs [18], and

neutron star-neutron star(NS-NS)orneutron star-black

hole(NS-BH)m ergersforshort-duration G RBs[19,20].

Thesem odelsand thedivision intotwoclassesofprogen-

itors are supported by observationsofsupernovae asso-

ciated with long-duration G RBs[12,13,14,21,22]and,

m ore recently,observationsofafterglowsand identi�ca-

tion ofhostgalaxiesforshort-duration G RBs[23,24,25,

26]. The end result in either scenario is the form ation

ofa stellar-m assblack hole [27]and,in either scenario,

theory predicts the em ission ofgravitationalradiation.

In theform ercase,gravitationalwaveswould resultfrom

the collapse ofa m assive star’score,while in the latter

case,gravitationalradiation would resultfrom theinspi-

ral,m erger,and ringdown phasesofthecoalescence.Re-

cently,there has been an observation-driven suggestion

ofa third classofG RBswhich could includeboth short-

and long-duration G RBs[28],butm oreobservationsare

needed to supportthissuggestion.

Dueto theexpected evolution oftheproposed progen-

itors,the redshiftdistribution oflong-duration G RBsis

thoughtto follow thestarform ation rateoftheUniverse

[29,30],and recentredshiftm easurem entstend to sup-

port this m odel,with the m easured G RB redshift dis-

tribution peaking at z >� 1 [31]. Long-duration G RBs

havealso been associated exclusively with late-typestar-

form inghostgalaxies[32].O n theotherhand,therecent

observationsofx-ray and opticalafterglowsfrom a few

short-duration bursts seem to suggestthat these G RBs

are located at lower redshifts relative to long-duration

G RBs[25,33],and thatshortburstsarefound in a m ix-

ture ofgalaxy types,including ellipticalgalaxies,which

have olderstellarpopulations. Allofthese observations

areconsistentwith thecurrently favored m odelsofG RB

progenitors. Although a large fraction ofG RBsare too

distantforany associated G W signalsto be detected by

LIG O ,itisplausiblethata sm allfraction occuratcloser

distances.Forexam ple,aredshiftofz = 0:0085,oradis-

tanceof35 M pc,hasbeen associated with long-duration

burst/supernova G RB 980425/SN 1998bw [12].Itisnot

unreasonable to expect that a few G RBs with no m ea-

sured redshiftscould havebeen located relatively nearby

4

aswell.Forshort-duration G RBs,therecentredshiftob-

servationshaveled to fairly optim isticestim ates[34,35]

foran associated G W observation in an extended LIG O

sciencerun.

In this paper,we present the results ofa search for

short-duration gravitational-wave bursts (G W Bs) asso-

ciated with 39 G RBsthatwere detected by gam m a-ray

satellite experim ents on dates when the S2,S3,and S4

science runs ofthe Laser Interferom eter Gravitational-

W ave Observatory (LIG O ) were in progress. Although

the theoreticalshapes ofthe G W burst signals result-

ing from the two progenitor scenarios are not known,

m any m odels predict that the G W signals would be

of short duration, ranging from � 1 m s to � 100 m s

[36,37,38,39,40]. The search m ethod presented here

targets such short-duration signals, and calculates the

crosscorrelation between two LIG O interferom eter data

stream s to look for these signals. A crosscorrelation-

based m ethod e�ciently suppresses uncorrelated tran-

sient noise in the data stream s,and at the sam e tim e

teststhatacandidateG W signalappearsin datafrom at

leasttwo interferom eters[41]. Previously,we presented

the results ofa search for a G W B associated with the

bright and nearby G RB 030329 [42]. Here,we present

analysism ethodswhich search forG W Bsassociated with

G RBsnotonlyon an individual-G RB basistotargetloud

G W Bs,butalso on a statisticalbasis.Thestatisticalap-

proach is sensitive to the cum ulative e�ectofany weak

G W signalsthatm ay be presentin the LIG O data.

Itisnoted herethatforthecom pactbinarycoalescence

m odels ofshort-duration G RBs,a subset ofthe associ-

ated inspiralwaveform s are wellm odelled, and that a

tem plate-based search forinspiralG W signalsassociated

with short-duration G RBs is currently being developed

using LIG O data.

II. LIG O S2,S3,A N D S4 SC IEN C E R U N S

TheLIG O interferom eters(IFO s)havebeen described

in detailelsewhere [43]. These detectors are kilom eter-

length M ichelson interferom eterswith orthogonalFabry-

Perot arm s,designed to detect im pinging gravitational

waves with frequencies ranging from � 40 Hz to sev-

eralkilohertz.Theinterferom eters’m axim um sensitivity

occursnear100 Hz to 200 Hz.There are two LIG O ob-

servatories:onelocated atHanford,W A (LHO )and the

other at Livingston,LA (LLO ).There are two IFO s at

LHO :one IFO with 4-km arm s(H1)and the otherwith

2-km arm s (H2). The LLO observatory has one 4-km

IFO (L1).Theobservatoriesareseparated by a distance

of3000 km ,corresponding to a tim e-of- ightseparation

of10 m s.

Each IFO consistsofm irrorsatthe endsofeach arm

which serve as test m asses. Data from each IFO is in

the form ofa tim e series,digitized at 16384 sam ples/s,

which records the di�erential length of the arm s and

which,when calibrated,m easuresthe strain induced by

frequency (Hz)210 310

)-1

/2(f

) (H

z1/

2hS

-2310

-2210

-2110

-2010

-1910

-1810

-1710

LLO 4km, S2 (2003.03.01)LHO 4km, S3 (2004.01.04)LHO 4km, S4 (2005.02.26)LIGO I SRD Goal, 4km

FIG .1: Progression ofLIG O sensitivities from S2 to S4 sci-

ence runs. For each run,the corresponding curve gives the

m agnitude ofthe noise spectraldensity,in strain-equivalent

units,forone ofthe IFO sduring a representative tim e inter-

valwithin the run. The solid curve gives the initialLIG O

design sensitivity goalas given in LIG O ’s Science Require-

m entsD ocum ent(SRD ).

a gravitational-wave.The responseofan IFO to a given

strainism easuredbyinjectingsinusoidalexcitationswith

known am plitudeinto thetestm asscontrolsystem sand

tracking the resulting signalsatthe m easurem entpoint

throughouteach run.Theresultisa m easurem entofthe

tim e-varying,frequency-dependent response function of

each IFO .

The LIG O S2 run was held from February to April

2003(59days),theS3run from O ctober2003toJanuary

2004 (70 days),and the S4 run from February to M arch

2005 (29 days). The sensitivity ofthe LIG O detectors

im proved signi�cantly between the S2 and S4 runs,and

approached theinitialLIG O design sensitivity duringthe

LIG O S4 run. The progression ofthe bestLIG O sensi-

tivity from theS2 to S4 runsisshown in Fig.1.Foreach

run,the corresponding curve in thisplotgivesthe m ag-

nitude ofthe noise spectraldensity,in strain-equivalent

units,for one ofthe IFO s during a representative tim e

intervalwithin the run.The solid curve givesthe initial

LIG O design sensitivity goalasgiven in LIG O ’sScience

Requirem entsDocum ent.Further,theduty factorofthe

threeIFO sincreased signi�cantly from theS2 to S4 run.

During theS2 run,theduty factorswere74% ,58% ,and

37% for the H1,H2, and L1 IFO s,respectively,while

during the S4 run,the duty factorswere 80.5% ,81.4% ,

and 74.5% ,respectively.

III. T H E G R B SA M P LE

Com pared to the 1990s,when BATSE was detecting

G RBs,theperiod from 2001to 2004,when LIG O had its

5

�rstthreescienceruns,wasa tim eofrelatively low G RB

detection rate. LIG O ’s S4 run coincided with a tim e

when Swift had just started operating and was m aking

its �rst G RB detections. There were 29 G RB triggers

duringtheS2run,11G RB triggersduringS3,and6G RB

triggersduringS4.TheseG RB triggerswereprovided by

the Third Inter-Planetary Network (IPN) [44], K onus-

W ind [45],HETE-2 ,INTEG RAL,and Swift,and were

distributed via the GRB Coordinates Network (G CN).1

O nly LIG O data which were ofscience m ode quality

were analyzed. These science m ode segm ents are data

collected when the interferom eterswere in a stable,res-

onantcon�guration. Additionally,data segm entswhich

were agged asbeing ofpoorquality were notincluded

in the analysis.Forexam ple,data segm entswhich were

known to havea high rateofseism ic transientswereex-

cluded from the analysis.Afterallthe data quality cuts

were m ade,there were 28 G RBs leftto be analyzed for

theS2 run,7 G RBsforS3,and 4 G RBsforS4,fora to-

talof39 G RB triggers.O fthese,22 G RBshad positions

well-localized to within afew arcm inutes,while17 G RBs

did not.These 17 G RBswere detected by eitherHETE

orIPN.In thecaseofHETE,no position m easurem ents

wereavailablewhile,in the case ofIPN,the G RBswere

notwell-localized.O fthe39G RBs,six had redshiftm ea-

surem ents,four ofwhich were atz > 1,and two fellin

the short-duration category ofbursts,i.e.had durations

� 2 seconds. Forthis analysis,due to the sm allsize of

thesam ple,wedid notattem pttodi�erentiatetheG RBs

accordingto theirobserved properties.Theuseofaclas-

si�cation schem e in a search can be done in the future

with a largerG RB sam ple.

Inform ation about m ost of the G RBs was collected

from the corresponding G CN circulars.The param eters

thatarerelevantforthisanalysisaretheG RB date and

triggertim e,and therightascension and declination.For

those HETE G RBswhich did nothave positions,infor-

m ation about the G RB triggertim e wasobtained from

the HETE website.2 A list ofthe G RBs analyzed and

relevantinform ation aregiven in TableI.

IV . D A TA A N A LY SIS

A . O n-source and o�-source data segm ents

Since G RBs have well-m easured detection tim es,the

search for short-duration G W signals can be lim ited to

tim e segm ents| called on-source segm entshere| sur-

rounding the G RB trigger tim es. Lim iting the search

to encom passonly these tim e segm ents signi�cantly re-

ducesthe num berofsearch trials,com pared to a search

which m akes use ofdata from the entire run. In case

1 http://gcn.gsfc.nasa.gov2 http://space.m it.edu/H ETE

ofa detection, such a reduction in trials translates to

a largersigni�cance for the detection com pared to that

which would resultfrom an untriggered search.

M aking use of on-source segm ents also m eans that

background estim ation can proceed by using data

stretches| called o�-sourcesegm entshere| which are

outside the on-sourcesegm ents,butwhich arestillclose

enough in tim eto theon-sourcesegm entsso thattheo�-

sourcedataaresim ilarin characterto,and representative

of,the on-sourcedata.

In thisanalysis,the length ofeach on-source segm ent

waschosen to be 180 seconds,with the �rst120 seconds

ofthe LIG O on-source data occurring before the G RB

triggertim e,and the last60 secondsoccurring afterthe

triggertim e. Thiswindow length islongerthan the ex-

pected tim e delay between a gravitational-wave signal

and the onsetofa G RB signal,which isofthe orderof

severalseconds [46,47,48],but which in certain m od-

elscan be aslargeas100 seconds[49].The largesearch

window alsotakesintoaccounttheuncertaintyin thedef-

inition ofthe m easured G RB trigger tim e,i.e. it takes

into accountthepossibility thatthetriggertim eused in

theanalysisoccurred beforeoraftertheactualstartofa

gam m a-ray burst signal. M any gam m a-ray lightcurves

show sub-threshold,precursorburstswhich occurbefore

the m easured G RB triggertim e,hence ourchoice ofan

asym m etricsearch window around the triggertim e.

For each G RB,a search for a G W signalwas carried

outusingdatafrom each pairofIFO sthatwasoperating

properlyatthattim e.Additionally,LHO -LLO on-source

pairs were analyzed only when G RBs had well-de�ned

positions,since position inform ation isnecessary to cal-

culate the LHO -LLO tim e-of- ight delay. After allthe

data quality cutswerem ade,therewere59 IFO -IFO on-

source pairsthat were analyzed. This num ber is larger

than thenum berofG RB triggersbecause,foreach G RB

trigger,itwaspossibletohaveup tothreeIFO pairspass

the data quality cuts. There were 35 H1-H2 on-source

pairsanalyzed,12 forH1-L1,and 12 forH2-L1.

