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Correlation between Gamma-Ray bursts and Gravitational Waves P. Tricarico∗, A. Ortolan, A. Solaroli and G. Vedovato I.N.F.N.,Laboratori Nazionali di Legnaro, via Romea, 4, I-35020 Legnaro, Padova, Italy L. Baggio, M. Cerdonio, L. Taffarello and J. Zendri Phys. Dept., Univ. of Padova and I.N.F.N. Sez. Padova, Via Marzolo, 8, I-35100 Padova, Italy R. Mezzena, G.A. Prodi and S. Vitale Phys. Dept., Univ. Trento and I.N.F.N. Gruppo Coll. Trento, Sez. Padova, I-38050 Povo, Italy. P. Fortini Phys. Dept., University of Ferrara, and INFN Sez. Ferrara, via Paradiso, 12, I-44100 Ferrara, Italy M. Bonaldi and P. Falferi 1 Centro di Fisica degli Stati Aggregati, I.T.C.-C.N.R.,Trento, Italy and 0 I.N.F.N. Gruppo Coll. Trento, Sez. Padova, I-38050 Povo, Italy. 0 (February 7, 2008) 2 Thecosmologicaloriginofγ-raybursts(GRBs)isnowcommonlyacceptedand,accordingtoseveral n a models for the central engine, GRB sources should also emit at the same time gravitational waves J bursts (GWBs). We have performed two correlation searches between the data of the resonant 5 gravitational wave detector AURIGA and GRB arrival times collected in the BATSE 4B catalog. No correlation was found and an upper limit hRMS ≤1.5×10−18 on the averaged amplitude of 1 gravitational waves associated with γ-ray bursts has been set for the first time. v 2 PACS numbers: 04.80.Nn, 98.70.Rz 2 0 1 I. INTRODUCTION redshifts close to z = 1 [3–6], implies the emission of a 0 hugeamountofelectromagneticradiation: infewseconds 1 Thirty yearsafter their discovery,Gamma Ray Bursts the GRBs release an energy of 1051 1054 erg. On the 0 − / (GRBs) are still the most mysterious objects in the Uni- other hand, this implies that the GRBs are rare events c verse,as their propertieshavenotyetbeen associatedto in a galaxy: in fact, as the rate of GRB events is of the q any well-known object, while for instance pulsars have order1/day, their rateamountsto 1event/106 yearsper - r been almost immediately identified as final objects of galaxy [7]. g the evolutionofstarsandquasarshavebeen wellframed The theoretical framework which gives a coherent ex- : v among galaxy nuclei in particular in the Seyfert class. planation of the experimental data set of GRBs is the Xi Unfortunately “standard” astrophysical objects do not so-called “fireball” model: an “inner engine” produces a exist for GRBs. The “non standard” compact astro- flux of relativistic energy of the order 1052 erg and it is r a physicalobjectswhichmightreproducetheexperimental extremely compact and clearly not visible; this relativis- characteristicofGRBsinvolveblack-holes(BH),neutron tic flux ofparticles is akindof“fireball”able to produce stars (NS) and massive stars. electromagneticradiationinaopticallythinshellofmat- The wide rangeofcharacteristicspreventsthe system- ter. The inner engine works for 1-3 days, producing the atic classification of GRBs. Their energy spectrum is afterglow,butthebulkoftheexplosionlaststypically10 continuous and non-thermal, and covers the range be- s (with peaks around 0.5 and 30 s and a variation of six tween 1 keV and 1 MeV; their emissionlasts from10 ms orders of magnitude, 10−3 103 s) when the major part ÷ to 103 s. The Burst And Transient Source Experiment of the energy is emitted. (BATSE)hasdetectedGRBswithaneventrateofabout The inner engine progenitors are likely to be coalesc- one per day between 1991 and 2000. ing binary systems, such as NS-BH or NS-NS systems, One of the most important results of observations is or single stars that collapse into BH in supernova-like that many GRBs are at cosmological distances [1]. This events (the so-called “collapsar” models [8]). If this is fact, recently confirmed also by the satellite BeppoSAX thecorrectexplanationoftheGRBs,thenwewouldhave [2], which has detected some optical counterpart with