The currently accepted model for gamma-ray burst phenomena involves the
violent formation of a rapidly rotating solar-mass black hole. Gravitational
waves should be associated with the black-hole formation, and their detection
would permit this model to be tested. Even upper limits on the
gravitational-wave strength associated with gamma-ray bursts could constrain
the gamma-ray burst model. This requires joint observations of gamma-ray burst
events with gravitational and gamma-ray detectors. Here we examine how the
quality of an upper limit on the gravitational-wave strength associated with
gamma-ray bursts depends on the relative orientation of the gamma-ray-burst and
gravitational-wave detectors, and apply our results to the particular case of
the Swift Burst-Alert Telescope (BAT) and the LIGO gravitational-wave
detectors. A result of this investigation is a science-based ``figure of
merit'' that can be used, together with other mission constraints, to optimize
the pointing of the Swift telescope for the detection of gravitational waves
associated with gamma-ray bursts.Comment: iop style, 1 figure, 6 pages, presented at GWDAW 200

Finn, Lee Samuel

Krishnan, Badri

Sutton, Patrick J.

English

arXiv

The currently accepted model for gamma-ray burst phenomena involves the violent formation of a rapidly rotating solar-mass black hole. Gravitational waves should be associated with the black-hole formation, and their detection would permit this model to be tested. Even upper limits on the gravitational-wave strength associated with gamma-ray bursts could constrain the gamma-ray burst model. This requires joint observations of gamma-ray burst events with gravitational and gamma-ray detectors. Here we examine how the quality of an upper limit on the gravitational-wave strength associated with gamma-ray bursts depends on the relative orientation of the gamma-ray-burst and gravitational-wave detectors, and apply our results to the particular case of the Swift Burst-Alert Telescope (BAT) and the LIGO gravitational-wave detectors. A result of this investigation is a science-based ``figure of merit'' that can be used, together with other mission constraints, to optimize the pointing of the Swift telescope for the detection of gravitational waves associated with gamma-ray bursts

Finn, L.

Krishnan, B.

Sutton, P.