The software used in this analysis is available in the

LIG O Scienti�c Collaboration’s CVS archives with the

tag m ultigrb r1 in M ATAPPS.3

B . D ata conditioning

Before the crosscorrelation between two LIG O data

stream swascalculated,thetim eseriesdatafrom each in-

terferom eterwasconditioned. Thisconsisted ofwhiten-

ing, phase-correction,and bandpassing from 40 Hz to

2000 Hz.The sam pling ratewasretained at16384 sam -

ples/s. W hitening wasdone to m ake sure the resulting

3 http://www.lsc-group.phys.uwm .edu/cgi-bin/cvs/viewcvs.cgi

/m atapps/src/searches/burst/m ultigrb

/?cvsroot= lscsoft& sortby= rev# dirlist

6

TABLE I:The G RB sam ple analyzed

LIG O G RB a UTC b G PSc durationd R.A.e D ecf Faveg

Faveg tim e delayh IFO i

run date tim e tim e (seconds) (degrees) (degrees) LHO LLO (seconds)

S2 030215 17 :11 :52 729364325:00 40 ::: ::: ::: ::: ::: H1,H2

030215a 11 :13 :32 729342825:00 30 ::: ::: ::: ::: ::: H1,H2

030215b 11 :16 :28 729343001:00 40 ::: ::: ::: ::: ::: H1,H2

030216 16 :13 :44 729447237:00 3 ::: ::: ::: ::: ::: H1,H2

030217 02 :45 :42 729485155:00 50 186:596 � 11:850 0:379 0:204 0:0078867 H2,L1

030218 11 :42 :38 729603771:00 200 ::: ::: ::: ::: ::: H1,H2

030221 07 :46 :14 729848787:00 ... ::: ::: ::: ::: ::: H1,H2

030223 09 :45 :06 730028719:00 10 ::: ::: ::: ::: ::: H1,H2

030226j

03 :46 :31:99 730266404:99 22 173:254 25:900 0:356 0:524 0:0059892 H1,H2,L1

030228 20 :26 :46 730499219:00 15 ::: ::: ::: ::: ::: H1,H2

030301 20 :27 :20 730585653:00 30 ::: ::: ::: ::: ::: H1,H2

030308 14 :06 :09 731167582:00 ... ::: ::: ::: ::: ::: H1,H2

030320a 10 :11 :40 732190313:00 80 267:929 � 25:317 0:317 0:418 0:0093172 H1,H2,L1

030320b 18 :49 :17 732221370:00 150 ::: ::: ::: ::: ::: H1,H2

030323a 08 :42 :24 732444157:00 5 297:250 � 12:500 0:269 0:131 0:0088762 H1,H2,L1

030323bk

21 :56 :57:60 732491830:60 25 166:525 � 21:900 0:533 0:336 0:0064593 H1,H2,L1

030324 03 :12 :42:80 732510775:80 45 204:296 � 0:317 0:148 0:288 0:0086716 H1,H2

030325 14 :15 :10 732636923:00 2 70:808 � 19:133 0:592 0:480 0:0039660 H1,H2,L1

030326 10 :43 :41 732710634:00 10 292:967 � 11:717 0:191 0:407 0:0094257 H1,H2,L1

030329 03 :31 :43 732943916:00 ... ::: ::: ::: ::: ::: H1,H2

030329al

11 :37 :14:67 732973047:67 22.8 161:208 21:517 0:265 0:051 � 0:0095090 H1,H2

030329b 15 :34 :15:35 732987268:35 65 160:626 � 48:572 0:635 0:665 � 0:0009927 H1,H2

030331 05 :38 :40:82 733124333:82 10 349:261 36:260 0:252 0:312 � 0:0057539 H1,L1

030405 02 :17 :28 733544261:00 5 248:275 � 24:150 0:565 0:377 0:0059975 H1,H2,L1

030406 22 :42 :07 733704140:00 65 285:429 � 68:083 0:598 0:551 0:0014338 H1,L1

030410 11 :23 :42 734009035:00 0.3 ::: ::: ::: ::: ::: H1,H2

030413 07 :34 :37 734254490:00 15 198:604 62:350 0:680 0:586 � 0:0031858 H2,L1

030414 13 :48 :27 734363320:00 40 119:887 � 48:583 0:702 0:653 0:0015308 H1,H2

S3 031108 14 :11 :01 752335874:00 22 66:729 � 5:930 0:278 0:313 � 0:0075264 H1,H2

031109a 11 :11 :48 752411521:00 59 327:765 20:203 0:336 0:464 � 0:0088324 H1,H2

031123 22 :41 :14 753662487:00 ... ::: ::: ::: ::: ::: H1,H2

031127a 18 :58 :58 753994751:00 10 ::: ::: ::: ::: ::: H1,H2

031127b 18 :59 :16 753994769:00 70 ::: ::: ::: ::: ::: H1,H2

031130 02 :04 :48 754193101:00 4 ::: ::: ::: ::: ::: H1,H2

031220 03 :29 :56:74 755926209:74 16.9 69:893 7:374 0:414 0:617 0:0068643 H1,H2

S4 050223m

03 :09 :06 793163359:00 23 271:390 � 62:481 0:676 0:596 0:0027031 H1,H2

050306 03 :33 :12 794115205:00 160 282:337 � 9:162 0:565 0:610 � 0:0013425 H1,H2,L1

050318n

15 :44 :37 795195890:00 32 49:651 � 46:392 0:528 0:293 0:0083075 H1,H2,L1

050319o 09 :31 :18:44 795259891:44 10 154:202 43:546 0:597 0:370 � 0:0070546 H1,H2,L1

aFor G R Bs with the sam e date,letters are appended to the date

to distinguish the G R Bs.bU TC tim e ofG R B trigger.cG PS tim e ofG R B trigger(seconds since 0h 6 Jan 1980 U TC.)dD uration ofgam m a-ray burst.eR ightA scension ofG R B.fD eclination ofG R B.gPolarization-averaged antenna factor forspeci�ed IFO site

(cf.Eq.9).hTim e-of- ightofG W signalbetween LH O and LLO .A positive

valuem eansthatthesignalarrived �rstatLLO ;a negative value

m eans thatthe signalarrived �rstatLH O .iInterferom eters which were analyzed.jz = 1:986.

kz = 3:372.lz = 0:168.

mz = 0:5915.

nz = 1:44.

oz = 3:24.

7

spectrum of the data was at instead of being dom i-

nated by low-frequency or high-frequency com ponents.

The procedure consisted ofusing one-second data units

to whiten the adjacentone-second data and,asa conse-

quence,rem oved any non-stationarity in thedata having

a tim e scalelargerthan one second.The whitening pro-

cedurealso rem oved known lines.

TheresponsefunctionsofthethreeLIG O interferom e-

terstoagiven G W strain signalarenotexactlythesam e.

A G W signalim pinging on thethreeinterferom eterswill

thusappearashavingslightly di�erentphasesin thecor-

responding tim eseriesdata (even aftercorrecting forthe

LHO -LLO tim e-of- ightdelay). Phase correction ofthe

tim e series data was therefore done to rem ove the dif-

ferencesthatcan be attributed to the di�erentresponse

functions ofthe interferom eters. The phase correction

process m ade use ofthe m easured,tim e-dependent,re-

sponsefunctionsofthe interferom eters.

C . M easuring the crosscorrelation statistic

The search m ethod consisted of a sim ple \binned"

search in which the 180-second conditioned on-source

tim e-series for each IFO was divided into tim e inter-

vals(orbins)and the crosscorrelation foreach IFO -IFO

tim ebin paircalculated.Crosscorrelation binsoflengths

25 m s and 100 m s were used to target short-duration

G W signals with durations of � 1 m s to � 100 m s.

These crosscorrelation lengthswere found,through sim -

ulations, to provide su�cient coverage ofthe targeted

short-duration G W signals. Using bins m uch shorter

than 25 m swould considerably increasethe trialsin the

search,and therefore decrease the signi�cance ofa can-

didate G W event, while using bins m uch longer than

100 m swould considerably dim inish thecrosscorrelation

strength ofsignals in the two data stream s due to the

increased duration ofnoise. The crosscorrelation,cc,is

de�ned as:

cc=

mX

i= 1

[s1(i)� �1][s2(i)� �2]

vuut

mX

j= 1

[s1(j)� �1]2

vuut

mX

k= 1

[s2(k)� �2]2

(1)

wheres1 and s2 arethetwo tim eseriesto becorrelated,

�1 and �2 are the corresponding m eans,and m is the

num berofsam plesin thecrosscorrelation,i.e.thecross-

correlation integration length m ultiplied by thesam pling

rate of16384 sam ples/s.The possible valuesofthe nor-

m alized crosscorrelation rangefrom � 1 to + 1.

The bins were overlapped by half a bin width to

avoid ine�ciency in detecting signals occurring near a

bin boundary.Thecrosscorrelation value wascalculated

for each IFO -IFO bin pair and, for each crosscorrela-

tion bin length used,the largest crosscorrelation value

| in the case ofan H1-H2 search | obtained within

the 180-second search window was considered the m ost

signi�cant m easurem ent for that search,for that cross-

correlation bin length,forthatIFO pair. In the case of

an H1-L1 or H2-L1 search,it was the largest absolute

valueofthecrosscorrelationsthatwastaken asthem ost

signi�cantm easurem ent.Thiswasdoneto takeinto ac-

countthepossibility thatsignalsatLHO and LLO could

beanti-correlated depending on the gravitationalwave’s

(unknown) polarization. In the sections that follow,a

referenceto the \largestcrosscorrelation",in the caseof

an LHO -LLO analysis,willalwaysm ean the largestab-

solutevalue ofcrosscorrelations.

Forthose G RBswhich had well-de�ned positions,the

position ofthe G RB in the sky atthe tim e ofthe burst

wasused tocalculatetheG W signal’stim e-of- ightdelay

between the LHO and LLO observatories. Each LHO -

LLO pairof180-second on-sourcesegm entswere shifted

in tim erelativeto each otherby thecorresponding tim e-

of- ightam ountbefore the crosscorrelationswere calcu-

lated. For those G RBs which were not well-localized,

only H1-H2 on-source pairs were analyzed. For these

G RBs,the m axim um uncertainty in theLHO -LLO tim e

delay is� 10 m s,which isofthe sam escaleasthe signal

durationstargeted by theanalysis,and such atim eo�set

between signalsatthe two interferom eterswould havea

considerablee�ecton the m easured crosscorrelation.

D . Post-trials distributions

To estim ate the signi�cance ofthe loudest event,i.e.

the largest crosscorrelation, that was found in an on-

source segm ent corresponding to a G RB and an IFO

pair,we used o�-source data within a few hours ofthe

on-source data to m easure the crosscorrelation distribu-

tion ofthenoise.Thisdistribution wasobtained foreach

G RB,foreach IFO pair,foreach crosscorrelation length

by applying the search (described in Sections IV B to

IV C)on theo�-sourcesegm ents.Thetotallength ofthe

o�-sourceregion wasaboutthree hourssurrounding the

on-sourcesegm ent.Each distribution wasconstructed by

collectingthelargestcrosscorrelation(orlargestabsolute

value ofcrosscorrelations,in the case ofH1-L1 and H2-

L1) from each 180-second segm ent ofthe o�-source re-

gion.Thispost-trialsdistribution takesinto accountthe

num berofe�ective trialsthatwasused in searching the

on-sourcesegm ent.

To obtain enough statisticsforeach distribution,tim e

shifts were perform ed such that the tim e series ofeach

IFO was shifted by m ultiples of180 seconds relative to

theotherIFO and two180-second stretchesfrom thetwo

IFO swerepaired ateach shift,m akingsurethattwo180-

second tim estretcheswerepaired only onceforeach dis-

tribution.The tim e shiftproceduree�ectively increased

the length ofthe o�-source data to about 50 hours or

m ore,typically.

