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Search for Gravitational Waves from Eccentric Binary Black Holes

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Title:
Search for Gravitational Waves from Eccentric Binary Black Holes
Creator:
Tiwari, Vaibhav
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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Language:
english
Physical Description:
1 online resource (139 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
KLIMENKO,SERGUEI GRIGORIEVICH
Committee Co-Chair:
TANNER,DAVID B
Committee Members:
WHITING,BERNARD F
MITSELMAKHER,GUENAKH
PILYUGIN,SERGEI S
Graduation Date:
5/2/2015

Subjects

Subjects / Keywords:
Average linear density ( jstor )
Black holes ( jstor )
Channel noise ( jstor )
Chirp ( jstor )
Gravitational waves ( jstor )
Information search ( jstor )
Laser interferometer gravitational wave observatory ( jstor )
Pixels ( jstor )
Signals ( jstor )
Waveforms ( jstor )
Physics -- Dissertations, Academic -- UF
binary -- eccentric -- gravitational -- ligo -- waves
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
The existence of gravitational waves (GW), which are ripples in space-time, is predicted by Einstein's theory of general relativity. Several interferometric detectors, such as LIGO and Virgo, have been constructed to detect gravitational-waves producedfrom violent astrophysical phenomena. The coalescence of compact binaries is one of the most promising source for the first detection of gravitational waves. So far, searches have been focused on the detection on circular binaries as it is believed the orbit of a binary circularizes by the time the emitted gravitational radiation enter LIGO/Virgo's sensitivity range. However, binaries with eccentric orbits could also be produced through channels, like dynamical interaction in galactic nuclei, when a significant fraction of them may maintain high eccentricities throughout their lifetime. These binaries have unique gravitational wave signatures and are not captured efficiently by searches designed for circular systems. The search was conducted for gravitational waves produced from merging eccentric binaries (eBBH) in the total mass range of 5 solar masses to 25 solar masses . The search was conducted on the data recorded by the LIGO/VIRGO detectors between May 2005 and October 2010. The search was performed using the Coherent WaveBurst algorithm which utilizes a likelihood approach to reconstruct GW events from data collected by multiple GW detectors. The background was estimated by performing the time-shift analysis, which ensures the absence of any GW events. For the reduction of background, model-based constraints were applied at the post-production stage (loose circular polarization constraint and reconstructed chirp mass constraint). No candidates were found that can be claimed as detection, and upper limits were estimated. Simulation studies were performed using eccentric binary waveforms for the estimation of upper rate limit density. I present the results of the search, including the sensitive volume and upper limits on the source rate density as a function of component mass. The upper limit on the source rate density, averaged over the component-mass plane, was measured to be 3.6 Myr-1 Mpc-3 . I developed and conducted the end to end analysis which involved organizational, data processing and post production stages. During the evolution of the search I made significant contribution in the development of regression analysis, event generator and tool for the reconstruction of chirp mass. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
Local:
Adviser: KLIMENKO,SERGUEI GRIGORIEVICH.
Local:
Co-adviser: TANNER,DAVID B.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-05-31
Statement of Responsibility:
by Vaibhav Tiwari.

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Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
5/31/2017
Classification:
LD1780 2015 ( lcc )

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SEARCHFORGRAVITATIONALWAVESFROMECCENTRICBINARYBLACKHOLESByVAIBHAVTIWARIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2015

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c2015VaibhavTiwari 2

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DedicatedtoSrilaPrabhupadaandhisfollowers. 3

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ACKNOWLEDGMENTS Firstandforemost,thisworkwouldnothavebeenpossiblewithouttheconstantguidanceofmyadvisorProf.SergeyKlimenko.Fromwordsofwisdomtopracticaladvices,IamgratefulthatIlearnedsomuchunderhisguidance.IamgratefultomyPhDcommittee,includingProf.GuenakhMitselmakher,Prof.BernardWhiting,Prof.DavidTannerandProf.SergeiPilyugin,forvolunteeringtobethepartofthecommitteeandofferingmevaluableadvice.IfeelindebtedtomyItalianfriends,GiulioMazzolo,GabrieleVedovatoandMarcoDrago,andtoValentinNecula,fortheendlesshelptheyprovidedduringmyPhD.AllthankstomywifeDeepti,forbeingaconstantsupportthroughmy,sometimepleasantandsometimearduous,journeyasagraduatestudent.AndtoAbhayforbeingthejoyofourlife.ThankstomyfriendsatKrishnaHousewithoutthemlifewouldbeempty.Tomyparentsandsiblings,fortheirunconditionalloveandselessservice.Tomyspiritualmaster,H.G.KalakanthaPr,tohelpmewriteeverythingwiththeinkofKrishnaConsciousness. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 13 CHAPTER 1GRAVITATIONALWAVES .............................. 15 1.1GeneralRelativity ............................... 15 1.2SourcesofGravitationalWaves ........................ 19 1.2.1ContinuousGravitationalWaves .................... 19 1.2.2InspiralGravitationalWaves ...................... 20 1.2.3BurstGravitationalWaves ....................... 20 1.2.4StochasticGravitationalWaves .................... 22 1.3EccentricBinaryCoalescence ........................ 22 1.3.1MassSegregationAroundMassiveCentralObject ......... 23 1.3.2FormationofBinary .......................... 25 1.3.3RatesforOneGalacticNuclei ..................... 26 1.4EccentricBinaryWaveforms .......................... 27 1.4.1WaveformsUsedintheeBBHSearch ................ 29 1.4.2DistributionofEnergy ......................... 32 2DETECTIONOFGRAVITATIONALWAVES .................... 35 2.1OverviewofGWdetectors ........................... 36 2.2NoiseSourcesinInterferometricGWDetectors ............... 37 2.2.1SeismicNoise .............................. 38 2.2.2MechanicalResonances ........................ 39 2.2.3ShotNoise ............................... 39 2.2.4RadiationPressure ........................... 39 2.2.5ThermalNoise ............................. 40 2.3TheLIGOandVirgoInterferometers ..................... 40 2.4DetectorResponse ............................... 41 3REGRESSIONOFLINEARANDBILINEARNOISEINLIGO .......... 44 3.1RegressionAnalysis .............................. 45 3.1.1RegressionwithWDM ......................... 46 3.1.2MultipleChannels ........................... 47 3.1.3Regulators ................................ 50 3.1.4Effectofregulatorsonprediction ................... 53 5

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3.2ApplicationsofWDMRegression ...................... 55 3.2.1Regressionofnoisewithlinearcoupling. ............... 55 3.2.2Regressionofnon-linearnoise .................... 58 3.2.3MonitoringofGravitationalWaveData ................ 60 4SEARCHALGORITHM ............................... 64 4.1Time-FrequencyAnalysis ........................... 65 4.2OverviewofthePipeline ............................ 67 4.2.1CoherentAnalysis ........................... 67 4.2.2DominantPolarizationFrame ..................... 68 4.2.3Un-constrainedLikelihood ....................... 69 4.2.4DualStreamAnalysis/PolarizationConstraint ............ 71 4.2.5Regulators ................................ 74 4.3ReconstructionofChirpMass(chirpcut) .................. 74 4.3.1Algorithm ................................ 75 4.3.2ErrorinReconstructedChirpMass .................. 76 4.3.3ApplicationofChirpMassCut ..................... 77 4.3.4EccentricBinary ............................ 80 4.4RunStaging ................................... 82 5SEARCHFORGRAVITATIONALWAVESFROMECCENTRICBINARYBLACKHOLES ........................................ 85 5.1DataQualityFlags ............................... 86 5.2Post-ProductionTuning ............................ 87 5.3CalibrationUncertainties ............................ 90 5.4BackgroundEstimation ............................ 92 5.4.1Time-ShiftAnalysis ........................... 93 5.4.2BackgroundSets ............................ 94 5.5Simulation ................................... 98 5.5.1EventGenerator ............................ 98 5.5.2MassRange ............................... 99 5.5.3RadialPlacement ............................ 101 5.5.4VisibleVolume ............................. 102 5.5.5StatisticalUncertainty ......................... 106 5.5.6StatisticalandSystematicUncertaintiesintheeBBHAnalysis ... 107 5.5.7SystematicErrors ............................ 107 5.6SignicanceofanEvent ........................... 108 5.6.1FalseAlarmRateDensity ....................... 109 5.6.2FalseAlarmDensity .......................... 111 6RESULTSOFTHecWBS5-S6/VSR1-VSR2-VSR3eBBHsearch ........ 113 6.1FalseAlarmDensityandLoudestEvents .................. 113 6.2EffectiveRadius ................................ 113 6.3UpperLimittoRateDensity .......................... 114 6

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6.4ComparisonwiththeCBCIMRsearch .................... 119 6.5EffectofEccentricityontheResults ..................... 121 6.6SummaryofResults .............................. 122 7FUTUREPROSPECTS ............................... 126 7.1SecondGenerationDetectors ......................... 126 7.2eBBHSearchintheADEEra ......................... 127 7.2.1DecreaseinCut-OffFrequency .................... 128 7.2.2IncreaseinSensitivityoftheDetectors ................ 128 7.2.3RequiredImprovementsintheAnalysis ............... 129 REFERENCES ....................................... 134 BIOGRAPHICALSKETCH ................................ 139 7

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LISTOFTABLES Table page 1-1Mergerratepergalacticnucleifordifferentvaluesofandblackholemassdistribution.Results ................................. 27 3-1Anexampleofimprovementinpredictionduetomulti-channelanalysis .... 50 3-2IncreaseinRMSofpredictionwithincreaseinnumberofwitnesschannels .. 50 3-3Effectofregulatorswhennoneofthewitnesschanneloutof16iscoupledwiththetargetchannel.TableshowstheRMSvalueoftheprediction. ..... 55 3-4Effectofregulatorswhenonlyonewitnesschanneloutof16iscoupledwiththetargetchannel.TableshowsRMSofresidualforrc=0.99. .......... 55 5-1Summaryofnetworks,live-timeandparameterspaceforeBBHsearch. .... 86 5-2CategorydenitionandtheirapplicationintheeBBHanalysis. ......... 86 5-3Live-time(days)fordifferentrunsaftertheapplicationofCAT2,CAT3andHVETOvetoes. ........................................ 87 5-4MaximumfractionalerrorontheamplitudeofthecalibratedGWstraininthe40Hz)]TJ /F1 11.955 Tf 12.62 0 Td[(2kHzbandwidth. .............................. 91 5-5Numberofinjectionsmadeineachrun. ...................... 107 6-1Listoftherstthreeloudestforeground(lagzero)events. ............ 113 6-2Upperlimittotheratedensityforvarioussourcemodels. ............ 116 6-3Comparisonoftemplate-basedCBCsearchandeBBHsearchperformedusingcWB,ontheS5/S6-VSR1/2/3scienceruns .................... 121 8