compactastrophysicalsystemsforwhichgravityplaysan importantrole,andgravitationalwaveswouldbeemitted [9–11]. This scenario has motivated us to investigate the be- ∗ havior of the gravitational wave detector AURIGA dur- Corresponding author, ing time spans which include the arrival time of a GRB e-mail: [email protected] 1 burst. The AURIGA detector [12] is an Al5056 reso- of the noise correlation of the filtered data. nant bar of about 2.3 tons with a typical noise temper- In what follows, “δ-like signal” means any g.w. signal ature of 7 mK and a bandwidth of about 1 Hz. The which shows a nearly flat Fourier transform at the reso- detector is sensitive to gravitationalwaves(g.w.) signals nance frequencies of the detector (913 and 931 Hz) over over 1 Hz bandwidth around each one of its two reso- a 1Hzbandwidth. Thereforethemetricperturbations ∼ nant frequencies i.e. 913 and 931 Hz. The sensitivity h(t)sensedbytheAURIGAdetectorcanbealargeclass of a g.w. detector can be conveniently expressed by the of short signals (of millisecond duration) including the quantity h , representing the minimum g.w. ampli- latest stable orbits of inspiralling NS-NS or NS-BH, the min tude detectable at SNR = 1, which for the AURIGA subsequent merging and the final ringdown [19] and the detector is h 2 5 10−19. The AURIGA sen- collapsar bursts [11] which could be expected signals as- min ∼ ÷ × sitivity is enough to detect NS-NS, BH-NS and BH-BH sociated with GRBs. mergers which take place within the Galaxy. Previous work on GRB-GWB association appeared in the literature, specifically about a single GRB trigger A. Coincidence search [13], and about a set of GRB triggers [14–16]; here we present an experimental upper limit on such an associa- The coincidence method has been successfully applied tion obtained with the AURIGA-BATSE data. to the search of coincident excitations of different g.w. Theplanofthepaperisasfollows: inSec. IIwediscuss detectors[20,21];muchworkhasbeendevotedtoexploit the two methods, coincidence and statistical searches, its potentialities and to develop robust estimates of the which we use for the association of the GRBs in the background of accidental, even in the presence of non BATSE catalog with GWBs. The results obtained with stationary event rates [17,21]. A g.w. candidate event the twomethods arepresentedin Sec. III. InSec. IV we is a local maximum of the filtered data corresponding analizeourresultsanddiscussthepossibilitiesopenedby to any excitation of the detector. From the AURIGA our methods for future searches. filtered output we extract a list of candidate events setting an adaptive threshold in their signal-to-noise ratio (SNR = 5). The event lists used in this analysis be- II. SEARCH METHODS long to periods of satisfactory performance of AURIGA as described in Ref. [22]. It is worth noticing that the The aim of our analysis is the search of an associa- AURIGA event search checks each event against the ex- tionbetweenGRBandGWB arrivaltimes withinatime pected signal template by means of a χ2 test [23]. The window W. The analysis has been carried out by means event lists contain the information needed to describe a of two different procedures which have been already dis- δ-like signal namely, its time of arrival (in UTC units), cussed in the literature: i) the correlation method [17] the amplitude ofthe Fouriertransform,andthe detector which is based on the coincidence between the arrival noiselevelatthattime. TheBATSEdataaretakenfrom time of candidate gravitational events selected over a the4BcatalogbyMeeganetal. availableonInternet[24] given threshold and the GRB trigger and ii) a statisti- andincludestriggertime(inUTCunits),rightascension cal method [16] which relies on a hypothesis test on a and declination, error box and other information about statistical variable representing the mean energy of the every GRB