As an exam ple,the post-trials distribution for G RB

050318,fortheH1-H2 IFO pairand forthe25-m scross-

8

Entries 3423Mean 0.3719RMS 0.02542

largest crosscorrelation0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

num

ber

of e

vent

s

1

10

210

310 Entries 3423Mean 0.3719RMS 0.02542(a)

largest crosscorrelation0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

frac

tion

of

even

ts

-310

-210

-110

1

with time shiftsno time shifts

(b)

FIG .2: (a) Exam ple ofa crosscorrelation post-trials distri-

bution forthe 25-m scrosscorrelation window,forthe H1-H2

IFO pair.D ata from o�-sourcesegm entsforG RB 050318 was

used. (b) Cum ulative distribution of(a),norm alized to the

totalnum ber of entries in the distribution. Both distribu-

tions with and without tim e shifts are shown,including the

statisticalerrors. The arrow pointsto the largestcrosscorre-

lation found in the on-source segm ent for G RB 050318. In

this exam ple,the largest crosscorrelation of0.36 has a local

probability of0.57.

correlation length,is shown in Fig.2. For com parison,

thecum ulativeplotshowsboth thedistribution obtained

with tim e shifts,and the distribution obtained without

em ploying tim e shifts.

Each resulting post-trialsdistribution wasused to es-

tim atethecum ulativeprobability thatthelargestcross-

correlationfound in thecorrespondingon-sourcesegm ent

could be due to noise. This was done by determ ining

whatfraction ofthedistribution wereatleastaslargeas

the loudest crosscorrelation found in the on-source seg-

m ent.Forexam ple,thesigni�canceoftheloudest25-m s

crosscorrelationfound in theH1-H2on-sourcesegm entof

G RB 050318,indicated by an arrow in Fig.2(b),can be

estim ated by using the plotted post-trials distribution.

This probability willbe referred to interchangeably in

this paperas the post-trials,orlocal,probability ofthe

on-source crosscorrelation statistic. This is also known

in the literatureasthe false alarm probability.

Since H1 and H2 are colocated,environm entaldistur-

bancescan give rise to correlated transientnoise in the

two interferom eters. The e�ectofthese correlated envi-

ronm entalnoiseon an H1-H2crosscorrelation were,how-

ever,suppressed by:thejudicioususeofdataqualitycuts

(cf. Section III),the applied data conditioning (cf. Sec-

tion IVB),and theuseofo�-sourcedataim m ediatelysur-

rounding theon-sourcedata to estim atethebackground

noise (cf. this section),which m ade it m ore likely that

the background would properly re ect the rate ofany

correlated noisein the on-sourcedata.

The cum ulative distribution of localprobabilites re-

sulting from the search of59 on-source segm entpairsis

shown in Fig.3forthe25-m scrosscorrelationlength,and

in Fig.4 for the 100-m s crosscorrelation length. Also

shown (bold dashed lines) is the expected distribution

undera nullhypothesis.Therewereno loud eventsthat

were notconsistentwith the expected distribution,and

wethereforeconcludethattherewasno loud G W signal

associated with any singleG RB in the sam ple.

V . STA T IST IC A L T EST S

As m entioned earlier, G W signals from individual

G RBsarelikely to beweak in m ostcasesdueto thecos-

m ologicaldistancesinvolved. Therefore,besidessearch-

ing forG W signalsfrom each G RB,wealso considerthe

detection ofa G W signature associated with a sam ple

ofseveralG RBs.Such approaches,�rstproposed in the

contextofG W sin [50],havealreadybeen used [51,52]to

analyzeresonantm assdetectordata using triggersfrom

the BATSE and BeppoSAX m issions.

W e use two di�erent statisticalm ethods to look for

a G W signature associated with a sam ple of m ultiple

G RBs. As one m ay expect,the statisticalperform ance

ofa m ethod willdepend on thenatureoftheunderlying

source population distribution. The two di�erentm eth-

odspresented herehavecom plem entarypropertiesin this

respect.The�rststatisticaltestpresented,thebinom ial

test,is m oste�ective when severaleventscontribute to

thetail,i.e.thesigni�cantend,oftheprobability distri-

bution ofa sam ple. M oreover,it is also e�ective when

thereisa singlesigni�canteventin thesam ple.Thesec-

ond test,therank-sum test,ism oree�ectiveatdetecting

thecum ulativee�ectofweakersignals,butitisnotvery

e�ectiveatdetecting a few largeeventswhich fallon the

tailofa probability distribution.

Sincethesignalstrengthstargeted by thesetwo m eth-

odsareslightly di�erent,theresulting signi�cancesfrom

thetwo m ethodscan bedi�erentwhen therearerealsig-

nalspresentin the sam ple.Ifa detection isclaim ed and

them oresigni�cantm easurem entfrom thetwostatistical

testsischosen,then the properstatisticaltreatm ent,in

orderto arriveata �nalsigni�cance,would beto im pose

a penalty factorforusing two statisticaltests to search

forthe cum ulativesignal.

9

−2 −1.5 −1 −0.5 0

100

101

102

log10(plocal

)

cum

ulat

ive

#eve

nts

dataexpectedneeded for ~1% CL

FIG .3: Cum ulative localprobability distribution resulting

from thesearch of59 IFO -IFO on-source pairsusing a 25-m s

crosscorrelation length. The m ost signi�cant excess is indi-

cated by thearrow.Theexpected distribution underthenull

hypothesisis indicated by the bold,dashed line. The excess

needed for a � 1% con�dence in the nullhypothesis is indi-

cated by thesolid line.Them axim um excessindicated by this

line is 15 eventsbecause only the 15 m ost signi�cant events

in the actualdistribution are tested.

A . Testing a probability distribution: T he binom ial

test

Underanullhypothesis,thedistribution oflocalprob-

abilitiesisexpected tobeuniform ly distributed from 0to

1. The m easured distribution oflocalprobabilities was

tested tosearchforan excesswhich m ayhavebeen dueto

thecum ulativee�ectofweak G W signals.In particular,

we searched the tailofthe distribution,or the sm allest

probabilities found in the on-source searches,by using

thebinom ialtest.To testthetailofa probability distri-

bution,one �rstm akesa choice asto how m any events,

n,in the tailwould be tested out ofthe totalnum ber

ofevents,N ,in the sam ple.In thisanalysis,there were

59 IFO -IFO on-source pairs,and the upper 25% ofthe

resulting probability sam ple,or the 15 m ost signi�cant

events,was tested. The probabilities ofthese n events

are then sorted according to increasing value,i.e. de-

creasing signi�cance: p1;p2;p3;:::;pi;:::;pn. Foreach of

these probabilities,pi,one calculatesthe cum ulative bi-

nom ialprobability,which isthe probability forgetting i

−2 −1.5 −1 −0.5 0

100

101

102

log10(plocal

)

cum

ulat

ive

#eve

nts

dataexpectedneeded for ~1% CL

FIG .4:Sim ilarto Fig.3,butusing a 100-m scrosscorrelation

length.

orm oreeventsatleastassigni�cantaspi:

P� i(pi)= Pi(pi)+ Pi+ 1(pi)+ Pi+ 2(pi)+ :::+ PN (pi)

(2)

= 1� [P0(pi)+ P1(pi)+ P2(pi)+ :::+ Pi� 1(pi)]

(3)

and wherePi(p)isthe binom ialprobability forgetting i

successesin N trials:

Pi(p)=N !

i!(N � i)!pi(1� p)N � i (4)

Here,N isthenum berofon-sourcesearches,which is59,

and \success" m eans getting ievents at least as signif-

icant as p. Note that ifthere is one loud event in the

sam ple,with p � 1,then it follows from Eqs.3 and 4

thatthe cum ulativebinom ialprobability is,

P� 1(p)= 1� (1� p)N (5)

� N p (6)

Thus,the binom ialtest is able to autom atically handle

the caseofa singleloud eventin the distribution.

Afterthecum ulativebinom ialprobability,P� i(pi),has

been calculated for each post-trials probability,pi,the

sm allestbinom ialprobabilityin thesetisidenti�ed.This

sm allestbinom ialprobability willpointto the m ostsig-

ni�cantexcessthatwasfound in searchingthetailofthe

probability distribution.

The m ostsigni�cantexcessthatwasfound by the bi-

nom ialtest in the tailof the distribution is indicated

10

by an arrow in Figs. 3 and 4. For the 25-m s dis-

tribution, the sm allest binom ialprobability found was

P� 9(p9 = 0:104)= 0:153.Thism eansthatthe binom ial

testfound thatthe m ostsigni�cantexcessin the tailof

thedistribution consisted ofnineeventswith localprob-

abilitiesp � 0:104,and thatthebinom ialprobability for

havingnineorm oreeventsatleastassigni�cantas0:104,

given 59 trials,is0:153.

In the case of the 100-m s distribution, the sm allest

binom ialprobability found wasP� 9(p9 = 0:112)= 0:207.

This m eans thatthe binom ialtestfound thatthe m ost

signi�cantexcessin thetailofthedistribution consisted

ofnineeventswith localprobabilitiesp � 0:112,and that

the binom ialprobability forhaving nine orm ore events

atleastassigni�cantas0:112,given 59 trials,is0:207.

Searching the tailofa post-trialsprobability distribu-

tion forthem ostsigni�cantexcessintroducesadditional

trialsto the search. W e thusneed to testthe m ostsig-

ni�cant excess found in the tailofeach localprobabil-

ity distribution against the nullhypothesis to properly

establish itslevelofsigni�cance.The expected distribu-

tion ofthe binom ialprobability statistic under the null

hypothesiswasobtained through sim ulations.Thesim u-

lationsconsisted ofrandom ly generating59num bersuni-

form ly distributed from 0 to 1 to sim ulate 59 post-trials

probabilities under the nullhypothesis. Then the sam e

binom ialtest that was applied to the actualpost-trials

probability distribution wasapplied to this distribution

ofrandom events to search for the m ost signi�cant ex-

cess in the 15 m ost signi�cant events in the tail. This

wasrepeated a m illion tim es,and thebinom ialprobabil-

ity ofthe m ostsigni�cantexcessfound in each trialwas

collected. The resulting distribution ofbinom ialprob-

abilities under the nullhypothesis,in e�ect,takes into

accountthenum beroftrialsused in searching thetailof

the post-trialsdistribution.

Resultsofthese sim ulationsshow that,underthe null

hypothesis,theprobability forgetting a m easurem entat

leastassigni�cantas0.153 thatwasfound in the 25-m s

search is0.48.In otherwords,underthenullhypothesis,

1 in 2.1 setsof59 on-sourcesearcheswillresultin a m ost

signi�cantexcesswith a binom ialprobability atleastas

signi�cantas0.153. Thisquanti�esthe conclusion that

the resultofthe 25-m ssearch isconsistentwith the null

hypothesis.

Sim ilarly,we �nd that,underthe nullhypothesis,the

probability forgetting a m easurem entatleastassigni�-

cantas0.207thatwasfound in the100-m ssearch is0.58.

In other words,under the nullhypothesis,1 in 1.7 sets

of59 on-source searcheswillresultin a m ostsigni�cant

excesswith a binom ialprobability atleastassigni�cant

as0.207.And,aswith the25-m sresult,thislevelofsig-

ni�cance forthe 100-m ssearch resultis consistentwith

the nullhypothesis.

Also shown in Figs. 3 and 4 isa curve indicating the

excessneeded fora � 1% con�dence in the nullhypoth-

esis.Ateach localprobability,thecurvegivesthecum u-

lative num ber ofevents needed to obtain a � 1% �nal

probability underthenullhypothesis,given 59 on-source

pairs.

B . M axim um likelihood ratio based tests

A m axim um likelihood ratio test [53]for detecting a

G W signatureassociated with a sam pleofm ultiple trig-

gers was derived in [54]. (It was shown there that [50]

is a specialcase of the m axim um likelihood ratio ap-

proach.) The m ethod proposed in [54]cannot be ap-

plied directly to theentireG RB sam pledescribed above

sincethelargestcrosscorrelation valueswereobtained in

di�erent ways for H1-H2 and H1-L1 (H2-L1) (cf. Sec-

tion IV C).In the following,we willonly use the largest

crosscorrelationsfrom H1-H2 on-source segm ents. This

reduces the total num ber of G RB on-source segm ents

used in thistestto 35.