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LISTOFFIGURES Figure page 1-1AnexampleofsimulatedcontinuousGW.AspinningneutronstarwithadeformitywillproducecontinuousGW ............................. 20 1-2AnexampleofansimulatedIMRgravitationalwave.TheseGWareproducedbycoalescingbinariesandhaveachirpingsignal ................. 21 1-3Anexampleofasimulatedburstgravitationalwaveproducedbyasupernova.ThesignatureoftheseGWisnotwellunderstood. ................ 21 1-4Anexampleofsimulatedstochasticgravitationalwave.TheyareunderstoodtobetheremnantoftheBigBangmuchlikecosmicmicro-wavebackground. 22 1-5TheevolutionofBahcall-Wolfcuspwithtime.Afteraboutone-halfrelaxationtime,thedensity ................................... 24 1-6Fittingfactorsbetweencircularandeccentricbinarywaveforms.Thelegendliststhemassofthecomponentsofthebinary. .................. 28 1-7Plotofwaveformforinitialeccentricityofe=.5fora1.4+1.4binarysystem .. 29 1-8TheFouriertransformofwaveformfor1.4+1.4binarysystemforeccentricitye=.1 .......................................... 30 1-9TheplotshowsradiatedpowerPn(e)forvariousharmonics(n)asafunctionofeccentricity. .................................... 33 2-1AGWtravelingperpendiculartotheplaneofthediagram,willdistortacircularringoftestparticlesintoanellipse ......................... 36 2-2Basicsub-systemsofaGWdetector.Imagecredit:iopscience[ 33 ] ....... 37 2-3Strainsensitivities,expressedasamplitudespectraldensitiesofdetectornoiseconvertedtoequivalentGWstrain. ......................... 38 2-4AerialviewoftheGWdetectors,locatedat,(top-left)Casina,Italy,(top-right)Livingston,USA(bottom)andHanford,USA.Imagecredit:[ 35 ] ......... 41 2-5p F2++F2andjFj=jF+jasafunctionofskylocationovertheearth'ssurfacefortheLHVnetwork.Detectorshavebeenassumedtobeequallysensitive. .. 43 3-1Anexampleoftime-frequencymapobtainedbyapplyingWDMtransformonLIGOdata ....................................... 48 3-2Predictionofseismicnoiseusing16coilcurrentchannel ............. 51 3-3EigenvaluedistributionofmatrixQforsomecases.Black-all16witnesschannelscarrythesame .................................... 52 9

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3-4Predictionofsimulatedmonochromaticnoiseusingdifferentregulatorswitht=1. ......................................... 54 3-5Performanceofvariousregulators(withthreshold0.5)insubtractionofseismicnoiseusing16CCchannels. ............................ 56 3-6Cleaningofpower-linesusingvoltagemonitor(H0:PEM-LVEA2 V1)asthewitnesschannels. .................................. 57 3-7Subtractionofmechanicalresonancesusing14accelerometersand2microphones. 58 3-8Removaloftheup-conversionnoisearound180Hzpowerlinewith16syntheticwitnesschannelsconstructedfrom16coilcurrentchannels ........... 59 3-9Removalofup-conversionaround113Hzcalibrationlineusingcoilcurrentchannelsandasimulatednoise. .......................... 61 3-10Removalofup-conversionaroundjitterpeaksusingcoilcurrentchannelsandaself-predictedcarrier. ............................... 62 3-11ThemaximumRMSvalueasafunctionofthechannelidenticationnumberandGPStime. .................................... 63 4-1Thetopplotsshow(!)andhighlightthedeningparameters:thewidthofthetopatregion(black)is2A ........................... 67 4-2Thegeometricalinterpretationofthelikelihoodanalysis ............. 70 4-3Phasetransformationofdetectorresponsetoachievethecondition ...... 73 4-4Orientationofand~forfourdifferentpixelsbelongingtoaGWsignal,basedonitspolarization. .................................. 74 4-5ThemodiedbasisforlikelihoodanalysisofGWusingregulators ........ 75 4-6InjectedvsReconstructedchirpmassforS6A(LHV)unmodeledsearch(withoutpolarizationconstraint) ................................ 77 4-7InjectedvsReconstructedchirpmassforS6A(LHV)unmodeledsearchusingEOBNRv2waveforms. ................................ 78 4-8ReconstructionofchirpmassfortheBigDogevent.Injectedchirpmassis4.962Mwhilethereconstructedchirpmassis4.800M ............ 79 4-9Energyfractionandellipticitydistributionforthe(leftplot)time-shiftanalysiswith1000lagsand(rightplot)simulation,fortheS6A(LHV)unmodeledsearch. 79 4-10ReconstructedchirpmassforS6A(LHV)time-shiftanalysiswith1000lags. .. 80 4-11Redlinerepresentsthecut,energyfractionlog10(sizeofcluster)>1.3 .... 81 10

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4-12(Left)Distributionofreconstructed/injectedchirpmass.OverallratioissmallerfortheeBBHwaveforms. .............................. 82 4-13Redlinerepresentsthecut,energyfractionlog10(sizeofcluster)>1.3 .... 83 4-14Effectofchirpcutoneccentricbinarywaveforms.Bluecurveisthedistributionofeccentricity ..................................... 84 5-1EffectofCategory3andHVETOvetoesontheeventsidentiedinthetime-shiftanaysis,forthedatafromtheL1H1H2networkduringS5-VSR1sciencerun 88 5-2EffectofCategory3andHVETOvetoesontheeventsidentiedinthetime-shiftanaysis,forthedatafromtheL1H1H2networkduringS6D-VSR3sciencerun 89 5-3Distributionofnetworkcorrelationcoefcientversuscoherentnetworkamplitude 90 5-4Distributionofsubnetworkcorrelationcoefcientversuscoherentnetworkamplitude 91 5-5Thedistributionofthecentralfrequencyofthepixelsusedtoreconstructtheeventsforthetime-shiftanalysis.PlotisforS6CLHdata. ............ 92 5-6Thedistributionofcentralfrequencyofthepixelsusedtoreconstructtheevents 93 5-7Datafromreferencedetectorareleftastheyare,whiledatafromotherdetectors 94 5-8DistributionofbackgroundeventsandtheirFAR,fordifferentdetectornetworksusedintheanalysis ................................. 97 5-9EccentricityofthebinarieswhenGWgeneratedbythemhaveanaverageorbitalfrequencyof32Hz .............................. 99 5-10Distributionofchirpmassfordifferentvalues.ThedistributionisfortheMWmodel,althoughtheeffectoftheSMBHmassisnotsignicantonthisdistribution. 100 5-11Injectionsmadeonthemasscomponentplane.Theinjections,ineachmassbin,arecloseinnumber. ............................... 101 5-12Effectonthedistributionofeccentricitywheninjectionsaremadefollowingdistribution ...................................... 102 5-13Reconstructedversusinjectedchirpmassforhighermassbinaries ....... 103 5-14Injectiondistributionfor=2andRd(21.764)=200Mpc.Fiducialradiusiscalculated ....................................... 104 5-15VisiblevolumeasafunctionofcoherentnetworkSNR(),forvariousdetectornetwork ........................................ 105 5-16Percentageuncertainty(absolutestatisticaluncertaintynormalizedbythetotalvisiblevolumeandmultipliedby100) ........................ 108 11

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5-17Percentageuncertainty(absolutestatisticaluncertaintynormalizedbythetotalvisiblevolumeandmultipliedby100) ........................ 109 5-18ThepercentagedifferencebetweentheestimatedeffectiveradiusfortheeBBHandtheEOBNRv2(100(1-REOBNRv2=ReBBH))waveform,fortheS6Arun. .. 110 6-1TheFADversuscoherentnetworkamplitudelot.Theplotincludeallthebackground 114 6-2TheeffectiverangemeasuredfortheS5L1H1H2searchoverthecomponentmassplane. ...................................... 115 6-3TheeffectiverangemeasuredfortheS6DL1H1V1searchoverthecomponentmassplane. ...................................... 116 6-4Theratelimitforthemassbinsonthecomponentmassplane.Tocalculatetheaverageratelimitofanastrophysicalmodel .................. 117 6-5TheSMBHnumberdensityasafunctionoftheSMBHmass.Whenintegratedover104Mto3109M,theareaunderthecurveis0.351Mpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3. ...... 118 6-6TheSMBHmassdensityasafunctionoftheSMBHmass.Whenintegratedover104Mto3109M,theareaunderthecurveis6.39105MMpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3. .. 119 6-7RecoveredvsinjectedSNRforthecWBsearchperformedusingEOBNRv2injectionsoverS6A-VSR2runfortheLHVdetectornetwork. ........... 122 6-8Plotshowsthevisiblevolumevesusthecoherentnetworkamplitude.Thereisnovisualdifferenceinthe ............................. 123 6-9Theefciencyversuseccentricityplot.Thereisnovisibledependenceoftheefciencyontheeccentricity ............................ 124 7-1Time-linefortheprogressiveimprovementofsensitivityoftheadvanceddetectors 127 7-2Centralfrequencydistributionoftherecoveredinjections.Thedistributionhasalowermeanforinjectionmadeoversimulated .................. 129 7-3Spectrumofaninjectedsignal(black)andreconstructedsignal(red).Leftplotisthereconstructioninthe .............................. 130 7-4ExpectedeffectiveradiusfortheeBBHsourcesduringearlyADE.TheanalysishasbeenperformedfortheLHVnetworkusingsimulatedstrain. ........ 131 7-5ReconstructionofaninjectedsignalusingWDMtransformwithfrequencyresolution.(Top)1Hzto128Hz,and(Bottom)2Hzto64Hz. ................ 132 7-6ChirpmassreconstructionalgorithmforeBBHanalysisduringADEwillnotbeoptimum. ..................................... 133 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySEARCHFORGRAVITATIONALWAVESFROMECCENTRICBINARYBLACKHOLESByVaibhavTiwariMay2015Chair:SergeiKlimenkoMajor:PhysicsTheexistenceofgravitationalwaves(GW),whichareripplesinspace-time,ispredictedbyEinstein'stheoryofgeneralrelativity.Severalinterferometricdetectors,suchasLIGOandVirgo,havebeenconstructedtodetectgravitational-wavesproducedfromviolentastrophysicalphenomena.Thecoalescenceofcompactbinariesisoneofthemostpromisingsourcefortherstdetectionofgravitationalwaves.Sofar,searcheshavebeenfocusedonthedetectiononcircularbinariesasitisbelievedtheorbitofabinarycircularizesbythetimetheemittedgravitationalradiationenterLIGO/Virgo'ssensitivityrange.However,binarieswitheccentricorbitscouldalsobeproducedthroughchannels,likedynamicalinteractioningalacticnuclei,whenasignicantfractionofthemmaymaintainhigheccentricitiesthroughouttheirlifetime.Thesebinarieshaveuniquegravitationalwavesignaturesandarenotcapturedefcientlybysearchesdesignedforcircularsystems.Thesearchwasconductedforgravitationalwavesproducedfrommergingeccentricbinaries(eBBH)inthetotalmassrangeof5Mto25M.ThesearchwasconductedonthedatarecordedbytheLIGO/VIRGOdetectorsbetweenMay2005andOctober2010.ThesearchwasperformedusingtheCoherentWaveBurstalgorithmwhichutilizesalikelihoodapproachtoreconstructGWeventsfromdatacollectedbymultipleGWdetectors.Thebackgroundwasestimatedbyperformingthetime-shiftanalysis,whichensurestheabsenceofanyGWevents.Forthereductionofbackground,model-based 13