MPG.PuRe

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITYClass. Quantum Grav. 20 (2003) S815–S820 PII: S0264-9381(03)62487-1Swift pointing and gravitational-wave bursts fromgamma-ray burst eventsPatrick J Sutton1,2,3, Lee Samuel Finn1,2,3,4 and Badri Krishnan1,3,51 Center for Gravitational Wave Physics, The Pennsylvania State University, University Park,PA 16802, USA2 Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA3 Center for Gravitational Physics and Geometry, The Pennsylvania State University,University Park, PA-16802, USA4 Department of Astronomy and Astrophysics, The Pennsylvania State University,University Park, PA 16802, USA5 Max Planck Institut fu¨r Gravitationphysik, Am Mu¨hlenberg 1, 14476 Golm, GermanyE-mail: psutton@gravity.psu.edu, lsf@gravity.psu.edu and badkri@aei.mpg.deReceived 23 April 2003Published 18 August 2003Online at stacks.iop.org/CQG/20/S815AbstractThe currently accepted model for gamma-ray burst phenomena involves theviolent formation of a rapidly rotating solar-mass black hole. Gravitationalwaves should be associated with the black-hole formation, and their detectionwould permit this model to be tested. Even upper limits on the gravitational-wave strength associated with gamma-ray bursts could constrain the gamma-rayburst model. This requires joint observations of gamma-ray burst events withgravitational and gamma-ray detectors. Here we examine how the qualityof an upper limit on the gravitational-wave strength associated with gamma-ray bursts depends on the relative orientation of the gamma-ray-burst andgravitational-wave detectors, and apply our results to the particular case of theSwift Burst-Alert Telescope (BAT) and the LIGO gravitational-wave detectors.A result of this investigation is a science-based ‘figure of merit’ that can be used,together with other mission constraints, to optimize the pointing of the Swifttelescope for the detection of gravitational waves associated with gamma-raybursts.PACS numbers: 98.70.Rz, 04.80.Nn1. IntroductionThe currently accepted model for gamma-ray burst phenomena involves the violent formationof a rapidly rotating approximately solar-mass black hole surrounded by a similarly massivedebris torus [1, 2]. A gravitational-wave burst is likely to be associated with the formation ofthis ‘central engine’, and the observation of such a gravitational-wave burst may reveal details0264-9381/03/170815+06$30.00 © 2003 IOP Publishing Ltd Printed in the UK S815S816 P J Sutton et alof the central engine that cannot be revealed through observations of the gamma rays alone.In this paper we examine how joint observations from the LIGO gravitational-wave detectors[3] and the Swift gamma-ray burst satellite [4] can be used to detect or place upper limits ongravitational-wave emission by gamma-ray burst events.Finn et al [5] have described how the cross-correlated output of two gravitational-wavedetectors, taken in coincidence with gamma-ray burst (GRB) events, can be used to detector place upper limits on the emission of gravitational-wave bursts (GWBs) by GRBs. Finnet al estimate that 1000 GRB observations combined with observations from the initial LIGOdetectors could produce an upper limit on the gravitational-wave strain associated with GRBsof approximately hRMS  1.7 × 10−22 at 95% confidence.In their original work Finn et al assumed that GRBs would be detected isotropically; i.e.,that the GRB detector had an isotropic antenna pattern. They did note, however, that the Swiftsatellite [4], a next-generation multi-wavelength satellite dedicated to the study of GRBs, doesnot have an isotropic antenna pattern and that this has potentially important consequences forthe ability of the combined GRB/GWB detector array to detect or limit the gravitational-waveflux on Earth owing to GRBs. Here we study this question specifically in the context of theSwift satellite and the LIGO gravitational-wave detectors; i.e., we determine, as a function ofSwift’s pointing, the sensitivity of the Swift/LIGO detector array to gravitational waves fromGRBs, and propose a figure of merit that can be used in Swift mission scheduling to optimizethe sensitivity of the Swift/LIGO array to the gravitational-wave flux from GRBs. We findthat the upper limit that can be placed on hRMS differs by a factor of 2 between best and worstorientations of the satellite.In section 2 we review the analysis of Finn et al [5], focusing on the statistic that is used todetect or bound the gravitational-wave strength associated with gamma-ray bursts. In section 3we extract the direction dependence of this statistic and evaluate it specifically for the caseof the LIGO detectors and the Swift Burst Alert Telescope. We conclude with some briefremarks in section 4.2. Observing a GRB–GWB associationIn this section we briefly review the analysis presented in [5].Consider a set of N GRB detections, each characterized by the direction to the sourceˆk and the arrival time τk of the burst at the Earth’s barycentre. A plane gravitational waveincident on the detector pair from the direction ˆk will lead to correlated detector responseswith a time lag equal totk = t (2)k − t (1)k , (1)where t (i)k is the arrival time of the burst at detector i, which depends only on τk, ˆk and thedetector location.Let si(t) be the calibrated output of gravitational-wave detector Di , which we assume toconsist of detector noise ni(t) and possibly a gravitational-wave signal