Letthe largestcrosscorrelation from the ith G RB on-

source segm entbe denoted as ccm ax;i. Ifwe do notuse

any prior probability distribution for the properties of

G W signalsassociated with G RBs,the m axim um likeli-

hood ratiodetection statisticissim ply theaverageofthe

largestcrosscorrelation valuesfrom the G RB set,

� =1

N G R B

X

i

ccm ax;i ; (7)

whereN G R B isthenum berofH1-H2G RB on-sourceseg-

m entsused.W e call� the sum -m ax statistic.

Tobuild in robustnessagainstinstrum entalnoisearte-

facts,such as short duration transients,we replace the

sum -m ax statistic,which wasderived for the idealcase

ofG aussian and stationary noise,by a non-param etric

counterpart.The on-sourceand o�-source largestcross-

correlation valuesare pooled into two separate setsand

the W ilcoxon rank-sum test[55]isused forthe nullhy-

pothesis that the two sets ofsam ples were drawn from

the sam eunderlying truedistribution.

The cum ulative distribution ofthe on-source and o�-

source largest crosscorrelations from the 100-m s search

are shown in Fig.5. Application ofthe rank-sum test

showsthatthesigni�canceofthenullhypothesisis0:64.

This im plies that one out of1:6 trials can show a false

positive detection atthissigni�cancethreshold.Assum -

ing that G RB triggers occur at a rate ofone per day,

one yearofobservation would contain approxim ately 10

collections of35 G RBs. In order to achieve a low false

detection probability,wewould requirea m uch lowersig-

ni�cance,such as� 0:01,in orderto rejectthe nullhy-

pothesis.

Asa furthercheck,wealso com putetheem piricalsig-

ni�cance ofthe on-sourcevalue of� with respectto the

set of o�-source � values. Values of the o�-source �

statisticwerecalculatedbypoolingthelargestcrosscorre-

lationsfrom theH1-H2o�-sourcesegm ents,then dividing

thispoolinto subsets,each ofwhich had N G R B num ber

ofelem ents. For each ofthese subsets,the � statistic

11

0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.260

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

largest crosscorrelation

prob

abili

ty

FIG .5: The cum ulative distributionsofthe on-source (solid

black with + m arker)and o�-source (solid gray)largest H1-

H2 crosscorrelations from the 100-m s search. The vertical

lines denote the locations of the m edians of the o�-source

(gray)and on-source (black,dashed)sam ples.

0.17 0.175 0.18 0.185 0.19 0.195

0.0010.003

0.01 0.02 0.05 0.10

0.25

0.50

0.75

0.90 0.95 0.98 0.99

0.9970.999

χ

Pro

babi

lity

Normal Probability Plot

FIG .6: Plot ofthe cum ulative distribution function ofthe

o�-source valuesofthesum -m ax statistic �.The dashed line

shows the best �t norm aldistribution. The solid horizontal

and verticallinesindicatethelocation oftheon-sourcevalues

of �on and the corresponding cum ulative probability. The

on-source value was�on = 0:1753,which yieldsa cum ulative

probability of0.64 or an em piricalsigni�cance of1 -0.64 =

0.36.

wascalculated using Eq.7.Theem piricalsigni�canceis

de�ned asthefraction ofo�-source� valuesgreaterthan

or equalto the on-source � value. The em piricalsig-

ni�cance hasm ore scatterthan a signi�cance com puted

with a known nulldistribution sinceweonly havea �nite

num berofo�-sourcevalues.However,thenum berofo�-

sourcevaluesin thisanalysisislargeenough thatwecan

ignorethe associated error.

The m ean and standard deviation of the o�-source

sum -m ax sam ple were 0:1744 and 0:0029, repectively.

The on-sourcevalue ofsum -m ax was�on = 0:1753.Fig-

ure 6 shows the distribution ofthe o�-source values of

the teststatistic. The em piricalsigni�cance ofthe null

hypothesisis0:36. Following the discussion above,itis

clearthatthisresultisconsistentwith no detection.

V I. SIN G LE-G R B LIM IT S

Sim ulationsweredonetoestim atethesensitivityofthe

search m ethod to incidentG W burstsignals. Thispro-

cess was lim ited by the fact that the theoreticalwave-

form s of the G W burst signals associated with G RBs

were not known. O ther unknown quantities were: the

polarization ofthe waves,the orientation ofthe source

relativeto the observer,and the redshiftsofm ostofthe

G RBs.Consciousoftheselim itations,weproceed to set

upperlim itson the root-sum -squaream plitude (hrss)of

G W burstsignalsincidenton theinterferom etersduring

the on-source tim es by using sim ulated waveform swith

burst-likecharacteristics,adding these waveform sto the

raw IFO datastream s,and m easuringtheresultingcross-

correlations.

The antenna response of an IFO to incident, inde-

pendentgravitational-wavestrains,h+ (t)and h� (t),de-

pends on the relative position ofthe source in the sky

and the polarization ofthe wave[56]:

h(t)= F+ (�;�; )h+ (t)+ F� (�;�; )h� (t); (8)

where (�;�) isthe position ofthe source relative to the

IFO ’s zenith and x-arm , respectively; is the polar-

ization angle ofthe gravitational-wave;and F+ (�;�; )

F� (�;�; )arethecorresponding \plus" and \cross" an-

tenna factors.Form ostoftheG RBsanalyzed,theposi-

tion,(�;�),wasknown.Thepolarization angle, ,how-

ever,was an unknown param eter for allofthe G RBs.

Sincetheantennafactorisused in thesim ulations,upper

lim its were not set for G RBs which did not have well-

de�ned positions. The polarization-averaged antenna

factorisde�ned as:

Fave(�;�)=

rF 2+ + F 2

�

2=

q F 2+

�

=

q F 2�

�

: (9)

W eused sine-gaussiansasthesim ulated waveform sfor

h+ (t)and cosine-gaussiansforh� (t)in Eq.8:

h+ (t) = h+ ;0 sin(2�f0t)exp

�� (2�f0t)

2

2Q 2

�

; (10)

h� (t) = h� ;0 cos(2�f0t)exp

�� (2�f0t)

2

2Q 2

�

; (11)

wheref0 isthecentralfrequency ofthesine-gaussian and

cosine-gaussian,h+ ;0 and h� ;0 aretheam plitudeparam e-

tersofthe+ and � polarization signals,respectively,and

12

Q is a dim ensionless constant which represents roughly

the num berofcycleswith which the waveform oscillates

with m ore than halfofthe peak am plitude. The root-

sum -squared(rss)am plitudeofh+ (t)and h� (t)isrelated

to these param etersvia:

sZ

jh+ (t)j2 dt � h+ ;0

rQ

4p�f0

for Q >� 3;(12)

sZ

jh� (t)j2 dt � h� ;0

rQ

4p�f0

for Q >� 3:(13)

Using these waveform sforh+ (t)and h� (t),we sim u-

lated circularly polarized G W wavesby setting the sine-

gaussian and cosine-gaussian am plitudes equalto each

other,h+ ;0 = h� ;0 � h0. To sim ulate linearly polarized

waves,we set h� ;0 = 0. In the discussion that follows,

wede�ne the hrss ofa sim ulated waveform as:

hrss =

sZ

(jh+ (t)j2 + jh� (t)j

2)dt : (14)

Sincethepolarization angle, ,wasnotknown forany

G RB,a random polarization anglefrom 0 to 360 degrees

was generated for each sim ulated waveform event. In

the case ofLHO -LLO sim ulations,the source position-

dependent di�erence in the polarization angles at LHO

and LLO | dueto thenon-aligned detectorarm s| was

taken intoaccount.Finally,beforeaddingtheattenuated

waveform givenbyEq.8intoanIFO ’sraw datastream ,it

was�rstcalibrated usingthem easured responsefunction

ofthe IFO .

Following the procedure outlined above, sim ulated

sine-gaussianswith di�erent frequenciesand hrss values

wereadded to each IFO on-sourcedata atknown tim es.

Random ness in the injection tim es ofthe order ofthe

crosscorrelation length was introduced to sim ulate the

factthat the relative tim e ofthe G W signalwithin the

180-second search window wasnotknown.Data with in-

jected signalswerethen conditioned using theprocedure

outlined in Section IV B. The search was then applied

to the data nearthe injection tim es| notto the entire

180-secondon-sourcesegm ent| to�nd thelargestcross-

correlationsaround the injection tim es.Thissim ulation

procedureresulted in the determ ination ofthe probabil-

ity density,p(ccjhrss),for m easuring a crosscorrelation,

cc,correspondingtoasignalinjected in an on-sourceseg-

m entwith a certain hrss value.

The m ethod used to set upper lim its on hrss follows

the standard recipe for setting frequentist upper lim its

[57].Ifp(ccjhrss)istheprobability density form easuring

a crosscorrelation,cc,in an on-source segm ent given a

signalwith acertain hrss value,then the90% upperlim it

curve can be constructed from the set (cc90;hrss),such

that,

0:90=

Z1

cc90

p(ccjhrss)d(cc) : (15)

crosscorrelation

h rss (

Hz−

1/2 )

0 0.2 0.4 0.6 0.8 110

−22

10−21

10−20

10−19

H1−H2 linear polarizationfitted curve, linear polarizationH1−H2 circular polarizationfitted curve, circular polarizationon−source crosscorrelation

FIG .7: Exam ples ofupper lim it curves that were used to

set upper lim its on hrss using linear and circular polariza-

tions. These were the upperlim it curvesfor the H1-H2 IFO

pair,forG RB 050306,using sine-gaussianswith Q = 8.9 and

f0 = 150 Hz.Theshaded regionsindicatethetotal1� uncer-

tainty in the hrss value.

Exam plesofupperlim itcurvesobtained through this

procedureareshownin Fig.7,with onecurvecorrespond-

ing to linear polarization, and the other curve corre-

sponding to circularpolarization.Thesecurveswereob-

tained using the H1-H2 on-sourcedata forG RB 050306;

150-Hz,Q = 8.9sine-gaussians;and a25-m scrosscorrela-

tion length.Each curveshowsthehrss valueofthesim u-

lated waveform versuscc90,the crosscorrelation valueat

which 90% ofthe m easured crosscorrelation valueswere

larger (see Eq.15). The data was �tted with a four-

param etersigm oid function,

cc90 = p1 +1� p1

p4�1+ exp[� p2(log10(hrss)� p3)]

� ; (16)

where param eter p1 de�ned the asym ptote of cc90 at

sm allvalues of hrss, p4 tracked the asym ptote of cc90at large values ofhrss (i.e. p4 � 1=asym ptote),p3 was

the value ofhrss which gave a m id-range value ofcc90,

and p2 de�ned the slopeofthe curve.The largestcross-

correlation found in theon-sourcesegm entisalso shown

in Fig.7(verticaldashed line).The90% hrss upperlim it,

before uncertainties,wasfound by evaluating the upper

lim itcurve,which isthe inverseofEq.16,atthe largest

on-sourcecrosscorrelation value found in the search.

The curvesin Fig.7 also show the estim ated total1�

uncertainty in the m easurem entofthe hrss values. The

uncertainty in thehrss valuescom esfrom m easured ran-

dom and sytem atic errorsin the calibration param eters

thatwereused tocalibratethesim ulated waveform s,and

13

)rss

log10(h-22 -21.5 -21 -20.5 -20 -19.5 -19 -18.5 -18

num

ber

of e

ntri

es

0

1

2

3

4

5S2S3S4

FIG .8: Progression ofhrss upper lim its from the S2 to S4

LIG O runs for linearly polarized sine-gaussian waveform s;

25-m scrosscorrelation.

)rss

log10(h-22 -21.5 -21 -20.5 -20 -19.5 -19 -18.5 -18

num

ber

of e

ntri

es

0

1

2

3

4

5

6S2S3S4

FIG .9: Progression ofhrss upper lim its from the S2 to S4

LIG O runs for circularly polarized sine-gaussian waveform s;

25-m scrosscorrelation.

also from thestatisticalerrorswhich com efrom thesim -

ulation procedure.Depending on which science run and

IFO pair is being considered,the total1� uncertainty

from allthese sources ranged from � 10% to � 13% .