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constraintswereappliedatthepost-productionstage(loosecircularpolarizationconstraintandreconstructedchirpmassconstraint).Nocandidateswerefoundthatcanbeclaimedasdetection,andupperlimitswereestimated.Simulationstudieswereperformedusingeccentricbinarywaveformsfortheestimationofupperratelimitdensity.Ipresenttheresultsofthesearch,includingthesensitivevolumeandupperlimitsonthesourceratedensityasafunctionofcomponentmass.Theupperlimitonthesourceratedensity,averagedoverthecomponent-massplane,wasmeasuredtobe3.6Myr)]TJ /F5 7.97 Tf 6.58 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3.Idevelopedandconductedtheendtoendanalysiswhichinvolvedorganizational,dataprocessingandpostproductionstages.DuringtheevolutionofthesearchImadesignicantcontributioninthedevelopmentofregressionanalysis,eventgeneratorandtoolforthereconstructionofchirpmass. 14

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CHAPTER1GRAVITATIONALWAVESIn1905,Einsteinpresentedthetheoryofspecialrelativity,inwhichtimeandspacewerepostulatedtobeintertwined.Tenyearslater,hepresentedtheoryofgeneralrelativity,whichprovidesaunieddescriptionofgravityasageometricpropertyofspaceandtime,changingitsstatusfromaforcetoanactoffalling.Spaceandtimearenotsimplyabstract,externalconcepts,butmust,infact,beconsideredmeasuredphysicalobservables.Generalrelativitypredictsthepresenceofspace-timerippleswhichtravelatthespeedoflight.TherstcalculationofgravitationalradiationisEinstein'sown,famouslyknownasquadrupoleformulaforgravitationalwave(GW)emission.Thisformulabearsthesamestatusingeneralrelativityasdipoleradiationformuladoesinclassicalelectromagnetism,showingthatgravitationalwavesarisefromacceleratedmassessimilartoelectromagneticwaveswhicharisefromacceleratedcharges.Gravitationalwavesarethedirectconsequenceofgeneralrelativityintheweakeldlimit.Eventhoughtherehasbeennodirectmeasurementsofgravitationalwaves,thereareindirectevidencesfortheexistenceofgravitationalwavesandvalidityofgeneralrelativityingeneral.ThemostcelebratedexampleistheHulse-Taylorpulsar,B1913+16.YearsofobservationhaveconrmedthatthedecayintheorbitofthebinarycomplieswiththelossoforbitalenergyandangularmomentumbyGWaspredictedbygeneralrelativity. 1.1GeneralRelativityGeneralrelativitydescribestheeffectofmassonthebehavioranddynamicsofspace-timecurvature,expressedinspace-timemetricg.AccordingtoEinstein'sequations[ 1 2 ], R)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2gR=8G c4T,(1)whereRistheRiccicurvaturetensor,RistheRicciscalar,Tisthestress-energytensor,andgisthemetrictensor.Themoststraightforwardstartingpointforany 15

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discussionofGWislinearizedgravity.Linearizedgravityisanapproximationtogeneralrelativitywhenthespace-timemetric,g,isexpressedasasmallperturbationoverMinkowskimatrix(), g=+h,jjhjj1,(1)where, =0BBBBBBB@)]TJ /F4 11.955 Tf 9.3 0 Td[(10000100001000011CCCCCCCA.(1)jjhjj1,meanscomponentsofthemetricperturbationtobemuchlessthan1.Einstein'sequationsarenon-linear;inlinearizedgravity,theequationsareexpandedforthesmallperturbationwhilekeepingonlytermswhicharelinearinh.Asaconsequence,indicesareraisedandloweredusingtheMinkowskimetric.Followingthisperturbativescheme,theChristoffelcoefcients,withonlyrstorderterminh,become, )]TJ /F11 7.97 Tf 7.32 4.94 Td[(=1 2(@h+@h)]TJ /F3 11.955 Tf 11.95 0 Td[(@h)=1 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(@h+@h)]TJ /F3 11.955 Tf 11.95 0 Td[(@h. (1) AsMinkowskimetricisusedtoraiseorlowerindices,spatialquantitydon'tchangesignonloweringorraisingofindiceswhiletimequantitydoes.FollowingEquation 1 ,theRiemanntensorcanbesimilarlyconstructed, R=@)]TJ /F11 7.97 Tf 7.31 4.94 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(@)]TJ /F11 7.97 Tf 7.31 4.94 Td[(=1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(@@h+@@h)]TJ /F3 11.955 Tf 11.95 0 Td[(@@h)]TJ /F3 11.955 Tf 11.95 0 Td[(@@h. (1) ThiscanbecontractedtowritetheRiccitensor, R=R=1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(@@h+@@h)]TJ /F12 11.955 Tf 11.96 0 Td[(2h)]TJ /F3 11.955 Tf 11.95 0 Td[(@@h(1) 16

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whereh=histhetraceofperturbationmetric,and2@=@x+@y+@2z)]TJ /F3 11.955 Tf 12.23 0 Td[(@2tistheD'alembertian.FurthercontractionleadstotheRicciscalar R=R=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(@@h)]TJ /F12 11.955 Tf 11.95 0 Td[(2h.(1)SubstitutingallthecomponentsintheEinstein'sequationsresultsin, G=R)]TJ /F4 11.955 Tf 13.16 8.09 Td[(1 2R=1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(@@h+@@h)]TJ /F12 11.955 Tf 11.95 0 Td[(2h)]TJ /F3 11.955 Tf 11.96 0 Td[(@@h)]TJ /F3 11.955 Tf 9.3 0 Td[(@@h+2h. (1) TheEinsteintensorcanbegreatlysimpliedwhenexpressedintracereversedperturbationh=h)]TJ /F4 11.955 Tf 12.19 0 Td[((1=2)h.TheEinstein'stensorintracereversedperturbationis, G=1 2)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(@@h+@@h)]TJ /F12 11.955 Tf 11.95 0 Td[(2h)]TJ /F3 11.955 Tf 11.96 0 Td[(@@h.(1)Finally,wearriveattheequationoflinearizedgravity: @@h+@@h)]TJ /F12 11.955 Tf 11.95 0 Td[(2h)]TJ /F3 11.955 Tf 11.96 0 Td[(@@h=16G c4T.(1)Einstein'sequationsaresymmetricinthetwoindices;hence,agaugeneedstobexedtoarriveatasolution.Gaugetransformationsarecoordinatetransformations,andageneralinnitesimalcoordinatetransformationcanbewrittenasx0=x+,where(x)isanarbitraryinnitesimalvectoreld.Thistransformationchangesthemetricvia h0=h)]TJ /F3 11.955 Tf 11.95 0 Td[(@)]TJ /F3 11.955 Tf 11.96 0 Td[(@,(1)andthetracereversedperturbationbecomes, h0=h)]TJ /F3 11.955 Tf 11.95 0 Td[(@)]TJ /F3 11.955 Tf 11.95 0 Td[(@+@.(1)Thegauge @h=0,(1) 17

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iscalledtheLorentzgauge.Fortheperturbationmetrictosatisfythisgauge,thefollowingequationshouldbesatised, 2=@h.(1)ThelinearizedEinstein'sequationsintheLorentzgaugeare, 2h=)]TJ /F4 11.955 Tf 9.3 0 Td[(16T(1)whichinthesourcefreeregionreduceto 2h=0.(1)Equation 1 admitsaclassofhomogeneoussolutionswhicharesuperpositionsofplanewaves,givenas: h(x,t)=ReZd3kAeikx,(1)wherek(!=c,~k),isthewave4-vectorandx(ct,~x),istheposition4-vector.kisthemagnitudeofthe3-wavevector~k.ForthetracereversedperturbationtosatisfytheLorentzgaugecondition,kAshouldbeequaltozero.Metricperturbationhastendegreesoffreedom,whichgetsreducedtosixintheLorentzgauge.Intheabsenceofsource,amorerestrictiveLorentzgaugecanbechosen,calledthetransverse-traceless(TT)gauge.IntheTTgauge,metricperturbationisperpendiculartothewave-vectorwithonlytwodegreesoffreedom.Aftertheseconditionsareapplied,thesolutionstotheabovewaveequationtakeonaparticularlysimpleform: h=exp(ikx)0BBBBBBB@00000h+h00h)]TJ /F4 11.955 Tf 9.3 0 Td[(h+000001CCCCCCCA,(1) 18

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whereh+andhrefertothetwodifferentindependentamplitudesthatcorrespondtothetwodegreesoffreedomremaininginthesystem.Thefrequency!isdictatedbythesourcemodel.Ifacoordinatesystemisafxedtothewave-frameandthephaseinformationisneglected,wearriveattheform: h=exp(kz)]TJ /F3 11.955 Tf 11.95 0 Td[(!t)0BBBBBBB@00000h+h00h)]TJ /F4 11.955 Tf 9.3 0 Td[(h+000001CCCCCCCAxy.(1)Thecomponentsofthewaveareseparatedintoanorthogonalpolarizationbasiscalledplusandcrossrespectively.Bothcomponentsaretransversetothepropagationofthewave,whichreducesthedotproductintheexponentialtothevalueof~kalongthepropagationdirection(heredenotedbyz).Thexandysubscriptsthenrefertotheprojectionofthepolarizationsonarbitraryxandyaxespositionedontheplaneorthogonaltothewavepropagation. 1.2SourcesofGravitationalWavesTherearemanyastrophysicalsettingsthatcanproducedetectableGW;theycanbebroadlycategorizedintofourcategories. 1.2.1ContinuousGravitationalWavesWhenasourceemitsGWcontinuouslyatanearconstantfrequencyforalongperiodoftime,i.e.fordays,monthsorevenyears,itiscalledasourceofcontinuousgravitationalwaves.Theseareusuallyrotationalsystems,whoserotationdeterminesthefrequencyofGW.RotatingneutronstarswhicharenotperfectlyaxisymmetricorarewobblingaroundtheirrotationaxesproduceGWinthehighfrequencyregions.Thesesourcesareexpectedtoproducecomparativelyweakgravitationalwavessincetheyevolveoverlongerperiodsoftimeandareusuallylesscatastrophicthansourcesproducinginspiralorburstgravitationalwaves.ContinuousGWareexpectedtomaintain 19

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axedfrequencyoveraperiodoftime,butspindownduetolossofangularmomentumshouldalsobeconsidered.Figure 1-1 showsanexampleofcontinuousGW. Figure1-1. AnexampleofcontinuousGW.AspinningneutronstarwithadeformitywillproducecontinuousGW.TheseGWhaveafairlyconstantandwell-denedfrequency. 1.2.2InspiralGravitationalWavesInspiralgravitationalwavesaregeneratedbycoalescingbinaries.Thesesystemsareusuallytwoneutronstars,twoblackholes,oraneutronstarandablackholewhoseorbitshavedecreasedoveraperiodoftime,duetothelossofangularmomentumbecauseoftheemissionofGW.Asthetwomassesrotatearoundeachother,theirorbitaldistancesdecreaseandtheirspeedsincrease.ThiscausesthefrequencyandamplitudeoftheGWtoincreaseuntilthemomentofcoalescence.Becauseofthis,theseGWarecharacterizedasthechirpingsignal.Eventually,thetwobinariesmergetoproducealargerspinningobject.ThethreestagesofthisastrophysicalphenomenonaretermedasInspiral,MergerandRingdown(IMR).Figure 1-2 showsanexampleofIMRGW. 1.2.3BurstGravitationalWavesBurstgravitationalwavesareexpectedtobesignalsofshortduration(oftheorderofsecondsorminutes).Therearenoknownoranticipatedsources,butthereare 20