triggered. gravitationaldetector at the GRB trigger. We label with t(i) the estimate arrivaltime of the i-th A The coincidence window plays an important role in candidateg.w. eventdetectedbyAURIGAandwitht(k) γ boththeanalysis. Infactthecoincidencewindowshould the trigger time of the k-th GRB detected by BATSE. be wide enough to hold the delay distribution between A coincidence between the i-th AURIGA event and the GRB and GWB. We notice that its “optimal” value de- k-th GRB trigger time is observed if t(i) t(k) W, pends on the astrophysical sources and on the g.w. de- | A − γ | ≤ where W is the coincidence window. tectorproperties. IfweassumethattheGRBsaregener- The coincidences found have to be compared with the atedby internalshocks in the fireball,the delaybetween accidental coincidence background due to chance. Two GRB and GWB is less than 1 second [10] but there are standardmethods toevaluatetheprobabilityofacciden- still some uncertainties. Moreover,the delay is widen by tals have been applied to the pair BATSE-AURIGA: i) the cosmological redshift. To be as much conservative performing thousands time-shifts of the arrival time of as possible on the distribution of the delays we choose one detector with respect to the other and looking for W = 5 s. This value turns out to be also consistent accidental coincidences at each shift [20]; ii) assuming with the requirements of the filtering procedures of the independent Poisson distribution of event times and us- AURIGA detector. In fact, the search for g.w. bursts ing the mean measured rates to estimate the accidental requires a filtering of the data by a Wiener-Kolmogorov rate n a filter matched to δ-like signals [18] and its characteristic time is the inverse of the detector bandwidth i.e. 1 s. ∆T ∼ n =N N , (1) Thistime,asweshallseebelow,establishesthetimescale a A γ T 2 whereN isthenumberofAURIGAeventsintheperiod A statistically significant difference between on- and A T,N thenumberofGRBeventsinthesameperiodand off-source populations clearly supports a GWB-GRB as- γ ∆T =2W is the total amplitude of the coincidence win- sociation. For each of the Non GRB trigger we compute dow (see Appendix A for a proof of this basic relation). X(t(γk)), k = 1,...,Non which forms the χon set of on- To avoid the problem of multiple coincidence within the source events, and construct a complementary set χoff same window we prefer to estimate the backgroundwith with Noff off-source events using windows before and af- W = 1 s and to scale our results with the help of Eq. ter the trigger. The sets χon and χoff are samples drawn (1) to W = 5 s. The consistency of the two estimates from the populations whose distributions we denote pon ensures that the arrival times of the AURIGA-BATSE and poff. The off-source events are taken in periods not pairaredistributedasPoissonrandompointsevenifthe correlated with the trigger time, at a distance in time AURIGA eventrate hasbeen foundto be notstationary greaterthan103 secondsfromGRBthetrigger,bothbe- [17,21]. fore and after it; this should be sufficient to have a fair sample of the off-source events as any GRB-GWB association is reasonably excluded. B. Statistical search For windows W greater than the Wiener filter characteristic time, the central limit theorem implies that poff Themethodofthestatisticalsearchhasbeenproposed is a normal distribution. for a pair of interferometric detectors by L. S. Finn and Now suppose that the GWBs fall within the window coworkers[16];herewe slightly modify their approachto W opened around the GRB trigger time and that the the case of a single resonant detector. The data we use SNRofthegravitationalsignalassociatedwiththeGRB are the AURIGA filtered data obtained by means of the (averagedoverthesourcepopulation)issmallerthanone, Wiener filter matched to a δ-like signal, without setting then pon is also a normal