However,for G RB 030217 and G RB 030226,the total

uncertainty was about � 22% for the H1-H2 and H1-

L1 IFO pairs,dueto largercalibration errorsduring the

tim es ofthose G RBs. The �nal90% hrss upper lim its

were obtained by adding the corresponding total1:28�

uncertaintiesto thevaluesobtained from theupperlim it

curves.

Theupperlim itsresultingfrom theuseofQ = 8.9sine-

gaussiansand a 25-m scrosscorrelation length,forG RBs

with well-localized positions,are listed in Tables III to

V forlinearly polarized waveform s,and in TablesVIto

VIIIfor circularly polarized waveform s. Corresponding

lim its from the use ofa 100-m s crosscorrelation length

arelisted in TablesIX to XI,and in TablesXIIto XIV.

TABLE II:Best90% hrss upperlim itsresulting from a search

ofG W signals from G RBs occurring during the three LIG O

runs;25-m scrosscorrelation analysis(Hz� 1=2)

Run hrss;90 hrss;90 f0

(circular) (linear)

S4 1:1� 10� 21 (050306) 3:6� 10� 21 (050223) 150 Hz

S3 8:5� 10� 21 (031109A) 2:9� 10� 20 (031109A) 250 Hz

S2 8:2� 10� 21 (030414) 3:1� 10� 20 (030329B) 250 Hz

It can be seen that the upper lim its for the two cross-

correlation lengthsdo notdi�erm uch forthe waveform s

that were used. The upper lim its for f0 = 250 Hz and

25-m scrosscorrelation length are plotted in Figs.8 and

9 forlinearand circularpolarizations,respectively. The

im provem entin sensitivity from theS2 to S4 runscan be

seen in theseplots.Thebestupperlim itsfrom thethree

sciencerunsaregiven in TableII.From theS2 to theS4

run,there was an im provem ent in sensitivity by about

an orderofm agnitude.

Itcan also beseen from Figs.8 and 9 that,form ostof

theG RB sourcepositions,thecircularpolarization lim its

arebetterthan the linearpolarization lim itsby abouta

factor of3.5. This is always true in the case ofH1-H2

upperlim itssincewaveform satthetwo co-aligned LHO

IFO swerealwaysin phase(aftercalibrations).ForLHO -

LLO upperlim its,thereweretwocases,G RB 030217and

030323a,in which the positionsofthe G RBsrelative to

theIFO sweresuch thatcircularlypolarizedwaveform sat

LHO and LLO weresu�ciently outofphaseso thatup-

perlim itsforcircularpolarization werenotdeterm inable

forthoseG RBs.

V II. C O N ST R A IN IN G G R B P O P U LA T IO N

M O D ELS

Theapproach ofcom bining m ultipleG RBsto look for

a G W signature associated with a sam ple ofG RBswas

described in Section V.Having established thatthe null

hypothesisisquitesigni�cant,i.e.,thatwecannotclaim

the detection of an association between G W s and the

G RB population ata high enough con�dence,weturn to

settingconstraintson theparam etersofG RB population

m odels.Them ethod issum m arized below and described

in detailin [58].

Fora pairofdetectors,itcan beshown thatonly three

scalarparam etersassociated with a G W signalaresu�-

cientto determ ine the distribution oflargestcrosscorre-

lations.Theparam etersarethem atched �ltering signal-

to-noise ratios(SNRs)ofthe strain signalsin individual

detectorsand theanglebetween thetwostrain signalvec-

tors(asde�ned by the Euclidean innerproduct).In the

following,a source population m odelis the joint proba-

bility distribution function ofthese threeparam eters.

O urapproach to putting constraintson sourcepopula-

14

tion m odelsfollowsthe standard frequentistupperlim it

procedure(cf.Section VI).In thiscase,letP (�jZsource)

be the m arginalcum ulative probability density function

ofthe sum -m ax statistic,�,given the population m odel

Zsource,and let�� be such thatP (��jZsource)= 1� �,

where0< � < 1,and 1� � isthedesired con�dencelevel.

Iftheobserved valueof� isgreaterthan orequalto ��,

thecorrespondingm odelZsource isaccepted.Itisrejected

when � < ��.To obtain the m arginaldistribution of�,

we �rst construct its conditionaldistribution for a set

ofN G R B values for the scalar param eters above,where

N G R B isthenum berofH1-H2 G RB on-sourcesegm ents.

Them arginaldistribution of� foragiven sourcepopula-

tion m odelcan then be estim ated by random ly drawing

values ofthe scalar param eters followed by drawing �

from the corresponding conditionaldistribution.

Since we useonly the H1-H2 pair,which areperfectly

aligned,the angle between the strain responses is zero.

Further,for narrowband signals,the SNR values for H1

and H2 can be related by the m easurable ratio oftheir

calibrated noisepowerspectraldensities(PSDs).Hence,

only oneparam eter,which wechoseto betheSNR,�,of

thesignalin H1,isrequired.Thus,thesourcepopulation

m odel,Zsource,issim ply theunivariatedistribution of�.

An additionalpointthatneedsto beaccounted foristhe

variation in the sensitivities ofH1 and H2,both within

the runs as wellas the signi�cant im provem ents from

one run to the next. This is done by �xing a �ducial

noise PSD,S(0)(f),and approxim ating the PSD ofH1

for each G RB as sim ply a scaled version ofit. W e set

the �ducialnoise PSD to the one corresponding to the

initialLIG O design sensitivity for the 4-km IFO s4 and

com pute the scalefactorata �xed frequency of200 Hz,

which was approxim ately where m ost PSDs had their

m inim um during the S2,S3,and S4 runs.

W e usethe theoreticalprediction ofthe observed red-

shift distribution of G RBs given in [59] to construct

Zsource (prediction forthe scenario ofstarform ation via

atom ichydrogencooling).An alternativeistosim plyuse

them easured redshiftdistribution but[31,60]show that

there isa signi�cantselection biasthata�ectsthe m ea-

sured redshifts for Swift and non-Swift G RBs,both of

which areused in ouranalysis.Them odelin [59]isvalid

forlong-duration G RBs,which areexpected to tracethe

m assive star form ation rate ofthe Universe. W e �t a

piecewise parabolic curve (with 3 pieces) to �gure 1 of

[59]and then usethesam esubsequentcalculationalsteps

given in [59]toobtain theredshiftdistribution fora ux-

lim ited detectorsuch asSwift.Fitting thestarform ation

rate with a sm ooth curve allows us to extend the red-

shiftdistribution reliably to very sm allvaluesofthered-

shift.Having obtained thedistribution,wedirectly draw

random valuesofthe redshift,z,from it. Each redshift

value isthen converted to the corresponding lum inosity

4 http://www.ligo.caltech.edu/� lazz/distribution /LSC D ata/

distance D (corresponding to a Friedm ann-Robertson-

W alker cosm ologicalm odelwith m = 0:3,� = 0:7

and H 0 = 72 km sec� 1 M pc� 1).

A sim ple m odel is used for the G W em ission from

G RBs. W e assum e that G RBs are standard candles in

G W thatem ita �xed am ountofenergy,E G W ,isotrop-

ically with sim ilar am ounts ofradiation in the two un-

correlated polarizations + and � . Further, neglecting

the e�ectofredshifton the signalspectrum ,we assum e

that the spectra ofthe received signals h+ and h� are

centered ata �xed frequency offo in a band thatissu�-

ciently narrow such thatthenoisepowerspectraldensity

isapproxim ately constantoverit.In thiscase,the SNR

isgiven by

� ’p2Fave

hrsspS(0)(fo)

; (17)

where we have expressed the SNR with respect to the

�ducialnoise PSD.Since the em ission is isotropic,the

energy em itted in gravitational waves is (cf. Section

VIIIA),

E G W ��2c3

G

D 2

1+ zf2oh

2rss : (18)

To convertthe lum inosity distance,D ,fora given G RB

into SNR �,weuse the norm alization

� =p2Fave�0

D 0

D

�1+ z

1+ z0

� 3=2

; (19)

where D 0 is chosen to be the m ost probable lum inos-

ity distance, at the corresponding redshift z0, and �0is the observed SNR for a G RB that occured at D 0

with an optim alsky location and the above properties

for h+ , h� and E G W . The redshift distribution pre-

dicted in [59]for Swift has a peak at z = 1:8,which

yields D 0 = 13:286 G pc. The acceptance-rejection rule

abovesim ply becom esan upperlim iton thevalueof�0.

Notethat,becauseofthescalingofnoisePSDsdiscussed

above,�0 should beunderstood astheSNR ofthestrain

response(foraG RB directlyabovethedetector)thatop-

eratesatdesign sensitivity. ForG RBsthatdo nothave

direction inform ation,random valuesforFave aredrawn

from a uniform distribution on the celestialsphere.

Finally,in term s ofthe upper lim it,�upper,obtained

on �0,wegetan upperlim iton E G W ,

E G W ��2c3

G

D 20

1+ z0f2oS

(0)(fo)�2upper : (20)

For z0 = 1:8, fo = 200 Hz, andpS(0)(fo) = 2:98 �

10� 23 Hz� 1=2

, we get E G W � 8:43 � 1055�2upper ergs

(� 47:3�2upper M � c2).

Figure 10 shows the 90% upper lim it con�dence belt

for�0.The on-sourcevalue ofsum -m ax was� = 0:1753

for the S2,S3,S4 G RB sam ple. Hence,�0 � 35:5 and

E G W � 5:96� 104 M � c2. This lim it is too high to be

15

0.168 0.17 0.172 0.174 0.176 0.178 0.18 0.182 0.1840

5

10

15

20

25

30

35

40

45

χ

ρ 0

hypothetical 35 GRBss2s3s4 GRBslast 10

FIG .10:Upperlim itcon�dencebeltsat90% con�dencelevel

on �0,theSNR atthem ostprobableredshiftforSwiftG RBs

given in [59].Thesolid lineisthecurveforallS2,S3,S4G RBs

that were used in the H1-H2 search (on-source � = 0:1753).

The dashed line is the curve when only the last 10 G RBs

from the above setare selected (on-source � = 0:1702). The

line with �lled circles is for a hypotheticalscenario with 35

G RBs, allwith an optim alsky location, and two identical

and constantsensitivity detectors.The shifting ofthe curves

horizontally is due to the change in the variance of� as the

num ber of G RBs is changed. For each value of�0, 10,000

valuesof� were drawn from itsm arginaldistribution.

ofany astrophysicalim portance. However,asdiscussed

later,future analysesm ay be able to im prove by orders

ofm agnitudeon thisresult.

Since the detectorsduring the S2 run were m uch less

sensitive than S4,one m ay expectthatdropping the S2

G RBs from the analysis can im prove the upper lim it.

Figure 10 shows the 90% level upper lim it belt ob-

tained for the case when only the last 10 G RBs,span-

ning the whole ofS4 and part ofS3,were retained in

the analysis. The corresponding value of � = 0:1702

yields an upper lim it of 24:6 on �0. Thus, we obtain

E G W � 2:86� 104 M � c2.Thisshows,asexpected,that

m aking judicious cuts on the sam ple ofG RBs can lead

to im provem ents in upper lim its. The upper lim it can

probably be im proved further by retaining only the S4

G RBs,butfora sm allnum berofG RBsthedistribution

of� used is not valid and a m ore accurate calculation

hasto be done.In Fig.10,wealso show theupperlim it

con�dence beltfora hypotheticalscenario thatislikely

forthe ongoing S5 run:a sam plesizeofabout35 G RBs

with the H1 and L1 detectorsoperating atdesign sensi-

tivity.Theim plicationsofthiscurvearediscussed in the

nextsection.

The con�dence beltconstruction outlined in this sec-

tion is for illustrative purposes only. In particular,we

havenottaken into accountfactorssuch as(i)changing

noise spectralshapes, (ii) red-shifting of the standard

candle (K -correction)and possible system atic errorsas-

sociated with the population m odelused. A m ore com -

prehensiveinvestigation isplanned forthe S5 data.

V III. D ISC U SSIO N

Thissearch isnotvery restrictivewith respecttom od-

els for astrophysicalsystem s which give rise to G RBs.