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Figure1-2. AnexampleofanIMRgravitationalwave.TheseGWareproducedbycoalescingbinariesandhaveachirpingsignal(increasingfrequencyandamplitudewithtime). hypothesesthatsomesystemssuchassupernovaeorgammarayburstsmayproduceburstgravitationalwaves.Thesewavesincludealargevarietyofpossiblewaveforms,whichcouldbewideornarrowbandinthefrequencydomain,thoughmuchisnotunderstood.Figure 1-3 showsanexampleofburstGW. Figure1-3. Anexampleofasimulatedburstgravitationalwaveproducedbyasupernova.ThesignatureoftheseGWisnotwellunderstood. 21

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1.2.4StochasticGravitationalWavesStochasticGWarebackgroundGW.Expectedtobegeneratedduringearlyuniverse,alargenumberofrandom,independenteventscombinetocreateacosmicGWbackground.TheBigBangisexpectedtobeaprimecandidatefortheproductionofthemanyrandomprocessesneededtogeneratestochasticGW.TheseGWwouldproduceacontinuousGaussiansignalandwillbesamefromeverypartofthesky.Figure 1-4 showsanexampleofstochasticGW. Figure1-4. Anexampleofsimulatedstochasticgravitationalwave.TheyareunderstoodtobetheremnantoftheBigBangmuchlikecosmicmicro-wavebackground. 1.3EccentricBinaryCoalescenceThecomponentofacompactbinaryiseitheraneutronstarorablackholewhichmoveonanorbitaroundthesystem'scenterofmass.Gravitationalwavesareemittedfromsuchobjectswheninvolvedinstronggeneralrelativisticprocessesandarehallmarksofcompactbinarycoalescence.Itisanticipatedthatabinarylosesenergytogravitationalradiationandasaresultcoalescetoasinglespinningobject.Thethreestagesofthisphenomenonareknownasinspiral,mergerandring-down.Itisbelievedthatiftheinitialorbitofthebinarysystemiselliptic,theorbitwillcircularizebecauseoftheradiationreaction,longbeforetheemittedGWentertheLIGO/VIRGOsensitivityband[ 3 ].Duetothis,amajorityoftheefforthasbeenputin 22

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searchesrelatedtobinariesincircularorbits.Recently,therehasbeenanincreaseininterestforthecaseofbinariesinellipticorbits.ManyscenarioshavebeensuggestedinwhichabinaryretainseccentricityevenwhentheemittedGWentertheLIGO/VIRGOsensitivityband.Thereareastrophysicalsettingsthatcouldpopulateeccentricmergers.Inthedensegalacticnucleiwithasuper-massiveblackhole,therecanberunawayprocesswhentwoblackholeshaveacloseencounter,followingwhichtheyreleaseasufcientamountofenergyinGW,toformaboundsystemandmergeintimenotenoughforbinarytocircularize[ 4 ].Acircularneutronstar-blackholebinarycangaineccentricityduetomasstransferfromNStoBHastheNSllsBH'sRochelobe[ 5 ].TheproductionofNS-BHbinaryrequiresthesystemtoremainboundaftertheoccurrenceoftwosupernovathatproducedtheobjects.Ithasbeensuggestedthatthesesupernovakickscanalsorenderthesystemhighlyeccentricwithamergerfollowingwithinmonths[ 6 ].Anotherproposedscenarioinvolveshierarchicaltripletswhichconsistofaninnerandanouterbinary.Iftheorbitalplanesoftheinnerandoftheouterbinarymakeamutualinclinationanglelargeenough,thenthetime-averagedtidalforceontheinnerbinarymayinduceoscillationsinitseccentricity.ThisisknownastheKozaimechanism[ 7 ].Yetanotherscenariothatcancreateinspiralingeccentricbinariesinvolvescompactstarclusters.AninterplaybetweenGW-induceddissipationandstellarscatteringinthepresenceofanintermediate-massblackholecancreateshort-periodhighlyeccentricbinaries[ 8 ].Thefollowingsectionsdescribetheformationofeccentricbinaryblack-holesingalacticnuclei(asdiscussedin[ 4 ]). 1.3.1MassSegregationAroundMassiveCentralObjectThepresenceofasupermassiveblackholes(SMBH)atthecenterofmostgalaxiesiswidelyinferredfromobservationssuchasvelocityprolesofgasnearthecore,surfacebrightnessproles,etc.[ 9 ].BecauseofthepresenceoftheSMBH,itisexpectedthatmasssegregationoccursinthecoreofgalacticnuclei.Whentwo 23

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membershaveadynamicalencounter,theyexchangeenergyandmomentum.Thereisastatisticaltendencyforthekineticenergyofthetwomemberstoequalizeduringsuchanencounter.Thisphenomenonisknownasequipartition.Equipartitionrequiresthelessmassivemembertomovefasterandtobepushedtoanouterradiusandviceversaforthemoremassiveobject.TheproblemofstardistributionaroundamassiveblackholewasrstaddressedbyPeeblesin1972[ 10 ].BahcallandWolf[ 11 ]derivedthepower-lawcuspwithrespecttoradius(/r)]TJ /F5 7.97 Tf 6.59 0 Td[(7=4,Figure 1-5 )byderivingtheFokker-Planckequationsforasphericallysymmetricdistributionoflow-massstars,ofthesamemass,aroundamassiveblackhole.Inasecondpaper,theyextendedtheiranalysistomultiple-masssystems,inwhichtheyfoundthatmoremassiveobjectsformasteeperpower-lawdensityprolethanthelow-massstars[ 12 ]. Figure1-5. TheevolutionofBahcall-Wolfcuspwithtime.Afteraboutone-halfrelaxationtime,thedensitydistributionreachesasteadystate.X-axisisSMBH'sinuenceradius;theunitofdistanceisparsec. 24

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Multipleworkshavesuggestedthatblackholeswithmassaround10Mshouldsegregatetotheinnerparsec[ 13 16 ].FortheeBBHanalysis,weacceptablackholemassfunctiondn=dM/M)]TJ /F11 7.97 Tf 6.59 0 Td[(,asoutlinedin[ 4 ]. 1.3.2FormationofBinaryArunawayencounterfortheformationofbinarywillrequirethetwoblackholestoloosesufcientenergysothattheyformaboundsystem.Energyandangularmomentumlostinsuchencountersisgivenas(intheunitsofG=c=1)[ 17 ], E85 12p 22M9=2 r7=2p,(1) L6M42 r2p,(1)whereforthecomponentmassm1andm2,M=m1+m2isthetotalmass,=m1m2=M2isthesymmetricmass-ratioandrpisthepericenterdistanceoftheformedbinary.Thestateofformedbinaryisgivenas, Enal=E0+E,E0=Mw2 2,(1) Lnal=L0+L,L0=Mbw,(1)whereE0andL0aretheenergyandangularmomentumoftheunboundsystem'scentralmotion.bistheimpactparameter,wistherelativevelocity,givenas w=r MSMBH r,(1)andristhedistancebetweenthegalacticcenterandthepositionofthebinary.Notallencountersformbinaries;thecomponentsshouldradiatesufcientenergytoformaboundsystem.Themaximumpericentredistance/impactparameterforwhich 25

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E>E0(Enal<0)is, rp,max=85 12p 22 E02=7M9=7,bmax=340 31=7M1=7 w9=7. (1) Forpracticalpurpose,weassumethatrpcanbebetweenrpcorrespondingtoinnermoststableorbitandrp,max.Foraparticularb,eccentricityoftheformedbinaryisgivenas, e0=s 1+2Enalb2w2 M3.(1)Allformedbinariesarehighlyeccentric,witheccentricityclosetoone.EccentricitydecayswithtimewithmanybinariesretainingeccentricitybythetimeGWemittedbythementertheLIGOsensitivityband.EccentricitycanbeevolvedtoaparticularfrequencyusingthePeter-Mathewsformulation[ 3 ] da de=12 19a e[1+(73=24)e2+(37=96)e4] (1)]TJ /F4 11.955 Tf 11.95 0 Td[(e2)[1+(121=304)e2],(1)wherea=rp=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(e)isthesemi-majoraxisoftheeccentricorbit. 1.3.3RatesforOneGalacticNucleiRatesofeccentricmergersforonegalacticnucleicanbeestimatedbycalculatingthenumberofencountersasingleblackholewillhave.Thisisapproximately(b2maxnBHw))]TJ /F5 7.97 Tf 6.59 0 Td[(1peryear.nBHistheblackholedensityandbmaxisthemaximumimpactparameteratwhichtheencounterwillresultinaboundsystem.Whenintegratedoverallmassesandradius,rateforonegalacticnuclei()]TJ /F5 7.97 Tf 7.32 -1.79 Td[(1GN)isgivenas, )]TJ /F5 7.97 Tf 7.31 -1.79 Td[(1GN=ZrmaxrmindrZMmaxMmindMZMMmindmd3)]TJ /F5 7.97 Tf 7.31 -1.79 Td[(1GN drdmdM4r2,(1)where, d3)]TJ /F5 7.97 Tf 7.32 -1.79 Td[(1GN drdmdM=b2maxvc(r)nm(r)nM(r)r2.(1) 26

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BlackholedensityinthecuspisgivenbyEquation 1 nm(r)/r)]TJ /F5 7.97 Tf 6.58 0 Td[(pm)]TJ /F5 7.97 Tf 6.58 0 Td[(3=2,pm=p0m=Mmax,p0.5)]TJ /F4 11.955 Tf 11.95 0 Td[(.6.(1)TheintegralisperformedbydiscretizingEquation 1 )]TJ /F5 7.97 Tf 7.31 -1.79 Td[(1GN=42i=NrXi=1rj=NMXj=1Mk=NmXk=1mb2max,ijkvc(ri)nm(ri)nM(ri)r2i,(1)where,r=(rmax)]TJ /F4 11.955 Tf 12.17 0 Td[(rmin)=Nr,M=(Mmax)]TJ /F4 11.955 Tf 12.16 0 Td[(Mmin)=NMandm=(Mmax)]TJ /F4 11.955 Tf 12.16 0 Td[(Mmin)=Nm.Table 1-1 lists)]TJ /F5 7.97 Tf 7.31 -1.79 Td[(1GNforsomevaluesofandblackholemasses, Table1-1. Mergerratepergalacticnucleifordifferentvaluesofandblackholemassdistribution.ResultsareforMilkyWay-likegalaxy,forwhichtheinuenceradiusextendsuptormax=GMSMBH=2,forthedispersionvelocityof=75Km/s Mmin(M)Mmax(M)MergerRateperGalaxy(yr)]TJ /F5 7.97 Tf 6.59 0 Td[(1) 25102.210)]TJ /F5 7.97 Tf 6.59 0 Td[(1035102.010)]TJ /F5 7.97 Tf 6.59 0 Td[(1025152.810)]TJ /F5 7.97 Tf 6.59 0 Td[(1035152.510)]TJ /F5 7.97 Tf 6.59 0 Td[(1025253.810)]TJ /F5 7.97 Tf 6.59 0 Td[(1035253.410)]TJ /F5 7.97 Tf 6.59 0 Td[(10 1.4EccentricBinaryWaveformsSofar,searcheshavefocusedonbinarieswithcircularorbits.But,ithasbeenshownthatthereisalossofefciency,whichincreaseswithincreasedeccentricity,ifcircularbinarywaveformsareusedinsearchingeccentricbinaries,usingthematch-lteringmethod[ 18 ].Figure 1-6 plotsthettingfactorbetweenthecircularandtheeccentricbinarywaveforms,accurateuptoquadrapolemoment.Fittingfactorisaninner-productofthetwowaveformsanditreducesseverelywithincreasingeccentricities.Figure 1-7 showsthewaveformproducedbyacircularbinary,andFigure 1-8 showstheFouriertransformofthewaveform.Itcanbeseenthattheellipticwaveformhasalotmorevariationcomparedtoitscircularcounterpart[ 19 ]. 27