distribution with mean any threshold on their amplitudes. If a signal enters the detector at time t0, its output can be written as µon =µoff +E 1 W dt hf(t tγ)2 y(t)=hf(t t )+η(t) , (2) "2W Z−W | − | # 0 − t where f(t t ) is the normalized signal template of the µoff + w E[h2] (W tw) , (6) 0 ≃ 8W ≫ Wienerfilt−er,hitsamplitude(i.e. f(0)=1),t itsarrival (cid:18) (cid:19) 0 time andη(t) is a stochasticprocess with zeromean and where E[ ] is the average over the astrophysical source · correlation population of GRBs. The basic idea behind the statisticalapproachis a hy- η(t),η(t′) =σ2f(t t′). (3) h i h − pothesis testing where the null hypothesis 0 to test is the equivalence of the off-source and on-souHrce distribu- Hereσ2 isthevarianceofthefilteroutputintheabsence h tions: of any signal and f(t) is a superposition of two expo- nentially damped oscillating functions [25] which can be 0 : poff(X)=pon(X). (7) approximately expressed as H Therejectionof clearlysupportsaGWB-GRBassoci- f(t) e−|t|/twcos(ω t)cos(ω t) (ω t 1) , (4) H0 ≈ 0 B 0 w ≫ ation. Sincepon andpoff arenormalandcoulddifferonly in their mean values, we can test by the Student’s where tw is the Wiener filter decaying time (i.e. the in- t-test [26]. H0 verseof the detector bandwidth), ω a center carrierfre- 0 The t statistic is defined from χon and χoff by quency and ω is an amplitude modulation frequency. B Typical values for the AURIGA detector are t 1 s, w ≃ µôn µôff NonNoff ω0 920 Hz and ωB 20 Hz. t= − (8) L≃et us define now t≃he random variable X, which rep- Σ rNon+Noff resents an averaged measure of the energy released by a Σ2 = (Non−1)σô2n+(Noff −1)σô2ff , (9) g.w. signal impinging on the detector at time t0 Non+Noff 2 − X(t )= 1 W dt y(t t )2 , (5) where µôn and µôff (σô2n and σô2ff) are the sample means 0 2W | − 0 | (variances) of χon and χoff, respectively. Z−W The expected value of t averaged on the source popu- where t is the center of the time window. If an associ- lation and on the filtered output of the detector is 0 ation between GRB and gravitational waves exists, the filteredoutputofthegravitationalwavesdetector,inpe- tw E[h2] NonNoff µ =E[t]= , (10) riods just prior the GRB (“on-source” population) will t 8W σ Non+Noff differ(statistically)fromtheoutputatothertimes(“off- (cid:18) (cid:19) r source” population). where σ =E[Σ]. 3 LetusconsidertheupperlimitonE[h2]intheassump- respectively. Thetwoestimatesoftheaccidentalnumber tion is true: in this case the most probable value of of coincidences are in good agreement and demonstrate 0 H E[h2] is zero. From Eq. 10 we get that the two event rates are uncorrelated. We conclude that the 2 coincidence found are due to chance. tw E[h2] Non+Noff µ = t,max 8W σ ≤ NonNoff (cid:18) (cid:19) r B. Statistical search = µt,max 2/Nγ (Non =Noff =Nγ) (11) (cid:26)µt,max/p√Non (Noff ≫Non) We have first tested the method using a Monte Carlo simulation by adding N signals at fixed SNR over a where µ is the upper limit. Assuming true and on t,max 0 H gaussiannoisegeneratedby meansofanoisemodelwith Noff Non, we have ≫ the sameparametersoftheAURIGA detector. Thesim- 8W µ ulateddetectoroutputisthenfedtothesamedatafilter- E[h2] h2 = t,max σ ; (12) ≤ max (cid:18) tw (cid:19) √Non iWngephraovceedgeunreesrautseedd1f2o0rtδh-leikAeUsiRgInGalAs wexitpherSimNRent3a,l2daatnad. the value of µ can be deduced by the selected con- 1 and 1000 with SNR 1 storing their true arrival times t,max fidence level. In this way we are able to set an upper t0. The signals have been superimposed to stationary limit on the average amplitude of gravitational signals gaussian noise and we have then formed the on and off associated with GRBs populations and calculated the t value. The probability P(t)thatthetvalueobtainedisduetochanceisreported −1 h2 1.4 10−18 2 W tw µt,max in Table I. For signal with SNR=3 and SNR=2 this RMS ≤ × 5s 1s 1.96 × probability is very small and the value of t is statisti- (cid:18) (cid:19) (cid:2) Non −1/2(cid:3) σ . (13) cthaellyotshigenrifihcaanndt,,fsohrowsiignngalas GwRithB-SGNWRB=ass1o,cwiaetihoanv.eOtno × 100 [5 10−19]2 increase the number of GRBs to 1000 to get a statistical (cid:18) (cid:19) × significance. Theχoff andχon setsrelativetotheAURIGA-BATSE data between 1997 and 1998 are given in the histograms III. RESULTS ofFig. 