Them ain assum ption wehavem adeisthattheG W em is-

sion islim ited in duration | we sum overperiodsofup

to 100 m s,which is m uch greater than the characteris-

tic tim esexpected forG W burstem ission in m ostG RB

m odels. G iven the LIG O sensitivity atthe tim e ofthis

search,it is not surprising that our experim entallim -

itsin thissearch do notplace signi�cantrestrictionson

the astrophysicalm odelsatpresent.However,given the

rapid developm ent ofthe �eld,it is not precluded that

the lim its presented here willprovide guidance to G RB

astrophysics in the near future. In any case,it is use-

fulto geta sensefortheinterplay between them easured

gravitational-wavestrain lim itsforindividualG RBsfrom

Section VIand astrophysicalm odels. So in this section

weprovidesom eastrophysicalcontextto ourexperim en-

tallim its.W e em phasizethatthe estim atesgiven below

areforillustration,and are notto be construed asm ea-

sured astrophysicallim its.

Thelocalgravitational-waveenergy ux in thetwo in-

dependentpolarizations,h+ (t)and h� (t),is[56,61]

dE

dA dt=

1

16�

c3

G

�_h2+ + _h2

�

�

(21)

which can be integrated overthe duration ofa burstof

gravitationalradiation and overaclosed surfaceto relate

thestrainsevaluated on thesurfaceto thetotalintrinsic

energy associated with a source within thisvolum e.For

a sourceatthecenterofa sphereofradiusratnegligible

redshift,then dA = r2d,asusual.

Since m any of the G RBs in the sam ple are found

to have signi�cant redshifts, it is useful to generalize

the above to cosm ologicaldistances. In this case, we

can use the lum inosity distance,D ,which by de�nition

relates the intrinsic lum inosity, L, of an isotropically

em itting source to the energy ux F at a detector by

L = F (4�D 2). For a non-isotropic em itter,we replace

the 4� by an integration oversolid angle. W e note that

F isby de�nition thelefthand sideofEq.21,and thein-

trinsiclum inosity isL = dE e=dte.To integratethisover

thesignalduration atthedetector,weusedt= (1+ z)dte.

Hence,the energy em itted in gravitationalradiation is,

E e =D 2

1+ z

Z

d

Z

F dt

=1

16�

c3

G

D 2

1+ z

Z

d

Z �_h2+ + _h2

�

�

dt (22)

Fornegligible redshifts,D = r.W e note thatD = D (z)

isitselfa function oftheredshift,and in generaldepends

on the cosm ologicalm odel.

16

Ifthe signalpoweratthe detectorsisdom inated by a

frequency fo,as is the case for the sine-gaussian wave-

form s introduced earlier,then Eq.22 can be written in

the approxim ateform

E e ��

4

c3

G

D 2

1+ zf2o

Z

d

Z�h2+ + h

2�

�dt ; (23)

which allowsadirectrelation between E e and theobserv-

able hrss (see Eqs.12 and 13). For sine-gaussian wave-

form s,the approxim ation is quite good for Q >� 3;the

errorisapproxim ately 1=(1+ 2Q 2).W ewillassum ehere

thatthe sim ulated waveform saree�ectively localto the

detectors.Speci�cally,the frequency fo isthe m easured

frequency (which isrelated to thesourcefrequency fe by

fo = fe=(1+ z)).O fcourse,som e fraction ofthe source

powerm ight be shifted in or out ofthe sensitive LIG O

band in frequency orexpanded in tim e beyond ourinte-

gration tim e.W e ignoreany such e�ecthere.

A . C ase I:Isotropic em ission

First,we consider a sim ple,but unphysical,exam ple

where the radiation is em itted isotropically,with equal

power in the (uncorrelated) + and � polarizations. In

thiscase,Eq.23 becom es

E iso ��2c3

G

D 2

1+ zf2oh

2rss : (24)

Then for a LIG O sensitivity for som e waveform repre-

sented by hrss,we m ight hope to be sensitive to a dis-

tance

D � 70M pc

�100Hz

fo

� �E iso

M � c2

� 1=2

�

10� 21 Hz� 1=2

hrss

!

(1+ z)1=2 (25)

foran isotropicsourcewhich em itsgravitational-waveen-

ergy E iso (in units ofsolarrestenergy)atdetected fre-

quency fo.

B . C ase II:Long-duration G R B s

For long-duration G RBs, we consider the scenario

where such events are associated with a core collapse,

perhapsinvolving a very m assiveprogenitor[62].G ravi-

tationalwaveem ission hasbeen sim ulated forsupernova

core collapse m odels for relatively light (� 10M � ) pro-

genitors,for exam ple,in Refs.[36,37]. These m odels

invoke axisym m etry,with linearly polarized strain that

isproportionalto sin2 �,where� istheanglewith respect

to the sym m etry axis.

Integrating overthe fullsolid angle,Eq.23 becom es

E sn �8�2c3

15G

D 2

1+ z

f2o h2rss

sin4�: (26)

W e then �nd an analogousexpression to Eq.25,

D � 1M pc

�100Hz

fo

� �E sn

10� 4M � c2

� 1=2

�

10� 21 Hz� 1=2

hrss

!

sin2� (1+ z)1=2 (27)

Asdescribed earlier,ourexperim entallim itscorrectlyac-

countfortheantennapattern associated with each G RB.

Hence,no additionalfactorsarerequired in theequation

aboveifonewereto usevaluesfrom thetablesofresults.

However,ifonewished,forexam ple,to apply a theoreti-

calhrss toa particularG RB,theantennafactorsforeach

G RB aregiven in TableI.

Core collapse sim ulations indicate that m ost of the

gravitationalradiation isem itted from the core bounce,

and that E sn should be at m ost 10� 7M c2 [37],or even

sm aller[36].Forthevery m assiveprogenitorsoften asso-

ciated with long-duration G RBs,the collapse processis

uncertain.W hetherthereisa bounce atall,orsim ply a

directcollapseto a black hole,depends[63]on them ass,

m etallicity,and angularm om entum oftheprogenitor.In

any case,thereisno reason to believethatthee�ciency

for converting the collapse into gravitationalradiation

increaseswith the progenitorm ass.

In fact, the situation for G W detection in this sce-

nario isespecially unprom ising.Itisnaturalto align the

sym m etry axis ofthe (rotating) core collapse with the

direction ofthe gam m a-ray beam . Hence,� = 0 would

be along the line ofsightto the detectors.Fora typical

gam m a-ray beam ing angle ofhalf-width � 10�,then at

best,where the detectors are at the edge ofthe beam ,

thiswould givea suppression factorof� 30.Finally,we

note thatlong-duration G RBsare distantobjects,with

m ean observed redshift of� 2:4.5 G iven their redshift

distribution,thesim ulationsto dateindicatethatdetec-

tion oflong-duration G RBsisunlikely ifcore bounce is

the dom inantradiation m echanism .

However, core collapse can potentially drive other

m echanism s m ore favorable for gravitationalradiation

detection. In particular,bar m ode instabilities are po-

tentially very e�cient radiators and do not su�er from

the unfavorable alignm ent noted above for axisym m et-

ric core bounces. Sim ilarly,core fragm entation during

collapse can lead to G W radiation from the inspiraling

fragm ents. Reference [63]has exam ined these possibili-

ties,and while the likelihood ofbarinstabilitiesorcore

fragm entation,alongwith theirdetailed properties,isun-

certain,the resulting gravitationalradiation isplausibly

detectablefora nearby G RB.In such cases,Eqs.29 and

30m ightbem oreappropriatedescriptionsoftheradiated

energy and distanceto which wecan detectthe source.

Thenearestknown G RB todateislong-duration burst

G RB 980425 at D = 35 M pc. From Eq.27,LIG O de-

5 http://swift.gsfc.nasa.gov

17

tection at35 M pcby them ethod described in thispaper

would require an e�ciency ofatleastE sn=M � c2 � 10%

for a 1M � system ,m uch larger than the e�ciency ex-

pected from conventionalcore collapse,butperhapsnot

unreasonablein caseofbarinstabilitiesorcorefragm en-

tation. Unfortunately,the data considered here did not

includeany such nearby events.Forexam ple,during the

(m ost sensitive) S4 run, the G RB sam ple consisted of

only 4 events,alllong-duration G RBs.Them ostnearby

of these with a m easured redshift was G RB 050223

(z = 0:5915) at D � 3:5 G pc. Assum ing linear polar-

ization,we can obtain an estim ate for sensitivity from

the90% upperlim itforG RB 050223from TableII.This

gives for E sn the value 1:6� 104 M � c2. This is in fact

very close to the source lum inosity m axim um ofc5=G

[64],which gives 2� 104 M � c2 ifsustained for 100 m s.

Thelargersam pleofG RBsin future runswillhopefully

include som elong-duration G RBsatsm allerredshift.

C . C ase III:Short-duration G R B s

Short-duration G RBs,to the extentthatthe popula-

tion isassociated with them ergerofcom pactbinary sys-

tem s,o�erseveralpotentially interesting characteristics.

First,such m ergersarefound to berelatively e�cientra-

diatorsofgravitationalradiation. Second,the em ission

pattern isnotexpected tobeproblem atic.M oreover,the

m easured redshifts to date indicate a signi�cant num -

berofrelatively low-z G RBs.(The averageredshiftwas

� 0:4 forthe2005sam pleof5 events.) Them ergersm ay

includeform ation ofa hyperm assiveneutron star[65]or

a black hole with associated ringdown [66]. Finally,the

m ergerG W em ission,which isbestsuited tothem ethod-

ology described in this paper,would be preceded by an

inspiralwhich ispotentially detectableby a sensitive,in-

dependentLIG O searchbased on m atchinginspiralwave-

form tem plates[67]. However,we have veri�ed thatthe

presentsearch,while notassensitive to inspiralsasthe

dedicated waveform tem plate-based search,can readily

detectinspiralem ission when thereissu�cientsignalto

background in individual25-m sor100-m sbins. In this

case,them axim um crosscorrelation occurswhen thefre-

quency ofthe inspiralradiation passesthrough the 100-

300 Hz range,where the detectorsensitivity isbest(see

Fig.1).Therefore,whilethissearch isuniquely sensitive

to the higherfrequency,short-duration,poorly m odeled

gravitationalwaves from the m erger phase,it also pro-

videsindependentinform ation on theinspiralphase.Re-

centestim ates[34,35]place the chance fordetection of

a BH-NS m ergeratup to � 30% fora yearofsim ultane-

ousLIG O and Swiftoperation,and � 10% fora NS-NS

m erger. Here,we provide an estim ate for a contrived,

butphysically m otivated,m odel.

W e suppose thatthe gravitational-waveem ission pat-

tern forthe m ergerfollowsthatofthe inspiral,thatis

h+ = ho f(t)1

2(1+ cos2 �); h� = ho g(t)cos� (28)

where � is m easured with respect to the axis orthogo-

nal to the plane of the inspiral orbit. The functions

f(t)and g(t)are orthogonalfunctions,forexam ple f(t)

could be the sine-gaussian form discussed earlier,while

g(t) is a cosine-gaussian;ho represents a constant am -

plitude. W hile the degree of gam m a-ray beam ing for

short-duration G RBsisstilluncertain,we suppose that

the gam m a raysarepreferentially em itted along the an-

gularm om entum axisofthem ergersystem .IftheEarth

is near the center ofthe gam m a-ray beam ,then � = 0

is along the line ofsight between detector and source,

which is a m axim um ofthe assum ed em ission pattern,

and theradiation willbecircularly polarized.Returning

toEq.23and integratingoverthefullsolid angle,we�nd

in thiscase(with � = 0):

E m erge �2�2

5

c3

G

D 2

1+ zf2o h

2rss : (29)

Rewriting thisforD ,asbefore,gives

D � 44M pc

�250Hz

fo

� �E m erge

M � c2

� 1=2

�

10� 21 Hz� 1=2

hrss

!

(1+ z)1=2 : (30)

The com m entsbelow Eq.27 concerning antenna factors

also apply here.

There has been substantialrecent progress in calcu-

lations ofgravitationalradiation production in various

typesofm ergers.Num ericalsim ulationsofNS-NS m erg-

ersgive[66,68,69]typicalvaluesoftheradiated energy

ofabout 0.5-1% ofthe totalm ass,or E � 0:01M � c2.