hi(t) produced by theGRB source:si(t) = ni(t) + hi(t). (2)Finn et al [5] defineS( ˆk, τk) = 〈s1, s2〉 :=∫ T0dt∫ T0dt ′s1(t(1)k − t)Q(t − t ′)s2(t(2)k − t ′), (3)as a measure of the cross-correlation of the two detectors corresponding to the GRBcharacterized by ( ˆk, τk). Here Q is a freely specifiable symmetric filter function, and TSwift pointing and gravitational-wave bursts from gamma-ray burst events S817is chosen large enough to encompass the range of possible times by which the gravitationalwaves from a GRB event may precede the gamma rays, which is typically thought to be oforder 1 s for GRBs produced by internal shocks and 100 s for GRBs produced by externalshocks [6, 7].The presence of a GWB in the data will increase the mean value of the cross-correlation(3) over its mean value when no GWBs are present. Finn et al [5] showed that this effectcan be used to detect or place upper limits on the gravitational-wave strain associated withGRBs by comparing the mean value Son of the cross-correlation statistic for a set of GRBobservations to the mean cross-correlation Soff computed using random ( ˆ, τ) not associatedwith any GRB. The expected difference between Son and Soff is simply the contribution due toGWBs from the GRB events; using (2), we haveSon − Soff = 〈h1, h2〉 =∫ T0dt∫ T0dt ′Q(t − t ′)h1(t(1)k − t)h2(t(2)k − t ′), (4)where the overline denotes an average over the population of GRBs. Since the right-hand sideof (4) is quadratic in the gravitational-wave strain, a measurement of or limit on the differencein cross-correlations leads directly to an estimate of or upper limit on the gravitational-wavestrain associated with GRBs.Here we are not interested in the absolute value of the upper limit that can be achieved,but rather in how that upper limit varies according to the relative orientation of the GWB/GRBdetector array. We shall extract this dependence in the next section.3. Direction dependence of the upper limitThe analysis described in [5] involved two gravitational-wave detectors. It assumed forsimplicity that these two detectors had identical isotropic antenna patterns and that each wassensitive to exactly the same gravitational-wave polarization. Here we relax all of theseapproximations; i.e., we properly account for the position and orientation of the gravitational-wave detectors, focusing particularly on the two LIGO detectors on the Earth, and thedependence of their sensitivity to the direction to the GRB source. Our result is an expressionfor the dependence of the upper limit on the population-averaged gravitational-wave strength〈h1, h2〉 as a function of the distribution of detected GRBs on the sky. We also combine thisresult with the directional sensitivity of the Swift detector to determine the dependence onSwift pointing of the upper limit on 〈h1, h2〉 that can be set by joint LIGO/Swift observations.The gravitational-wave contribution hi to the output of the ith LIGO detector is a linearfunction of the physical gravitational-wave strain hab(t, x),hi(t) = hab(t, xi) dabi , (5)where xi is the gravitational-wave detector’s location and dabi is the detector response function.For interferometer i with arms pointing in the directions given by unit vectors ˆXi, ˆYi , the latterisdabi = 12(Xai Xbi − Y ai Y bi). (6)It is convenient to decompose the gravitational wave into its two polarization states,hab(t) = h+(t)+ab( ˆ) + h×(t)×ab( ˆ). (7)See [8] for one choice of the polarization tensors Aab( ˆ). Lacking any detailed model for thegravitational waves that may be produced in a GRB event, we assume that (i) the waves haveequal power in the two polarizations (h+(t)h+(t ′) = h×(t)h×(t ′)) and (ii) the two polarizationsare uncorrelated (h+(t)h×(t ′) = 0).S818 P J Sutton et alWe focus attention now on the mean gravitational-wave contribution 〈h1, h2〉 to the cross-correlation statistic (3). For GWBs arriving from the direction ˆ, it can be shown that〈h1, h2〉 = ρGWB( ˆ|d1, d2)∫ T0dt∫ T0dt ′Q(t − t ′)h+(t)h+(t ′), (8)whereρGWB( ˆ|d1, d2) ≡∑A=+,×dab1 Aab(ˆ)dcd2 Acd(ˆ) (9)describes the direction dependence of the sensitivity of the gravitational-wave detector pair tothe GWB.To complete the evaluation of 〈h1, h2〉 consider the fraction of GRB detections thatarise from different patches on the sky. Since the intrinsic GRB population is isotropic, thedistribution of detection on the sky depends entirely on the directional sensitivity of the GRBdetector. Let the fraction of GRB detections in a sky patch of area d2 ˆ be given byρGRB( ˆ| ˆ′, nˆ′) d2 ˆ, (10)where ˆ′ is the direction in which the GRB detector is pointed, and nˆ′ describes the rotationof the satellite about its pointing direction6. It can then be shown that the upper limiton the squared gravitational-wave strain averaged over the observed GRB population whenthe orientation of GWB and the GRB detectors are given by ( ˆ′, nˆ′, d1, d2) is inverselyproportional toζ( ˆ′, nˆ′, d1, d2) =∫d2 ˆρGRB( ˆ| ˆ′, nˆ′)ρGWB( ˆ|d1, d2). (11)Clearly ζ can be regarded as a figure of merit that describes how capable the gravitational-wave/gamma-ray burst detector combination is at identifying GWBs associated with GRBsas a function of the detector orientations. This figure of merit may be normalized to have amaximum of unity; however, regardless of the normalization[ζ( ˆ′, nˆ′, d1, d2)/ζ( ˆ′′, nˆ′′, d1, d2)]−1 (12)is the ratio of the upper limits on the squared gravitational-wave amplitude that can be attainedby orienting