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Figure1-6. Fittingfactorsbetweencircularandeccentricbinarywaveforms.Thelegendliststhemassofthecomponentsofthebinary. Thethreemajoreffectsoftheeccentricityarea)decreaseinthedurationofthewaveform,b)modulationofthewaveformamplitudeandc)anoverallincreaseintheamplitude.Forhigheccentricity,thewaveformoftheeccentricbinarywillgreatlycontrastwiththecircularbinaryduetothepresenceofsharppeaks.Thepeaksappearbecauseoftheincreaseinthegravitationalradiationthatoccursduringtheperiastronpassage.Theeccentricbinarieswillemitaseriesoflongrepeatedbursts,aftertheirformation,followedbyacontinuouspowerfulchirpwithasignicantsignal-to-noiseratio[ 20 21 ].Theorbitofanellipticbinaryisexpectedtocircularizeinduetime,beforeitmerges.But,forhighlyeccentricbinaries,whichmergequicklywhileretainingtheeccentricity,theendbehaviorwillbeaplungeinsteadofachievinganinnermoststablecircularorbit[ 22 23 ].Theeccentricbinarywaveformshavebeenobtaineduptoanaccuracyof3PNcorrectionsandarevalidforalarge-enoughbinaryseparation.Theconservative3PN 28

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Figure1-7. Plotofwaveformforinitialeccentricityofe=.5fora1.4+1.4binarysystem;thebottominsetshowswaveformatearlytimeswheneccentricityisstilllarge.Thetopinsetshowsthewaveformatlatetimes,whentheeccentricityismuchreduced. dynamicsofaneccentricsysteminthequasi-Keplerianrepresentationhavebeenderived[ 24 ],andenergyandmomentumuxeshavealsobeendeterminedforthesameorder[ 25 ].Inthestrongeldlimit,therehavebeendevelopmentsincalculatingwaveformsusingnumericalrelativity[ 26 ]. 1.4.1WaveformsUsedintheeBBHSearchAtthetimeoftheeBBHanalysis,onlyoneimplementabletoolwasavailableforthegenerationofcompleteeccentricbinarywaveforms.Themodelisbasedonmappingthebinarytoaneffectivesingleblackholesystem,describedbyaKerrmetric,therebyincludingcertainrelativisticeffects,suchaszoom-whirl-typebehavior.MassandtotalangularmomentumofthebinaryareidentiedwiththemassandspinparametersoftheeffectiveKerrspacetimeandtheorbitalangularmomentumandenergywiththatofthe 29

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Figure1-8. TheFouriertransformofwaveformfor1.4+1.4binarysystemforeccentricitye=.1;thesignalistakentoabruptlystartwhenforb=13.3Hz,forcomparisontheinsetshowsFouriertransformforwaveformwithacircularorbitforthesamemassesofbinary. geodesics.ThisapproachhastheadvantageofreproducingthecorrectorbitaldynamicsintheNewtonianlimitandgeneral-relativistictestparticlelimit,whilestillincorporatingstrong-eldphenomenasuchaspericenterprecession,framedraggingandzoom-whirldynamics[ 27 ].TheequationsforanequatorialgeodesicinaKerrspace-timewithmassManddimensionlessspinacanbewrittenusingBoyer-Lindquistcoordinates.Totherstorder 30

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theyare, _= ~ER20)]TJ /F4 11.955 Tf 11.96 0 Td[(2M2a~L=r,Q_=1 R20[~LQ+2M2a=r],_r=QPr=r2,_Pr=1 r2Q2)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(r3)]TJ /F4 11.955 Tf 11.96 0 Td[(M3a2+M(2Ma)]TJ /F4 11.955 Tf 11.95 0 Td[(1)+P2rQ r3M2a2)]TJ /F4 11.955 Tf 11.96 0 Td[(Mr, (1) whereR0=r2+2M3a2=r+M2a2,=r2)]TJ /F4 11.955 Tf 10.11 0 Td[(2Mr+M2a2,Pr=r2_r=(Q),isthepropertime,theoverdotmeanstime-derivatives.~Eand~Laretheenergyandangularmomentumofthegeodesic.Theseequationsareappliedbyexpressingtheparametersofabinaryinthecenterofmass(COM)frameandevolvingthemusingEquation 1 .Thedissipativetermisquadrupoleradiationterm,givenbyEquation 1 _~E=)]TJ /F4 11.955 Tf 13.8 8.09 Td[(G 5c5...Iij...Iij_~L=)]TJ /F4 11.955 Tf 10.87 8.08 Td[(2G 5c5zijIik...Ijk. (1) InEquation 1 ,isthereducedmassandIijisthereducedquadrupolemoment.Iij,...Iijarewrittenasthefunctionsofr,,Pr,~E,~L.Missettothetotalmassofthebinary.Contributionfromorbitalenergyisignored.Spinais~L=M2+aBH,intheCOMframe.OnlyspinlesssystemsareconsideredandaBHissetequaltozero.Followingthis,Equation 1 isevolvednumerically.Consideringonlylinearterms,inthetransverse-tracelessgauge,thecomplexgravitationalwavestrainhopt,atadistanced,fromanoptimallyorientedsourceisrelatedtothetimederivativeofquadrupolemomentas, hopthopt++ihopt2G dc4Ixx+iIyx.(1) 31

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Thewaveformsobtainedusingtheaboveequationisstitchedtoamergerandringdownstage.Themodeliscircularlypolarizedandisnotstrictlyvalidforeccentricbinaries,butitdoesprovideareasonablygoodapproximationtotheresultsfromnumericalsimulations[ 28 29 ].Comparisonsofwaveformsforstrongeldsinglepericenterpassagesandcapture-merger-ringdownforlargeeccentricitieshavebeenmadewithnumericalrelativitysimulations.But,allperturbativeapproachesfailforlargeeccentricitiesandLIGO-relevantseparations.Numericalrelativity(NR)cannotbeappliedtothefullIMRcoveringtheLIGOband.Hence,thereispresentlynootherwaytoestimateaccuracy.Itwould,inprinciple,bepossibletouseNRforpericenterpassage,andperturbativetechniquesfortherestoftheorbit,butnosuchcodepresentlyexiststhatcandothis.Amoredetailedexplanationispresentin[ 27 ]. 1.4.2DistributionofEnergyIftheangularfrequencyofacircularbinaryis!sp Gm=R3(m=m1+m2isthetotalmassandRistheorbitalradius),becausethemostdominantmodeisn=2,thecorrespondinggravitationalwavefrequencyis!GW=2!s.Thisisnotthecasewitheccentricbinaries,whentheradiatedpowerisdistributedoverseveralmodes.Withtheincreaseineccentricity,thenumberofmodesmakingsignicantcontributionalsoincreases.ThepowerradiatedbyaneccentricbinaryaccurateuptothequadrupoletermisgivenbyEquation 1 P(e)=32G42m3 5c5a5f(e),f(e)1 (1)]TJ /F4 11.955 Tf 11.96 0 Td[(e2)7=21+73 24e2+37 96e4.(1)Whereaisthesemi-majoraxisandeistheeccentricityofthebinary.Thecontributiontoradiatedpowerfromamodenisgivenby,Equation 1 [ 30 ] Pn=32G42m3 5c5a5g(n,e),g(n,e)=n6 96a4A2n(e)+B2n(e)+3C2n(e))]TJ /F4 11.955 Tf 11.95 0 Td[(An(e)Bn(e).(1) 32

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A,BandCarefunctionsoftheBesselfunctionofe,givenas, An=a2 n[Jn)]TJ /F5 7.97 Tf 6.59 0 Td[(2(ne))]TJ /F4 11.955 Tf 11.96 0 Td[(Jn+2(ne))]TJ /F4 11.955 Tf 11.96 0 Td[(2e(Jn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(ne)Jn+1(ne))],Bn=b2 n[Jn+2(ne))]TJ /F4 11.955 Tf 11.96 0 Td[(Jn)]TJ /F5 7.97 Tf 6.59 0 Td[(2(ne)],Cn=ab n[Jn+2(ne))]TJ /F4 11.955 Tf 11.96 0 Td[(Jn)]TJ /F5 7.97 Tf 6.58 0 Td[(2(ne))]TJ /F4 11.955 Tf 11.96 0 Td[(e(Jn+1(ne)Jn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(ne))]. (1) Theaboveequationsareforclosedellipticalorbit,whichisnottrueforthecaseofcoalescingbinaries.Buttheyshouldremainvalidwhentheeldisnotstrong.Figure 1-9 plotsdistributionofemittedpowerfordifferentvaluesofn. Figure1-9. TheplotshowsradiatedpowerPn(e)forvariousharmonics(n)asafunctionofeccentricity.TheGWfrequencyofaharmonicisgivenas!n=n!0.(Topleft)e=0.2,(topright),e=.5,and(bottom)e=.7.Pnisnormalizedtothevaluefore=0. 33

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Withthedecreaseineccentricity,then=2modebeginstodominatethecontributiontotheradiatedpower.Thefrequencycutoffofinitialdetectorsisaround50Hz.ItisexpectedthatmostofthebinarieswillcircularizeorwillhaveloweccentricitybythetimefrequencyoftheemittedGWreachthatfrequency.Nevertheless,highermodesofGWfromamassivesystem,athigheccentricity,willenterthedetectorsensitivityband. 34