These sim ulationsindicate thatthe frequency spectrum

can bebroad,rangingfrom afew hundred Hzto� 2kHz.

Perhaps the m ost interesting case is BH-NS m ergers.

Very recent calculations [70, 71, 72]indicate radiative

energies ranging from � 10� 4 to � 10� 2 of the total

m ass,where the rangeislikely to re ectthe very di�er-

entinitialconditionsassum ed in the sim ulations.W hile

there are no short-duration G RBsin the S4 sam ple,we

can use typicalupper lim its on hrss from Table VI as

an indication ofsensitivity. For exam ple a 1:4M � NS

plus 10M � BH binary system would have m erger G W

em ission atfrequenciesstarting atabout400 Hz.Ifthis

system wereto radiate1% ofitsrestenergy into gravita-

tionalradiation at400 Hz,thedistancesensitivity would

beD � 5 M pc.Thesearch would alsobesensitiveto the

inspiralem ission from this system at lower (� 200 Hz)

frequency.

D . P rospects

Here we discuss the future prospects for science run

S5 and beyond. At the sensitivity for science run S4,

the prospectsfordetection areclearly dom inated by the

possibility ofa nearby G RB.W hile thisdistancescaleis

18

guided by the discussion above,we are prepared to be

surprised by new m echanism sforG W em ission. Never-

theless,we expect detection ofindividualG RBs to de-

pend in no sm allparton the appearance ofa \special"

event.Thus,a data sam ple which includesa largenum -

berofG RBsisespecially im portant.Forsciencerun S5,

the LIG O detectorswillbe operating atdesign sensitiv-

ityand fullycoincidentwith Swiftoperation.Thisshould

yield over100 G RBs,including som e with redshiftm ea-

surem ents.And clearly,thesearch radiuswillincreasein

proportion to im provem entsin theLIG O strain sensitiv-

ity.

TheresultspertainingtotheG RB population obtained

in Section VIIwillcertainly im prove forthe S5 run and

in future observations with Advanced LIG O .To m ake

an estim ate,we look at the various factors involved in

Eq.20 for the upper lim iton E G W . As a reference,we

usethelim itobtained hereusingallS2,S3and S4G RBs.

Since m ostfactorsin Eq.20 com e assquares,m oderate

im provem entsin each hasa signi�cantoveralle�ect.

Sincethedirection to each G RB willbeknown,itm ay

be possible to selecta subsam ple of,say,35 G RBsfrom

the sam ple in S5 (i.e., about the sam e num ber as the

whole ofS2,S3 and S4) such thatFave

�’ 1=

p2,the

m axim um possible. Further,assum e thatwe use H1-L1

crosscorrelations. Figure 10 shows the con�dence belt

forthe case of35 optim ally located G RBsand a pairof

identicaldetectors.O necan expectto getan upperlim it

of’ 10 on �0 with thiscurve,which isa factorof� 3:5

betterthan the currentlim iton �0.

W ithout altering other param eters of the analysis,

therefore,we can expect 3:52 or,in round num bers,a

factorof� 10 im provem entin the upperlim iton EG WforS5.Additionalim provem entsare possible by im pos-

ing a cutbased on m easured redshifts,in addition to the

cuton sky positions,and by reducing thesearch interval

from the currentvalue of180 seconds. Looking beyond

S5,the m ost obvious source ofim provem ent would be

the � 10 factorofim provem entin the strain noise level

when Advanced LIG O com es online around the m iddle

ofthe next decade. This translates into an additional

factorof� 100 reduction in the upperlim it.W hen Ad-

vanced LIG O com es online,there m ay be a worldwide

network ofG W detectorsofcom parable sensitivity. Be-

sidesallowing a m ore uniform sky coverage,resulting in

a largersam ple ofG RBs with optim alorientation,net-

work analysism ethods[73,74,75]thatm akem oreopti-

m aluse ofdata from m ultiple detectorscan be used to

increasethebasesensitivity ofthem ethod.Finally,with

enough G RBs,we could separately analyze the class of

long and short duration bursts. Since the m ost proba-

ble redshift for short-duration G RBs is expected to be

inherently sm aller,we could obtain signi�cantly tighter

constraintson theenergy em itted in gravitationalwaves

from thisclassofG RBs.

The discussion above was con�ned to a particular

m odelfor G RB redshift distribution and G W em ission.

Furtherwork isneeded to develop m oregeneralanalysis

m ethodsthatcan beapplied to a widervariety ofm odels

and that take better account ofprior inform ation from

existing observations.

IX . SU M M A R Y A N D C O N C LU SIO N

W e searched for gravitational-wave bursts, target-

ting short G W signals with durations from � 1 m s to

� 100 m s,associated with 39 G RBsthatwere detected

bygam m a-raysatelliteexperim entswhiletheS2,S3,and

S4sciencerunsoftheLIG O experim entwerein progress.

To takeinto accounttheunknown onsettim eoftheG W

signalrelative to the G RB triggertim e,the search cov-

ered 180 seconds ofdata surrounding the G RB trigger

tim es. These 180-second data segm ents from the dif-

ferentIFO swere crosscorrelated to probe forcorrelated

signals.W esearched foran association on an individual-

G RB basis,and also applied di�erentstatisticalteststo

search forthecum ulativee�ectofweak G W signals.W e

found no evidence forgravitational-waveburstem ission

associated with theG RB sam pleexam ined using thedif-

ferentsearch m ethods.

Using sim ulated Q = 8.9 sine-gaussian waveform sand

the direction-dependent antenna response ofthe inter-

ferom etersto a G W source,weobtained upperlim itson

the root-sum -squaream plitude oflinearly polarized and

circularly polarized gravitationalwavesfrom each of22

G RBs with well-localized positions. Associating these

lim itswith theenergy radiated by theG RB sourcesinto

gravitationalradiation is inherently speculative at this

stage ofdevelopm entofthe �eld and depends crucially

on the astrophysicalscenario one adopts for the G RB

progenitors. The m ost favorable cases considered here

suggestthattheLIG O sensitivity forrun S4 would allow

sensitivity to a solarm ass-equivalentofradiated G W en-

ergy to distancesoftensofM pc.

The sam ple ofG RBs was com bined to set an upper

lim iton theG W energy em itted using a sim plestandard

candle m odeland a theoreticalredshift distribution of

G RBs. Although the upper lim itobtained isnotastro-

physically im portant,a straightforward and realistic ex-

trapolation tofutureobservationssuggeststhatthislim it

can beim proved by ordersofm agnitude.Itm ay bepos-

sibleto seta sub-solarm asslim itwhen Advanced LIG O

com esonline.Thiswould putusin an astrophysicallyin-

teresting regim esinceatleastonem odel[76]predictsan

energy lossof0.2 solarm assesforlong-duration G RBs.

It is opportune that Swift willbe operating and de-

tecting G RBs atthe tim e when the �fth science run of

LIG O ,S5,willbe in progress. The goalfor the S5 run

isto collectoneyearofcoincidentLHO -LLO data atthe

design sensitivity. G iven the SwiftG RB detection rate,

we anticipate an S5 sam ple ofm ore than 100 G RB trig-

gersthatcan be used to furtherprobe forgravitational

radiation associated with G RBs.Itishoped thata large

G RB sam plewillincreasethechancesfor�nding such an

association.

19

A cknow ledgm ents

Theauthorsgratefully acknowledgethesupportofthe

United States NationalScience Foundation forthe con-

struction and operation oftheLIG O Laboratory and the

ParticlePhysicsand Astronom y Research Councilofthe

United K ingdom ,theM ax-Planck-Society and theState

ofNiedersachsen/G erm any for support ofthe construc-

tion and operation oftheG EO 600detector.Theauthors

alsogratefullyacknowledgethesupportoftheresearchby

these agenciesand by the Australian Research Council,

the NaturalSciences and Engineering Research Coun-

cilofCanada, the Councilof Scienti�c and Industrial

Research ofIndia,the Departm entofScience and Tech-

nology ofIndia,the Spanish M inisterio de Educacion y

Ciencia,The NationalAeronauticsand Space Adm inis-

tration, the John Sim on G uggenheim Foundation, the

Alexander von Hum boldt Foundation, the Leverhulm e

Trust,theDavid and LucilePackard Foundation,theRe-

search Corporation,and theAlfred P.Sloan Foundation.

Thisdocum enthasbeen assigned LIG O Laboratory doc-

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21

TABLE III:S4 90% upperlim itson hrssofQ = 8.9 linearly polarized sine-gaussians,in unitsof10� 21

Hz� 1=2

;25-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

050223 5.5 ... ... 3.6 ... ... 4.1 ... ... 6.9 ... ... 11.7 ... ... 25.8 ... ...

050306 7.8 6.4 12.0 5.2 5.2 8.8 5.6 6.3 9.5 9.0 12.6 16.0 16.4 24.5 30.4 31.4 61.9 82.4

050318 7.9 10.2 15.4 6.0 7.0 10.7 6.0 9.3 11.9 9.5 16.7 19.8 15.8 30.2 35.0 33.4 55.3 66.7

050319 6.6 6.8 8.3 4.7 4.9 5.7 5.4 6.1 6.2 8.1 11.1 11.0 15.5 21.1 19.8 29.7 36.9 34.9

TABLE IV:S3 90% upperlim itson hrssofQ = 8.9 linearly polarized sine-gaussians,in unitsof10� 20 Hz� 1=2;25-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

031108 6.5 ... ... 3.6 ... ... 3.6 ... ... 4.2 ... ... 6.7 ... ... 19.7 ... ...

031109a 4.8 ... ... 2.9 ... ... 2.9 ... ... 3.6 ... ... 6.0 ... ... 14.7 ... ...

031220 5.7 ... ... 3.3 ... ... 3.0 ... ... 3.7 ... ... 6.3 ... ... 14.7 ... ...

TABLE V:S2 90% upperlim itson hrssofQ = 8.9 linearly polarized sine-gaussians,in unitsof10� 19

Hz� 1=2

;25-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

030217 ... ... 4.4 ... ... 2.2 ... ... 1.0 ... ... 1.6 ... ... 4.4 ... ... 10.2

030226 7.7 3.5 5.4 3.4 1.6 2.2 1.00 0.68 0.63 1.3 1.1 0.81 2.6 2.4 1.4 7.1 6.6 2.7

030320a 7.2 2.1 7.1 2.5 1.1 2.2 0.69 0.58 0.71 1.0 1.1 1.3 1.6 2.9 3.1 3.8 6.0 5.6

030323a 5.1 3.1 6.4 2.5 1.7 2.9 1.1 0.99 1.5 1.7 2.3 3.3 2.6 6.1 7.2 6.0 11.4 13.4

030323b 4.6 1.8 5.2 1.7 0.94 1.8 0.64 0.45 0.81 0.92 0.82 1.5 1.3 1.8 2.4 3.0 3.5 4.8

030324 9.2 ... ... 4.7 ... ... 1.6 ... ... 2.0 ... ... 3.3 ... ... 7.9 ... ...

030325 2.8 1.7 3.0 1.3 0.80 1.5 0.55 0.48 0.76 0.89 1.0 1.5 1.3 2.0 2.4 3.2 4.9 5.3

030326 10.2 3.9 9.6 4.4 2.1 3.7 1.4 0.94 1.2 2.0 1.6 1.9 3.1 3.4 3.1 8.4 8.1 6.3

030329a 4.6 ... ... 2.4 ... ... 1.1 ... ... 1.8 ... ... 3.0 ... ... 7.6 ... ...

030329b 2.8 ... ... 1.1 ... ... 0.31 ... ... 0.55 ... ... 0.89 ... ... 2.0 ... ...

030331 ... 3.4 ... ... 1.6 ... ... 0.85 ... ... 2.0 ... ... 3.4 ... ... 8.0 ...

030405 2.1 1.4 3.1 1.0 0.80 1.3 0.34 0.42 0.51 0.59 0.76 0.97 0.87 2.0 2.2 2.0 4.8 4.5

030406 ... 1.2 ... ... 0.67 ... ... 0.42 ... ... 0.77 ... ... 1.7 ... ... 4.4 ...