the GRB satellite as ( ˆ′, nˆ′) versus ( ˆ′′, nˆ′′). To the extent that, e.g., the GRBdetector orientation ( ˆ′, nˆ′) can be manipulated on orbit, choosing orientations that maximizeζ will lead to larger signal contributions (4) to the cross-correlation and thus more sensitivemeasurements of the gravitational-wave strength associated with GRBs.Let us now consider the special case of the Burst Alert Telescope (BAT) on the Swiftsatellite [4] and the LIGO gravitational-wave detectors [3]. The BAT is a wide field-of-viewcoded-aperture gamma-ray imager that will detect and locate GRBs with arc-minute positionalaccuracy. Its sensitivity to GRBs depends on the angle λ between the line of sight to the GRBand the BAT axis, as well as the rotational orientation of the satellite about the BAT axis. TheBAT sensitivity averaged over the azimuthal angle as a function of λ has been evaluated bythe BAT instrument team. It has the approximate form7ρSwift ={2 cos λ − 1 + 0.077 sin [13(1 − cos λ)] λ ∈ [0, π/3],0 otherwise. (13)For the purposes of illustration we will use this azimuthal-angle averaged expression for theBAT sensitivity.6 The detailed form of ρGRB depends on the detailed construction of the GRB detector. We consider the case of theSwift Burst Alert Telescope here.7 See the BAT section of the Swift homepage, http://swift.gsfc.nasa.gov/science/instruments/bat.html.Swift pointing and gravitational-wave bursts from gamma-ray burst events S819–180 –120 –60 0 60 120 180–90–60–300306090Degrees East LongitudeDegrees North Latitude+x10.90.80.70.60.50.40.30.30.310.90.80.70.60.50.4Figure 1. Figure of merit ζ (11) for Swift pointing in Earth-based coordinates, produced byconvolving the LIGO sensitivity pattern ρGWB (9) with the Swift sensitivity function ρSwift (13).The figure of merit is nowhere zero, having a range of [0.25, 1.00] and an all-sky average of 0.56.The + and × mark the locations of the LIGO Livingston and LIGO Hanford detectors.Convolving ρGWB (9) for the LIGO detector array (see [9]) with the Swift sensitivityfunction ρSwift as in equation (11) gives the figure of merit ζ for the Swift pointing, which isshown in figure 1. The figure of merit is nowhere zero, varying by a factor of approximately 4between best (near zenith of detectors) and worst (near planes of detectors at 45◦ from arms)orientations of Swift; this translates into a factor 2 difference in amplitude sensitivity. Theall-sky average of the figure of merit is 0.56 times the maximum value.4. ConclusionIn this paper, we have evaluated how the quality of an upper limit on the gravitational-wavestrength associated with gamma-ray burst observations depends on the relative orientation ofthe gamma-ray burst and gravitational-wave detectors, with particular application to the SwiftBurst-Alert Telescope (BAT) and the LIGO gravitational-wave detectors. Setting aside otherphysical and science constraints on the Swift mission, careful choice of BAT pointing leadsto an upper limit on the observed GRB population-averaged mean-square gravitational-wavestrength a factor of two lower than the upper limit resulting from pointing that does not takethis science into account.AcknowledgmentsWe are grateful to Margaret Chester, Shiho Kobayashi and John Nousek for useful discussions.LSF and PJS acknowledge NSF grant PHY 00-99559. LSF, PJS and BK acknowledge theCenter for Gravitational Wave Physics, which is supported by the NSF under cooperativeagreement PHY 01-14375, PJS acknowledges the support of the Natural Sciences andEngineering Research Council of Canada and BK acknowledges the support of the AlbertEinstein Institut.S820 P J Sutton et alReferences[1] Me´sza´ros P and Rees M J 1993 Astrophys. J. 405 278[2] Rees M J and Me´sza´ros P 1994 Astrophys. J. 430 L93[3] Sigg D 2001 Class. Quantum Grav. 19 1429[4] Webpage http://swift.gsfc.nasa.gov/[5] Finn L S, Mohanty S D and Romano J D 1999 Phys. Rev. D 60 121101[6] Sari R and Piran T 1997 Astrophys. J. 485 270[7] Kobayashi S, Piran T and Sari R 1997 Astrophys. J. 490 92[8] Allen B and Romano J D 1999 Phys. Rev. D 59 102001[9] Althouse W E, Hand S D, Jones L K, Lazzarini A and Weiss R LIGO-P000006-C-E

Swift Pointing and Gravitational-Wave Bursts from Gamma-Ray Burst Events

The currently accepted model for gamma-ray burst phenomena involves the violent formation of a rapidly rotating solar-mass black hole. Gravitational waves should be associated with the black-hole formation, and their detection would permit this model to be tested. Even upper limits on the gravitational-wave strength associated with gamma-ray bursts could constrain the gamma-ray burst model. This requires joint observations of gamma-ray burst events with gravitational and gamma-ray detectors. Here we examine how the quality of an upper limit on the gravitational-wave strength associated with gamma-ray bursts depends on the relative orientation of the gamma-ray-burst and gravitational-wave detectors, and apply our results to the particular case of the Swift Burst-Alert Telescope (BAT) and the LIGO gravitational-wave detectors. A result of this investigation is a science-based 'figure of merit' that can be used, together with other mission constraints, to optimize the pointing of the Swift telescope for the detection of gravitational waves associated with gamma-ray bursts

Swift Pointing and Gravitational-Wave Bursts from Gamma-Ray Burst Events

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