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CHAPTER2DETECTIONOFGRAVITATIONALWAVESTherehavebeenongoingeffortsforthedetectionofGWsincetheirexistencewaspredicted.DetectionofGWwillbeadirecttestofthegeneraltheoryofrelativity.Thereconstructedwaveformswillprovideinsightintothedynamicsoftheastrophysicalphenomenongeneratingthem.Detectionsoveranextendedamountoftimewillhelpconstructaskymapofthelocationofthesources.ThedevelopmentofgravitationalwavedetectorswaspioneeredbyJosephWeberintheearly1960s[ 31 ].Hedevelopedtherstgravitationalwavedetector:weberbars.Madeofaluminum,weberbarswerecylindersdesignedtosetinmotionbecauseofanincidentGW.Sincethen,sensitivityoftheweberbarshasimproved.Presentlytherearemanyresonantbarsinoperation,suchasALLEGRO,EXPLORERandNIOBE.Duetoitsabilitytomeasuresmalllengthchanges,theMichelsoninterferometerhaslongbeenknownasaforemostinstrumentforthedetectionofGW.OneoftheinitialproposalstousealaserinterferometerasaGWdetectorwasdoneintheearly1960sbyWeber.Fowardattemptedthisbyusingaretro-reectortoreectabeamtoabeamsplitterandusedactivecontrolforlockingtheinterferometertoafringe.Heobtainedaspectralstrainsensitivityof210)]TJ /F5 7.97 Tf 6.59 0 Td[(16Hz)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2[ 32 ].Sincethen,multipleimprovementshavebeensuggested,whichincludepowerrecycling,squeezingetc.Longbase-linedetectorshavebeenconstructedinLivingston,LAandHanford,WA.Eachofthesedetectorhasarmlength4km.TheHanforddetectorhadanadditionaldetectorofarmlength2kmuntil2007,whichoperatedinparalleltotheprimarydetector(LIGO).InEuropea3kmdetectorhasbeenbuiltinCasina,Italy(Virgo)anda600mdetectorhasbeenbuiltinSarstedt,Germany(Geo).A3kmdetectoriscurrentlybeingbuiltinJapan(KAGRA)andthereisaproposaltobuildadetectorinIndia(INDIGO). 35

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2.1OverviewofGWdetectorsFigure 2-1 describesthebasicprinciplefortheGWdetectionusingaMichelsoninterferometer.AnincidentGWwillcausestrainpatternonthetestmassesarrangedasacircle.Forthetestmasses,intheplanetransversetotheincidentdirection,GWswilldeformthecircletoanellipse.TheorientationofthemajorandminoraxesoftheellipsewilldependontheshapeoftheGW. Figure2-1. AGWtravelingperpendiculartotheplaneofthediagram,willdistortacircularringoftestparticlesintoanellipse.ApassingGWasacertainphase,willcontracttestmassesalongonediagonal,whileexpandingitalongtheorthogonaldiagonal.Thesamewilloccurforthetwodiagonals,butvice-versa,afteraphasechangeof=4.Duetothis,aphaseshiftwillbeaccruedbythelightineacharm,therebychangingtheinterferencepatternproducedbythetwolaserbeams. Figure 2-2 depictsaninterferometerinadetailedmanner,outliningtheimportantsub-systems.LIGOoperatesa10Wlaserbeamoffrequency1064nm.TheInputModeCleanertransmitsthepre-stabilizedbeam,lteringoutanylightnotinthefundamentalGaussianspatialmode(higherspatialmodesproducedbeforethelaserbeamisincidentontheinterferometer).ThefrequencyofthisbeamisthenmatchedwiththefrequencyoftheFabry-PerotMichelsonarmcavity,usingaFaradayisolator.Toincreasetheamountoflaserpoweratthebeamsplitterandintheinterferometerarms,apowerrecyclingmirrorisusedtodirectthelightreectedfromtheFabry-Perot 36

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Figure2-2. Basicsub-systemsofaGWdetector.Imagecredit:iopscience[ 33 ] Michelsonbackintotheinterferometer.Theprimarybeamisthensplitatthebeamsplitter,whichtravelsalongthetwoarmcavities.Thecavitiesstorethebeamsandincreasetheeffectivepathlength.Finally,thebeamscombineatthebeamsplitter.Thephotodiodeattheasymmetricportreadstheoutput.Thesetupissuchthatthetwobeamsinterferedestructively. 2.2NoiseSourcesinInterferometricGWDetectorsGWdetectoroutputiscontaminatedbyavarietyofnoisesources.Thesenoisesourcescanbebroadlyseparatedintotwocategorized:displacementnoiseandsensingnoise.Displacementnoisescausemotionsofthetestmasses.Noisesourcessuchasseismicnoise,mechanicalresonances,etc.,comeinthiscategory.Sensingnoises,ontheotherhand,arephenomenathatlimittheabilitytomeasurethosemotions;theyarepresentevenintheabsenceoftestmassmotion.Shotnoise,thermalnoise,etc.areincludedinthiscategory.Figure 2-3 showsstrainofvariousdetectors.Strainherebearstheconventionalmeaningandisameasureofthedifferenceofthelengthofthe 37

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twoarmsofthedetector.Followingisthedescriptionofsomeoftheimportantnoisesources. Figure2-3. Strainsensitivities,expressedasamplitudespectraldensitiesofdetectornoiseconvertedtoequivalentGWstrain. 2.2.1SeismicNoiseSeismicnoiseisduetothemotioncausedbecauseoftheearth'ssurfacedrivenbywind,oceanwaves,humanactivityandlow-levelearthquakes.Thisnoiseisdominantonlyupto40Hzanddiesdownrapidly.Thisnoiseislteredbyisolationstacksandpendulumsanditismeasuredusingseismometersandaccelerometers.Powerdensityofseismicnoiseisroughlygivenas: 38

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S(f)/8><>:const.1Hz10Hz(2) 2.2.2MechanicalResonancesIsolationstacks,pendulumsandsuspensionsub-systemshavetheirownnaturalmodes,whichalsoshowsupinthestrain.Theseresonancescanbefoundinanyfrequencyregion.Theseresonancescanfurthercouplewithseismicnoisetoproduceupconvertednoise.Thesenoiseareinherenttotheinstrument,theireffectcanonlybereducedbyimprovingthesystemitself.Theseresonancescanbetrackedeasilybytheaccelerometers. 2.2.3ShotNoiseShotnoiseisdominantover100Hzanditarisesasphotonsdonothitthemirrorsatregulartimeintervals,causinguctuationinthemeasurementofthepositionofthemirrors.TheuctuationisfrequencydependentandisexpressedinEquation 2 ,wherePinisthepowerofthelaserbeam,isthewavelengthofthelaser,andsisthearmcavatitystoragetime. S(f)=~ Pinc 1+f 4s2!.(2)Apparently,thestraightforwardwaytoreducetheshotnoiseisbyincreasingPin.Thisincrease,though,comesatthecostofincreasingtheradiationpressure,duetothebeamsonthemirrors. 2.2.4RadiationPressureThisnoiseisdominantupto50Hz.Thelaserbeamfallingonthemirrorprovidesaforce,whichcausesuctuationinmirrorineacharm.Theuctuationsinthetwoarmsareuncorrelatedandcombinetoproduceapowerdensityof, S(f)=1 mf2L~Pin 23c,(2) 39

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wheremisthemassofthemirror,andListhelengthofthecavity. 2.2.5ThermalNoiseThermalnoiseliesinthecategoryofsensingnoiseandlimitsourabilitytodetectGW.Itcontaminatestheoutputbetweenfrequencies100and200HzandisduetotheBrownianmotionoftheatomsofthedetectormirrorsandsuspensions.Thermalnoiseduetothesuspensions,usedfortheisolationofthetestmassmirrors,arecalculatedfromtheFluctuation-Dissipationtheorem[ 34 ].Thefrequency(f)-dependentthermaldisplacementnoisex(f)isgivenbyEquation 2 x2(f)=kBT 23mff2o(f) f4o2(f)+(fw2o)]TJ /F4 11.955 Tf 11.96 0 Td[(f2)2,(2)whereTistheambienttemperature,misthemassofthependulum,(f)isthemechanicallossangleofthependulum,foistheresonantfrequency,kBistheBoltzmann'sconstantandwoisthebeamradius. 2.3TheLIGOandVirgoInterferometersLIGOandVirgoarekilometer-scale,power-recycledMichelsoninterferometerswithFabry-Perotarms.Thedetectorsaremostsensitiveinthefrequencyregion60Hz-300Hz,reachingstrainsofaslowas10)]TJ /F5 7.97 Tf 6.59 0 Td[(23=p Hz.TherearetwoLIGOdetectors.TherstoneislocatedinLivingston,Louisiana(L1)andtheotheroneinHanford,Washington(H1).Bothofthesedetectorshavearmlengthsof4km.Until2008,Hanfordalsohadasecondarydetectorwhichsharedonearmwiththeprimarydetector(H2).Thesecondarydetectorhadanarmlengthof2km.VirgohasadetectorlocatedinCasina,Italy(V1).Thisdetectorhasarmlengthof3km.Figure 2-4 showstheaerialviewsofthesedetectors.Figure 2-3 plotsthestrainsensitivitiesoftheLIGO-VirgodetectorsduringtheS6-VSR2/3run.TheeBBHsearchhasbeenconductedonthedatacollectedbytheLIGOandVirgodetectorsovertheS5-VSR1andS6-VSR2/3runs.S5-VSR1runwasconductedfrom 40

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Figure2-4. AerialviewoftheGWdetectors,locatedat,(top-left)Casina,Italy,(top-right)Livingston,USA(bottom)andHanford,USA.Imagecredit:[ 35 ] May2005andOctober2007,whileS6-VSR2/3runwasconductedfromJuly2009toOctober2010. 2.4DetectorResponseTheoutputstrainw(t)isthesumofdetectornoisen(t)anddetector'sresponse(t)totheGW, w(t)=(t)+n(t).(2) 41

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IntheTTgauge,thedetectorresponseiswrittenas (t)=F+h+(t)+Fh(t),(2)whereF+andFarecalledtheantennapattern.Antennapatterndependonthelocationofdetectorsontheearth'ssurfaceandaregivenbyEquation 2 F+(,)=1 2(1+cos2)cos2cos2 )]TJ /F4 11.955 Tf 11.96 0 Td[(cossin2 sin2F(,)=1 2(1+cos2)cos2sin2 )]TJ /F4 11.955 Tf 11.96 0 Td[(cossin2 cos2. (2) Variables,and aretheEulerAngleofthetransformationbetweendetectorframeandthewaveframe,withdetectorarmsalignedalongthexandtheyaxes.Figure 2-5 plotsp (F2++F2)asafunctionoftheskylocation.TheratioF=F+isalwayslessthanoneandrepresentsthesensitivityofthenetworktowardsthecross-polarizedcomponentofGW.ThisvariableisalsoplottedinFigure 2-5 42

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Figure2-5. p F2++F2andjFj=jF+jasafunctionofskylocationovertheearth'ssurfacefortheLHVnetwork.Detectorshavebeenassumedtobeequallysensitive. 43