030413 ... ... 1.6 ... ... 0.85 ... ... 0.50 ... ... 0.89 ... ... 2.3 ... ... 4.4

030414 1.4 ... ... 0.91 ... ... 0.32 ... ... 0.39 ... ... 0.70 ... ... 1.6 ... ...

22

TABLE VI:S4 90% upperlim itson hrssofQ = 8.9 circularly polarized sine-gaussians,in unitsof10� 21

Hz� 1=2

;25-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

050223 1.6 ... ... 1.1 ... ... 1.2 ... ... 2.0 ... ... 3.5 ... ... 6.7 ... ...

050306 2.2 1.4 2.6 1.5 1.1 1.8 1.6 1.4 2.0 2.6 2.6 3.3 4.5 5.0 6.2 8.5 14.2 17.6

050318 2.2 2.2 3.1 1.6 1.5 2.2 1.6 1.9 2.4 2.6 3.5 4.0 4.6 6.1 6.9 8.8 11.1 13.1

050319 1.8 1.8 2.3 1.4 1.3 1.6 1.5 1.7 1.8 2.4 3.1 3.0 4.3 5.5 5.2 8.2 10.0 9.9

TABLE VII:S3 90% upperlim itson hrssofQ = 8.9 circularly polarized sine-gaussians,in unitsof10� 21

Hz� 1=2

;25-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

031108 19.0 ... ... 11.3 ... ... 10.9 ... ... 12.5 ... ... 20.4 ... ... 51.5 ... ...

031109a 14.7 ... ... 8.8 ... ... 8.5 ... ... 10.6 ... ... 17.3 ... ... 42.2 ... ...

031220 14.4 ... ... 10.1 ... ... 8.9 ... ... 10.8 ... ... 18.4 ... ... 42.7 ... ...

TABLE VIII:S2 90% upperlim itson hrssofQ = 8.9 circularly polarized sine-gaussians,in unitsof10� 20

Hz� 1=2

;25-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

030226 22.2 11.0 18.0 9.2 5.0 6.9 2.9 2.1 1.9 3.7 3.3 2.6 7.1 7.1 4.1 20.3 20.3 7.2

030320a 21.9 7.0 26.6 7.3 3.6 7.9 2.0 1.9 2.2 2.9 3.3 4.1 4.6 9.5 10.1 10.7 17.3 16.1

030323a 16.1 ... ... 7.9 ... ... 3.6 ... ... 5.6 ... ... 7.9 ... ... 18.5 ... ...

030323b 13.4 4.9 15.5 4.9 2.5 5.1 1.8 1.2 2.3 2.7 2.4 4.0 3.7 5.1 6.6 8.5 9.2 12.3

030324 28.0 ... ... 13.3 ... ... 4.3 ... ... 5.6 ... ... 9.4 ... ... 22.2 ... ...

030325 9.0 4.3 9.5 4.0 2.0 4.2 2.0 1.2 2.4 3.1 2.8 4.4 4.3 5.3 6.7 10.2 12.2 15.0

030326 29.7 15.1 39.9 12.4 8.1 14.9 4.0 3.5 4.8 5.8 5.8 7.6 9.6 12.1 11.7 24.2 25.8 19.7

030329a 13.8 ... ... 7.3 ... ... 3.3 ... ... 5.1 ... ... 8.2 ... ... 21.6 ... ...

030329b 8.8 ... ... 3.2 ... ... 0.90 ... ... 1.5 ... ... 2.4 ... ... 5.9 ... ...

030331 ... 7.1 ... ... 3.5 ... ... 1.8 ... ... 4.3 ... ... 7.3 ... ... 17.4 ...

030405 6.2 3.4 8.2 2.9 2.0 3.4 0.99 1.1 1.3 1.6 2.0 2.5 2.5 5.1 5.4 5.9 11.3 10.7

030406 ... 2.8 ... ... 1.5 ... ... 0.90 ... ... 1.8 ... ... 4.0 ... ... 10.0 ...

030413 ... ... 4.1 ... ... 2.2 ... ... 1.3 ... ... 2.4 ... ... 6.0 ... ... 11.0

030414 4.1 ... ... 2.6 ... ... 0.82 ... ... 1.1 ... ... 1.9 ... ... 4.6 ... ...

23

TABLE IX:S4 90% upperlim itson hrssofQ = 8.9 linearly polarized sine-gaussians,in unitsof10� 21

Hz� 1=2

;100-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

050223 5.6 ... ... 4.1 ... ... 4.8 ... ... 8.0 ... ... 14.5 ... ... 30.9 ... ...

050306 6.9 6.7 12.6 4.9 5.8 9.1 5.6 7.6 10.4 9.1 13.8 17.3 16.0 28.0 34.0 30.0 74.1 91.8

050318 7.4 9.7 12.5 5.9 7.4 10.3 6.4 9.9 11.8 10.7 17.5 17.9 18.4 33.2 34.1 33.3 63.4 64.5

050319 5.5 6.0 9.6 4.6 4.6 7.2 5.2 6.5 8.4 8.8 11.4 14.4 15.2 21.3 25.1 30.1 34.7 48.3

TABLE X:S3 90% upperlim itson hrssofQ = 8.9 linearly polarized sine-gaussians,in unitsof10� 20 Hz� 1=2;100-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

031108 6.0 ... ... 3.6 ... ... 3.8 ... ... 4.5 ... ... 7.9 ... ... 20.1 ... ...

031109a 4.4 ... ... 2.7 ... ... 2.9 ... ... 3.5 ... ... 6.1 ... ... 15.1 ... ...

031220 5.0 ... ... 3.0 ... ... 3.0 ... ... 4.1 ... ... 7.0 ... ... 15.8 ... ...

TABLE XI:S2 90% upperlim itson hrssofQ = 8.9 linearly polarized sine-gaussians,in unitsof10� 19

Hz� 1=2

;100-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

030217 ... ... 4.0 ... ... 2.0 ... ... 0.94 ... ... 1.5 ... ... 4.1 ... ... 9.5

030226 7.3 3.1 5.3 3.2 1.5 2.1 1.1 0.65 0.62 1.4 1.0 0.85 2.6 2.4 1.4 7.1 6.5 2.7

030320a 6.7 2.3 6.8 2.5 1.3 2.3 0.76 0.67 0.70 1.0 1.2 1.4 1.6 3.6 3.5 4.1 7.2 5.8

030323a 5.3 2.7 5.6 3.0 1.5 2.5 1.2 0.86 1.4 1.8 2.2 3.0 2.7 5.5 7.0 6.4 10.0 12.4

030323b 5.1 1.8 4.9 2.0 0.95 1.7 0.77 0.47 0.79 1.1 0.90 1.6 1.6 1.9 2.5 3.9 3.7 5.0

030324 8.7 ... ... 4.6 ... ... 1.5 ... ... 2.0 ... ... 3.7 ... ... 8.0 ... ...

030325 2.9 1.5 3.4 1.4 0.78 1.6 0.63 0.46 0.90 1.0 1.00 1.9 1.5 1.9 2.9 3.7 4.6 6.6

030326 9.0 3.0 7.4 4.2 1.8 3.1 1.3 0.81 0.98 1.9 1.5 1.8 3.7 3.1 2.9 8.6 6.8 5.7

030329a 4.4 ... ... 2.5 ... ... 1.2 ... ... 2.1 ... ... 3.0 ... ... 8.6 ... ...

030329b 2.6 ... ... 1.2 ... ... 0.34 ... ... 0.56 ... ... 0.94 ... ... 2.2 ... ...

030331 ... 3.5 ... ... 1.7 ... ... 0.97 ... ... 2.1 ... ... 4.1 ... ... 10.3 ...

030405 2.3 1.2 2.6 1.3 0.76 1.1 0.46 0.40 0.47 0.73 0.73 0.90 1.2 1.8 1.9 2.7 4.4 4.0

030406 ... 1.2 ... ... 0.73 ... ... 0.45 ... ... 0.87 ... ... 1.9 ... ... 5.0 ...

030413 ... ... 1.7 ... ... 0.94 ... ... 0.61 ... ... 1.1 ... ... 2.9 ... ... 5.4

030414 1.3 ... ... 0.89 ... ... 0.30 ... ... 0.43 ... ... 0.74 ... ... 1.7 ... ...

24

TABLE XII:S4 90% upperlim itson hrssofQ = 8.9 circularly polarized sine-gaussians,in unitsof10� 21

Hz� 1=2

;100-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

050223 1.7 ... ... 1.3 ... ... 1.5 ... ... 2.4 ... ... 4.4 ... ... 8.3 ... ...

050306 2.0 1.5 2.6 1.5 1.2 1.9 1.7 1.7 2.2 2.8 3.1 3.7 4.9 6.0 7.0 9.3 16.3 19.1

050318 2.2 2.1 2.8 1.7 1.6 2.2 1.9 2.1 2.4 3.0 4.0 4.2 5.5 6.9 7.4 10.3 12.7 14.0

050319 1.7 1.6 2.5 1.4 1.3 1.9 1.6 1.7 2.2 2.6 3.2 3.8 4.7 5.7 6.7 9.1 10.3 12.8

TABLE XIII:S3 90% upperlim itson hrssofQ = 8.9 circularly polarized sine-gaussians,in unitsof10� 21

Hz� 1=2

;100-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

031108 18.4 ... ... 11.5 ... ... 11.8 ... ... 14.0 ... ... 23.2 ... ... 61.0 ... ...

031109a 13.5 ... ... 8.5 ... ... 8.7 ... ... 11.3 ... ... 19.0 ... ... 47.6 ... ...

031220 12.1 ... ... 9.4 ... ... 8.8 ... ... 11.6 ... ... 20.5 ... ... 49.1 ... ...

TABLE XIV:S2 90% upperlim itson hrssofQ = 8.9 circularly polarized sine-gaussians,in unitsof10� 20

Hz� 1=2

;100-m scrosscorrelation length.

100 Hz 150 Hz 250 Hz 554 Hz 1000 Hz 1850 Hz

G RB

date H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1 H1-H2 H1-L1 H2-L1

030226 22.1 9.6 16.7 9.8 4.7 6.1 3.1 2.0 1.9 4.1 3.2 2.6 7.8 7.2 4.5 21.2 19.6 8.0

030320a 21.0 7.4 24.7 7.7 4.1 7.6 2.2 2.2 2.4 3.3 4.0 4.5 5.3 11.4 10.8 12.3 21.7 18.3

030323a 16.7 ... ... 8.8 ... ... 4.0 ... ... 6.3 ... ... 9.3 ... ... 21.8 ... ...

030323b 14.8 4.8 14.4 5.9 2.6 4.8 2.2 1.3 2.4 3.3 2.5 4.4 4.7 5.3 7.4 10.9 10.4 14.1

030324 27.0 ... ... 13.9 ... ... 4.7 ... ... 6.3 ... ... 10.7 ... ... 24.7 ... ...

030325 9.7 3.7 9.9 4.6 2.0 4.5 2.2 1.2 2.8 3.5 2.6 5.4 5.2 5.1 8.4 12.3 12.5 19.1

030326 28.3 11.0 28.6 13.0 6.3 10.9 4.3 2.9 3.7 6.3 5.0 6.2 10.6 10.6 10.2 26.4 23.2 18.3

030329a 13.7 ... ... 7.8 ... ... 3.6 ... ... 5.8 ... ... 9.5 ... ... 24.8 ... ...

030329b 8.1 ... ... 3.3 ... ... 1.0 ... ... 1.7 ... ... 2.8 ... ... 6.7 ... ...

030331 ... 7.4 ... ... 3.7 ... ... 2.1 ... ... 4.9 ... ... 8.6 ... ... 20.6 ...

030405 7.1 3.1 6.8 3.7 1.9 2.9 1.3 1.1 1.2 2.1 1.9 2.3 3.3 5.0 5.2 7.8 11.6 10.4

030406 ... 2.8 ... ... 1.7 ... ... 1.0 ... ... 2.0 ... ... 4.6 ... ... 11.6 ...

030413 ... ... 4.3 ... ... 2.4 ... ... 1.5 ... ... 2.8 ... ... 7.3 ... ... 13.5

030414 4.1 ... ... 2.7 ... ... 0.91 ... ... 1.3 ... ... 2.2 ... ... 5.2 ... ...