2andFig. 3,wherethenongaussiantailsaredue to the nonstationarityof the AURIGA noise. The value TheAURIGAdatausedinthetwoanalysisarerelative of the Student’s t-test obtained is t =0.58, which corre- to the years 1997 and 1998, and the number of GRBs sponds to a probability of 0.28 that it is due to chance. which fall into the AURIGA data taking periods is 120. As we conclude that there is no evidence of an association of GRB-GWB, we can put a constraint on gravitational radiation emitted from GRBs averaged over A. Coincidence search the source population, h . The value of µ can RMS t,max be found setting a confidence level C.L. and solving the Withinthewindowof5secondswehavefound2events equation: in coincidence. This experimental result has to be com- µt,max pared with the number of coincidences due to chance. f (t)dt=C.L. , (14) d Theshifts methodconsistsof104 time shifts oftheco- Z−∞ incidence window; for each one we compute the number where f (t) is the distribution function for the Student’s of coincidences; if the candidate GWB arrival times and d t withddegreesoffreedom. Notice thatforhighd,f (t) the GRB triggers can both be modeled as Poisson ran- d tendstoanormalcurvewithzeromeanandunitaryvari- dom points [27], the number of accidental coincidences ance. IfwechooseC.L.=95%andsetd=Non+Noff 2, is fitted to a Poisson curve; from the fit we get the ex- − we get µ =1.65. Using Eq. 13, we set the following pected number of coincidences due to chance. The re- t,max upper limit on the averaged g.w. amplitude associated sults are summarized in Fig. 1. From the fit we ob- with GRBs: tain a value of mean expected accidental coincidences of naA=no2t.h57er±a0p.0p4ro.ach to evaluate the number of acci- hRMS ≤1.5×10−18. (15) dental coincidences is given by Eq. 1, that holds in case of Poisson random points [27]. The total number IV. DISCUSSION of AURIGA events and GRB triggers are respectively N =26816, N =120, assuming ∆T =2W =10 s and A γ T =1.32 107 s we get n =2.4 0.2. The error on n Thetwosearchmethodsreportedinthispapershowno a a × ± can be easily estimated assuming the Poisson statistics evidenceofacorrelationbetweenγ-rayburstsandgravi- for the fluctuations on N and N i.e. √N and N tational waves: First, the two coincidences found with A γ A γ p 4 the coincidence search have no statistical significance P ( k ,t )=e−nta/T(nta/T)ka. (A1) and can be reconduced to chance. Next, the statistical { a a} k ! a method doesn’t lead to a statistically significative value For m not overlapping windows, it can be demonstrated of the Student’s t. However we were able to put an up- that in the limit of n , T and n/T constant, perlimitongravitationalsignalsassociatedtotheGRBs → ∞ → ∞ averagedoverthesourcepopulation,h 1.5 10−18. the probability of k1 points in t1 , ..., km points in RMS { } { ≤ × t is [27] Theexistenceofburst-likeexcitations(usuallyreferred m } as non-gaussian noise) in the AURIGA data is well known, and here we deal with this noise selecting the P ( k ,t ,..., k ,t ) = m e−nti/T(nti/T)ki 1 1 m m periodsoftimewhenthedetectorwasoperatinginasat- { } { } ki! i=1 isfactory way. The vetoing procedure of data dominated Y m bynongaussiannoisehasbeensetbytheAURIGAdata = P ( k ,t ) (A2) i i analysis [28] and the resulting duty cycle is about 40 % { } i=1 of the total operating time. Care was also taken to cope Y with non stationarities of the AURIGA noise that give showing that the events ki points in ti and kj points { } { rise to the non gaussian tails in Fig. 2 and Fig. 3 ex- in tj are independent for every i and j. Substituting } cluding such tails from the fits we used to estimate the m=Nγ, ti =2W =∆T and n=NA we get statistical variable t. However, it is important to notice that even with the above limitations the analysis P {k1,∆T},...,{kNγ,∆T} = has reacheda levelofsensitivity whichis astrophysically (cid:0) N ∆T k Nγ(cid:1) 1 of some interest. In order to get better upper limits we =e−NγNA∆T/T A . (A3) T k ! havealsoexploredthepossibilityofusinginouranalysis (cid:18) (cid:19) i=1 i Y the incoming direction of GRBs but we have found no significant improvements (see Appendix B). We can now evaluate the probability P(k) to have k To increase the confidence in our results we have ap- coincidences, that can be obtained taking into account, plied the non-parametric Mann-Whitney u-test [29] to at fixed k, all the possible sets [ki]={k1,···,kNγ} with the sample sets of the statistical search, obtaining a sec- the constrain Nγ k =k; we get i=1 i ond confirmation of the null hypothesis. Ranking the elements of the union set χoff χon in increasing order, P(k)= P P k ,∆T ,..., k ,∆T we get a statisticalparameter z⊕=1.59,smaller than the { 1 } { Nγ } critic value z =1.95 imposed by the fixed C.L. of 95 %. X[ki] (cid:0) (cid:1) The upper limit we have set can be improved in the =e−NγNA∆T/T NA∆T k Nγ 1 future simply by the increasing of common data taking T k ! periods of AURIGA (N increases) and the new exper- (cid:18) (cid:19) X[ki]iY=1 i on iment HETE-II (High Energy Transient Explorer [30]) N ∆T k Nk =e−NγNA∆T/T A γ . (A4) which is going to substitute by now the wasted BATSE T k! satellite inthe GRB search. Another possibility to reach (cid:18) (cid:19) more astrophysically interesting sensitivities is the up- Rearranging the factors in Eq. A4 we get a Poisson dis- grade of the AURIGA detector which is now in progress tribution as in Eq. 1. The last equality can be easily [31]. The predicted sensitivity and bandwidth of the demonstrated using the multinomial expansion relation AURIGA detector equipped with the new read out sys- [32] temwouldberespectivelyh 8 10−20andt−1 10 Hz. This sensitivity, togethmeirn w≈ith×an enhancemwen≈t of N k N xki noise stationarity and duty cycle of the detector, would x =k! i (A5) i k ! correspond to the lowering of the upper limit hRMS of Xi=1 ! X[ki]Yi=1 i about 2 order of magnitude in one year of correlation analysis. and substituting xi =1, i=1 N. ··· APPENDIX A: APPENDIX B: To derive Eq. 1 let us consider n points random As the incomingdirectionofGRBs is known,one may distributed in a time interval [0,T]. The probability wonder if a selection of the GRBs based on a cutoff on P ( k ,t ) that k points lies in the time window t = theAURIGAantennapattern,averagedontheunknown a a a a t { t is}given by the binomial distribution [27] with polarizations of the GWB, could increase the sensitivity 2 1 pro−bability p = t /T that a single point lie in t . If of our analysis. a a n 1 and t T, using the Poissontheorem we get Theantennapatternofaresonantbardetectorisgiven a ≫ ≪ by 5 F(θ)=1 kˆ ˆz 2 =sin2(θ) , (B1) − · (cid:16) (cid:17) where kˆ and ˆz are unitary vectors parallel to the GRB 140 direction and the antenna bar axis and θ is the angle betweenkândˆz. Sourcesthatfalloutsidethetwocones, 120 which are defined by the equation kˆ ˆz cos(ξ), have a figure pattern F(θ) sin2(ξ) and th·ere≥fore the average 100 ≥ energy associated with these g.w. sources is 15 dΩ s 80 Eξ[h2] F2(θ) nt ∝ 16 4π u ZΩξ Co 60 75 1 1 = sin(ξ)+ sin(3ξ)+ sin(5ξ) , (B2) 64 6 50 (cid:20) (cid:21) 40 where the solid angle Ω is defined by F(θ) F(ξ). ξ ≥ Moreover,asGRBs areisotropicallydistributedoverthe 20 sky, the cutoff on the figure pattern decreases the number of available GRB: Nξ = N sin(ξ). Therefore the on on net effect of this cutoff on