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CHAPTER3REGRESSIONOFLINEARANDBILINEARNOISEINLIGOLIGOoutputgetscontaminatedbyavarietyofenvironmentalnoises.Thepresenceofnoisecanobscurethealreadyfaintgravitationalwavesignalinthedataandhencethedetection.InadditiontotheLIGO'sGWchannel,datafromthousandsofPhysicalEnvironmentalMonitors(PEM)arecollectedtocharacterizethecouplingbetweentheGWchannelandtheenvironment.Manyenvironmentaldisturbancesrecordedinthedetectorsareduetothelinearcouplingtotheenvironment,forexample,powergridharmonics(powerlines).Also,thedetectordatacontaintheup-conversionnoise,whichisproducedbytheinterferenceoftwoormoreenvironmentalnoises.Mostprominentexampleoftheup-conversionnoiseisthebi-linearcouplingoftheseismicnoisewhichappearsasside-bandsaroundthepowerlinesandcalibrationlines.Toimprovethedataqualityandincreasethedetectionsensitivity,severalmethodshavebeenusedtoidentifyandultimatelyremovetheenvironmentalnoisefromthedetectordata[ 36 ].Theexamplesincludetheremovalofseismicnoiseusingthearrayofseismometers[ 37 ],removalofthelinenoisewiththeniteimpulseresponse(FIR)lters[ 38 ]andremovalofthecross-talkbetweenanauxiliarychannelandthechannelofinterestbyusingthefrequency-domainlinearregression[ 39 ].Thischapteraddressestheproblemofthenoiseregression(predictionandcancellation)byconstructingaspecialbankoftheWiener-Kolmogorovlters.TheenvironmentalnoisecontributiontotheGWchannel(targetchannel)ispredictedbylteringdatafromoneormorePEMchannels(witnesschannels)simultaneously.Tocapturethefrequencydependentcorrelationbetweenthetargetandthewitnesschannelsandforbetterestimationofthelterparameters,theregressionanalysisisperformedinthetime-frequency(wavelet)domain.Introductionofmultiplewitnesschannelsinpredictingthetargetchannelnoiseincreasestheeffectivenessoftheconstructedlterbanks.Thestrengthofthepresentedregressionanalysisliesnotonly 44

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inpredictingthelinear,butalsotheup-conversionenvironmentalnoise.ItisachievedbycreatingthesyntheticwitnesschannelsfromPEMandsimulatedchannels,whichmimicthephysicalprocessoftheup-conversion.Thisbecomesusefulastherearenowitnesschannelsthatmeasurebilinearnoisedirectly.Theproposedregressionmodelisexpectedtobemoreapplicablewhenanewgenerationofadvanceddetectorsbecomeoperable. 3.1RegressionAnalysisInapplicationtotheregressionofLIGOdataweconsidertheWiener-Kolmogorov(WK)lters[ 40 ].Givensampleddatafromaselectedauxiliarychannelwmeasuringenvironmentalnoise(witnesschannel),aniteimpulseresponse(FIR)lteracanbeconstructedtopredictthenoisecontributionxintotheGWchannelh(targetchannel).Theoutputofthelterisgivenbytheexpression x[i]=LXj=)]TJ /F5 7.97 Tf 6.58 0 Td[(Lajwi+j,(3)whereajaretheltercoefcientsand2L+1isthelterlength.Thelterisobtainedbyminimizingthemean-squareerror 2=N+LXi=L h[i])]TJ /F5 7.97 Tf 19.84 14.94 Td[(LXj=)]TJ /F5 7.97 Tf 6.59 0 Td[(Lajwi+j!2,(3)whereNisthenumberofdatasamplesusedfortheestimation(training)ofthelter.TheltercoefcientsajarecalculatedbysolvingtheWiener-Hopf(WH)equations Rxxa=ptx.(3)whereptxisthevectorwith2L+1componentsrepresentingthecross-correlationsbetweenthewitnesschannelwwiththetargetchannelhandRxxis(2L+1)(2L+1)auto-correlationmatrixforthewitnesschannel.ThematrixRxxispositivedeniteandthereforenon-singular,yieldingauniquesolutionfortheltercoefcients.Usually,thesolutionisobtainedwiththeLevinson-Durbinalgorithm[ 41 ]withouttheexplicitinversion 45

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ofRxx.ThepredictednoiseiscalculatedbyusingEquation 3 andcanbesubtractedfromthetargetchannel,thusreducingthenoiseintheGWdata.AspecialcaseoftheWienerlteristhelinearpredictionerrorlter,whichisobtainedbyminimizingthefollowingequation 2=N+LXi=L t[i])]TJ /F5 7.97 Tf 27.71 14.95 Td[(LXj=)]TJ /F5 7.97 Tf 6.58 0 Td[(L,j6=0ajti+j!2.(3)Inthiscase,thedatasamplei=Lispredictedbythesurroundingsamplesinthesametimeseries.Thisiswellsuitedforthepredictionofaquasi-monochromaticnoiseinthetargetchannel.TheRMSvalueofthewitnesschanneldatacanbebrokendownintocoupledanduncoupledparts.Thetargetandwitnesschanneldataarewhitenedbeforebeingpresentedtotheanalysis,e.g.,hasaRMSvalueof1.Followingthis,onecanwrite, 1=r2c+r2u,(3)whererc=ruistheRMSofthecoupled/uncoupledpartinthewitnessdata. 3.1.1RegressionwithWDMTheinterferometerdataspanawidefrequencyband(0-8kHz)andalargedynamicrange.Therefore,constructionoftheWKltersinthetimedomainrequireslonglters.Forexample,toregresspowerlineharmonics(15inthe0-1kHzband)aWKlterwithapproximately10000coefcientsshouldbeconstructed.Apartfromthecomputationalcomplexityassociatedwiththeinversionofthe1000010000matrix,oneneedtodetermineaccurately10000ltercoefcients,whileonly30parameters(amplitudesandphasesof15harmonics)arerelevant.Also,theWKlterisaffectedbythespectralleakage,whichmayfailtocapturealldetailsofthecorrelationbetweenthewitnessandthetargetchannels.Tosolvetheseproblems,weproposetousethefastWilson-Daubechies-Meyertransformation(WDM)[ 42 ]tosplitdataintothefrequencysub-bands(time-frequencymap).Figure 3-1 showsanexampleofthe 46

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WDMtime-frequencymapforasegmentofdatafromtheHanforddetector.TheWDMtransformationisorthonormal,invertibleandhasalowspectralleakage,whichmakesitauniquetoolfortheregressionanalysis.Thedataineachfrequencysub-band(WDMlayer)areatimeseriesusedfortheconstructionofasingleWKlterasdescribedinthebeginningofthissection.Therefore,insteadofonelongWKlter,abankofmuchshorterWKltersisconstructed.Eachlteristrainedindividuallytocapturedetailsofthetarget-witnesscorrelationinthecorrespondingsub-bandwithjustafewltercoefcients.Forexample,atypicalanalysisinthefrequencyband0-2048Hzinvolvestheconstructionof2048independentWKlterswith11coefcientseach.Therefore,insteadoftheinversionofaverylargematrix,onehastoinvert2048muchsmallermatrices,whichsignicantlyreducesthecomplexityoftheregressionproblem.ByapplyingtheinverseWDMtransform,cleandatah)]TJ /F4 11.955 Tf 13.1 0 Td[(xinthetimedomaincanbeobtained. 3.1.2MultipleChannelsTheGWchannelgetscontaminatedbytheenvironmentalnoiseenteringfromdifferentphysicallocationsanddirections.Itisacommonlyencounteredscenariowhenaparticularenvironmentalnoisecoupledtothetargetchannelismeasuredbythewitnesschannelssituatedatdifferentlocations.Ingeneral,whenmakingapredictionoveraperiodoftime,datafrommultiplewitnesschannelsareexpectedtoprovidemorecompleteinformationabouttheenvironmentandimprovetheestimationoftheltercoefcients.Forexample,theseismometersareinstalledatvariouskeylocationsinthegroupsofthreethroughouttheLIGOsites,measuringseismicnoisealongspeciedx,yandzdirections.Therefore,apredictionoftheseismiccontributiontothetargetchannelwouldrequireasimultaneoususeofallthreeseismicchannels.Theregressionanalysis 47

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Figure3-1. Anexampleoftime-frequencymapobtainedbyapplyingWDMtransformonLIGOdata(notalllayershavebeenshownforclaritypurpose).Layershavebandwidthof1Hz.Regressionanalysisisperformedforeachlayerseparately.Oncethepredictioniscalculatedandsubtractedfromeachfrequencysub-bandofthetargetdata,theinversetransformisappliedtobringthedatabacktothetimedomain. addressesthemultiplewitnesschannelcasebyextendingEquation 3 : 2=N+LXi=L h[i])]TJ /F5 7.97 Tf 19.83 14.95 Td[(LXj=)]TJ /F5 7.97 Tf 6.59 0 Td[(Lajxi+j)]TJ /F5 7.97 Tf 20.78 14.95 Td[(LXk=)]TJ /F5 7.97 Tf 6.58 0 Td[(Lbkyi+k)]TJ /F5 7.97 Tf 21.41 14.94 Td[(LXm=)]TJ /F5 7.97 Tf 6.58 0 Td[(Lcmzi+m+!2. (3) 48

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Wherehisthetargetchannel,x,y,zarethewitnesschannelsanda,b,caretheltercoefcients.ThecorrespondingWiener-Hopfequationis Q266666664abc377777775=266666664ptxptyptz377777775,Q266666664RxxCxyCxzCyxRyyCyzCzxCzyRzz377777775.(3)whereRxx=CxyareHermitianToeplitzmatricesofthewitnessauto/crosscorrelationandphx,phy,phzarethecross-correlationvectorsbetweenthewitnessandthetargetchannels.Theoutputofthemulti-channellterisgivenby: x[i]=LXj=)]TJ /F5 7.97 Tf 6.59 0 Td[(Lajxi+j+LXk=)]TJ /F5 7.97 Tf 6.59 0 Td[(Lbkyi+k+LXm=)]TJ /F5 7.97 Tf 6.58 0 Td[(Lcmxi+m.(3)Anexampleofimprovementinpredictionduetothemulti-channelregressionispresentedinTable 3-1 .AnotherexamplereleatedtodetectornoiseisshowninFigure 3-2 .Itshowstheregressionofseismicnoiseusingthecoilcurrent(CC)channels.TheCCchannelsmeasurethecurrentintheelectromagnetsdrivingsmallmagnetsattachedtothetestmasses.Themainfunctionoftheseelectromagnetsistocounteracttheseismicmotionandholdthetestmassesinplace.Sinceseismicnoiseatthesitesusuallyaffectsnearbytestmasses,atagiventime,onlyasmallsubsetoutoftotal16CCchannels(positionedattheinitialandendtestmass)exhibitscorrelationwiththeGWchannel.Therefore,theregressionwithasingleCCchannelisnoteffective.Onecanconstruct16regressioncasesandapplythemconsecutively,oneCCchannelatatime.However,asFigure 3-2 shows,thebestperformanceisobtainedwiththemulti-channelregressionwhenall16channelsareusedtoconstructasingleregressionlter.Multi-channelanalysis 3 presentstwoshortcomings.1)First,ifwitnesschannelswithhighlycorrelateddataareincluded,itwillmakethematrixQrankdecient;e.g., 49