the expected value of t in Eq. 0 0 1 2 3 10 is Square strain amplitude (h2x1036) 75 1 1 FIG.2. Gaussian fit oftheoff-sourceset ofthestatistical µξt =µt64 sin(ξ)+ 6sin(3ξ)+ 50sin(5ξ) × search. (cid:20) (cid:21) sin(ξ)1/2 . (B3) TABLEI. ResultsofaMonteCarlosimulation ofthesta- × tistical search. The column P(t) is the probability that the The function µξ/µ is a continuously increasing function estimated t is dueto chance. t t in the range ξ [0,π/2], and µξ/µ = 1 at ξ = π/2 (i.e. SNRof thegenerated signals Non t P(t) ∈ t t the whole solid angle). We must conclude that a cutoff ontheGRBdirectiondoesnotenhancethesensitivityof 3 120 8.1 <10−9 the statistical search. 2 120 3.4 4×10−4 1 120 0.4 3×10−1 1 1000 3.6 10−4 4 10 [1] C. A.Meegan et al., Nature355, 143 (1992). [2] E. Costa et al.,Nature387, 783 (1997). [3] S. Kulkarniet al., Nature393, 35 (1998). 3 10 [4] M. R.Metzger et al.,Nature 387, 878 (1997). [5] S. G. Djorgovski et al.,GCN notice 139 (1998). [6] P. M. Vreeswijk et al.,astro-ph/9904286 (1999). s [7] T. Piran, Physics Reports 314, 575 (1999). unt102 [8] S.E.Woosley,inProceedingsoftheFifthHuntsvilleCon- o C ference on Gamma-Ray Bursts, editedbyR.M.Kippen, R. S. Mallozzi, and V. Connaughton (AIP, Conf. Proc., 2000), Vol. 526, pp.555–564, astro-ph/9912484. [9] C. S. Kochanek and T. Piran, Astrophys. J. 417, L17 10 (1993). [10] P.MeszarosandM.J.Rees,Astrophys.J.Lett.430,L93 1 TABLEII. Resultsofthefitsontheχoff andχon setswith 0 1 2 3 4 5 a gaussian distribution. Number of accidental coincidences N µ σ FIG. 1. Poisson fit of the data obtained with the shifts method. off 2206 (0.97±0.02)×10−36 (0.37±0.02)×10−36 on 120 (0.99±0.06)×10−36 (0.33±0.06)×10−36 6 tic Processes, 3rd ed.(Mc Graw Hill, Singapore, 1991). [28] L. Baggio et al., Int.J. Mod. Phys.D 9, 251 (2000). [29] H. B. Mann and D. R. Whitney, Ann. Math. Stat. 18, 50 (1947). [30] Internet: http://space.mit.edu/HETE. [31] J. P. Zendri et al., in Proceedings of the XIV SIGRAV 10 CongressonGeneralRelativityandGravitationalPhysics (World Scientific, Singapore, 2000), in press. [32] M. Abramowitz and I. E. Stegun, Handbook of mathe- matical functions (Dover, New York,1972), p. 823. s nt [33] J. Weberand B. Radak,NuovoCimento 111, (1996). u o C 5 0 0 1 2 3 Square strain amplitude (h2x1036) FIG.3. Gaussian fit of theon-source set of thestatistical search. (1994). [11] C. L. Fryer, S. E. Woosley, and A. Heger, Astrophys. J. (2001), accepted and in press, astro-ph/0007176. [12] G. A. Prodi et al., in Proceedings of the Second Edoardo Amaldi conference on gravitational waves, edited by E. Coccia, G. Pizzella, and G. Veneziano (World Scientific, Singapore, 1998), Vol. 4, pp.148–158. [13] L. Amatiet al.,Astron. Astrophys.138, 605 (1999). [14] P.Astoneetal.,Astron.Astrophys.Suppl.Ser.138,603 (1999). [15] G. Modestino et al., in Frontier Objects in Astrophysics andParticlePhysics,editedbyGiovannelliandMannoc- chi(World Scientific,Singapore, 1998), p.295. [16] L.S.Finn,S.D.Mohanty,andJ.D.Romano,Phys.Rev. D 60, (1999), gr-qc/9903101. [17] G. A.Prodi et al.,Int.J. Mod. Phys. D 9, 237 (2000). [18] A. Ortolan et al., in Proceedings of the Second Edoardo Amaldi conference on gravitational waves, edited by E. Coccia, G. Pizzella, and G. Veneziano (World Scientific, Singapore, 1998), Vol. 4, pp.204–215. [19] E.E.FlanaganandS.A.Hughes,Phys.Rev.D57,4535 (1998). [20] E. Amaldi et al.,Astron. Astrophys.216, 325 (1989). [21] Z. A. Allen et al. Phys. Rev. Lett. 85, 5046 (2000), astro-ph/0007308. [22] L. Baggio et al., Nucl. Phys. B (Proc. Suppl.) 70, 537 (1999). [23] L. Baggio et al.,Phys. Rev.D 61, 102001 (2000). [24] C. A. Meegan et al., internet preprint at http://www.batse.msfc.nasa.gov/. [25] V.C. Visconti et al.,Phys. Rev.D 57, (1998). [26] G. W. Snedecor, Statistical methods, 5th ed. (The Iowa StateUniversity Press, Iowa, 1962). [27] A.Papoulis,Probability,RandomVariables,andStochas- 7