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Table3-1. Anexampleofimprovementinpredictionduetomulti-channelanalysis.Case(a)iswhenthecouplingis20Hzharmonicnoisebetweenthewitnessandthetargetchannelwithrc=0.7,andcase(b)iswhencouplingisbroadbandnoisebetweenthetargetandthewitnesschannelswithrc=0.6. No.ofchannels124816 (a)RMSofresidual0.4050.2960.2190.1650.127 (b)RMSofresidual0.8000.6840.5510.4220.309 Table3-2. IncreaseinRMSofpredictionwithincreaseinnumberofwitnesschannels.Thereisnocorrelationbetweenwitnessandtargetchannels,andincreaseinRMSisduetotheover-tting(witnesschannelsmakingpredictionevenintheabsenceofcorrelation). Numberofchannels124816 RMSofprediction.070.090.130.190.26 asdiscussedbefore,thedatafromthreeseismometerscanbeveryhighlycorrelated.2)Second,evenintheabsenceofthecouplingbetweentargetandwitnesschannels,theRMSvalueofthepredictionincreaseswithincreaseofthenumberofwitnesschannels.InthecaseofMwitnesschannelscontainingwhitenoise,increaseinRMSisproportionaltop M.IncreaseinRMSisduetoover-ttingoftheregressionanalysis(Table 3-2 ). 3.1.3RegulatorsBothshortcomingscanbeaddressedbyusingregulators.TheEquation 3 canberewrittenas 266666664abc377777775=O)]TJ /F5 7.97 Tf 6.59 0 Td[(1O266666664chxchychz377777775,266666664000010002377777775,(3)whereOistheorthogonalmatrixthatdiagonalizesQandi(i=0,1,2...)aretheeigenvaluesarrangedinthedecreasingorder.Theeigenvaluedistributioncapturestheprincipalregressioncomponentsandidentiesastrongcorrelationinthewitnessdata.Figure 3-3 plotstheeigenvaluedistributionsfordifferentregressioncases.Becausethewitnessdataarewhitenedbeforeperformingtheanalysis,themeanoftheeigenvalue 50

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Figure3-2. Predictionofseismicnoiseusing16coilcurrentchannel(discussedlater).Blackcurveistheoriginalpowerspectrum.Redcurveisobtainedbyconditioningthedatausingmultichannelanalysis.Bluecurve(purplecurvewithadifferentorderofapplication)isobtainedbyconditioningthetargetdatamultipletimesbyeachofthecoilcurrentchannel(Equation 3 ). distributionisalwaysequaltoone.Wheni>>1,thisisanindicationofstrongnarrow-bandnoiseartifactsinthewitnessdata.Respectively,wheni<<1,thewitnessdataishighlycorrelatedandthematrixQhasalowerrank.Thesmalleigenvaluesshouldberegulatedtoavoidun-physicalsolutionsoftheWHequation.Theregulated 51

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Figure3-3. EigenvaluedistributionofmatrixQforsomecases.Black-all16witnesschannelscarrythesameharmonicnoisealongwithadditionalwhitenoise.Red-allthe16witnesschannelscarrywhitenoise,noneofthewitnesschannelsarecorrelatedwitheachother.Green-allthe16witnesschannelscarrywhitenoiseandallarecorrelatedwitheachother. solutionisobtainedbymodifyingthematrix, 2666666666666664abc3777777777777775=O)]TJ /F5 7.97 Tf 6.59 0 Td[(1O2666666666666664chxchychz3777777777777775,266666666666666400000000000p00 000C00000C3777777777777775t(3) 52

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wheretheeigenvaluesbelowthethresholdtasettosomeconstantC.Wedistinguishthreeregulationschemes:hard(1=C=0),mild(C=0)andsoft(C=p).TheproposedregulatorshandletheproblemoftherankdeciencyinthematrixQbysuppressingtheeigenvectorswithsmalleigenvaluestherebypreventingun-physicalsolutions.Inthecaseofnocorrelationbetweenthewitnessandthetargetchannels,theregulatorsalsoconstrainthesystemtoreducetheRMSoftherandomnoiseintheoutputofthelter. 3.1.4EffectofregulatorsonpredictionTheregressionparametersandthetypeoftheregulatortousedependonthetypeofproblemonhand.Thecouplingoftheenvironmentalnoisetothetargetandthewitnesschannelsusuallychangeswithtime.Hence,nogeneralprescriptioncanbemadeontheapplicationoftheregulationschemes.Below,wediscussguresofmeritforsomefrequentlyencounteredcases.Inalldiscussedcases,thetargetandthewitnesschannelsdifferbyanadditionalGaussiannoiseinthewitnesschannel.1)Allthreeregulatorsareverycloseinperformanceforthepredictionofquasi-monochromaticnoise.Figure 3-4 comparestheregressionofsimulatedmonochromaticnoisewithdifferentregulators.Asonlyfeweigenvaluesaresignicantinthiscase,allregulationschemesremovethenoisepeak.Theeffectoftheover-tuningisclearlyvisiblearoundthepeakwhennoregulatorisused.Thehardregulatoristhebestchoiceforsuchcases.2)Inthecasewhenallwitnesschannelsmeasureindependentbroadbandnoise,theeigenvaluedistributionisGaussian.Ifnoneofthewitnesschannelsiscorrelatedwiththetargetchannel,thegoalistoreducetheRMSvalueoftheprediction.Table 3-3 showstheperformanceofvariousregulators.Inthiscaseusingahardregulatorwithahighthresholdworksthebest.Onthecontrary,ifthecorrelationexistsonlybetweenonewitnesschannelandthetargetchannelatatime,regulatingalargenumberofeigenvectorsresultsinreducedperformance,asshowninTable 3-4 .Asoftregulator 53

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Figure3-4. Predictionofsimulatedmonochromaticnoiseusingdifferentregulatorswitht=1.The99.5Hzlinehasbeencleanedusingthehardregulator,the101.5Hzlinehasbeencleanedusingthemildregulator,the103.5Hzlinehasbeencleanedusingthesoftregulator,andthe105.5Hzlinehasbeencleanedwithnoregulators. ornoregulatorworksbestforthiscase.Thiscaseisonlyrelevantwhenthecorrelationbetweenthewitnessandthetargetchannelisstrong.Ifthecorrelationisweak,duetothecontaminationbyuncorrelatedwitnesschannels,regressionanalysisdoesnotperformwell;e.g.,with16witnesschannelsandoneofthemcorrelatedwiththetargetchannelwithrc=.6,theRMSoftheresidualis0.8.3)InthecasewhenallthewitnesschannelshaveGaussiannoisepresentandmorethanonearecorrelatedwiththetargetchannel,thereissplittingintheeigenvalue 54

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Table3-3. Effectofregulatorswhennoneofthewitnesschanneloutof16iscoupledwiththetargetchannel.TableshowstheRMSvalueoftheprediction. Threshold0.250.50.751.01.251.5 Hard0.260.260.230.180.120.05Mild0.260.260.240.200.180.17Soft0.260.260.240.240.180.18 Table3-4. Effectofregulatorswhenonlyonewitnesschanneloutof16iscoupledwiththetargetchannel.TableshowsRMSofresidualforrc=0.99. Threshold0.250.50.751.01.251.5 Hard0.1520.1520.3810.6170.8110.953Mild0.1520.1520.2620.3520.3960.410Soft0.1520.1520.1610.2100.2830.362 distribution(Figure 3-3 ).Alltheregulatorsworkequallywellinthiscase.Forinstance,when8outof16witnesschannelsarecorrelatedwiththetargetchannelwithrc=.6,theRMSofresidualiscloseto.4foralltheregulatorsandthresholdvaluesofuptot=1.5. 3.2ApplicationsofWDMRegressionThereisnoonesingleprescriptionfortheWDMregression.Eachregressioncasedependsonasetofavailablewitnesschannelsandtypeofthenoisetobepredicted.Below,weconsiderseveraltypicalregressioncases. 3.2.1Regressionofnoisewithlinearcoupling.OneofthemostvisiblenoiseartifactsinLIGOdetectorsarepowerharmonicsshowingupinthenoisespectrumasthequasi-monochromaticlinesatmultiplesof60Hz.Theycanbemonitoredwiththevoltagemonitorsandmagnetometers.Oneorseveralvoltagemonitorscanbeusedaswitnesschannels.Figure 3-6 showsthenoisespectrumbeforeandaftertheapplicationoftheWDMregressionanalysis.Note,thatthelineadjacentto120HzinFigure 3-6 (middle)isnotremovedbecauseitisnotmonitoredbythevoltagechannels.OtherfrequentlyvisiblefeaturesintheLIGOnoisespectrumareduetothemechanicalresonancesoftheseismicstacks.Theselinesaremuchwiderthanthe 55

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Figure3-5. Performanceofvariousregulators(withthreshold0.5)insubtractionofseismicnoiseusing16CCchannels.Red-Noregulator,Green-Softregulator,Blue-Mildregulator,andCyan-Hardregulator.Effectofregulatorisclearlyvisible.ItcanbededucedthatonlyoneCCchanneliscorrelatedwiththeGWchannelatatime. 56

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Figure3-6. Cleaningofpower-linesusingvoltagemonitor(H0:PEM-LVEA2 V1)asthewitnesschannels.Blackcurveistheoriginalspectrum(H1:LDAS-STRAIN);redcurveisobtainedaftersubtractingtheprediction. powerlines,theycanstillbeefcientlyremovedbyusingregulatedregressionwithmultiplewitnesschannels(Figure 3-7 )Amoredifcultlinearregressioncaseisthepredictionofthebroadbandnoise,likeseismicnoise.Figure 3-2 showstheexampleoftheseismicnoiseregressionusing16coilcurrent(CC)witnesschannels(seeSection 3.1.2 ).WedonotobserveasignicantrankdeciencyofthewitnessmatrixindicatingthatthereisnosignicantcorrelationbetweentheCCchannels.Asexpected,inthiscase,thebestregressionisobtainedwhennoregulatorisused. 57

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Figure3-7. Subtractionofmechanicalresonancesusing14accelerometersand2microphones.Blackcurveistheoriginalspectrum(H1:LDAS-STRAIN);redcurveisobtainedaftersubtractingpredictionusingamildregulator(withthreshold1). 3.2.2Regressionofnon-linearnoiseSomenoiseartifactsinLIGOareproducedbytheinterferenceoftwo(bi-linearnoise)ormorenoisesourcesinthedetector.Themajorityofthebi-linearnoisecasesareduetotheup-conversionofthelowfrequencyseismicnoise,whichisobservedastheside-bandsaroundthepowerlines,calibrationlines,violinmodesandothernarrow-bandfeaturesinthenoisespectrum.TherearenodirectwitnesschannelsinLIGOtomonitorsuchbi-linearnoise.AstheCCchannelsprovideagoodmeasureofthelowfrequencyseismicnoise,theycanbeusedalongwiththeotherchannels 58

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tosynthesizethebi-linearwitnesschannels.SuchsyntheticwitnesschannelsareobtainedbymultiplyingtheCCtimeseriesbythecarriertimeseriesrepresentinganarrow-bandspectralartifact.Forexample,whenthecarrierisoneofthepowerlines,thebi-linearwitnesschannelcanbeobtainedasaproductofthetimeseriesfromaCCchannelandavoltagemonitorormagnetometer.Figure 3-8 showstheexampleoftheup-conversionnoiseregressionaround180Hzpowerlinewith16syntheticwitnesschannelsconstructedfromtheCCchannelsandonevoltagemonitorchannel.Usually,suchcasesneedtoberegulatedtoavoidtheregressionartifacts. Figure3-8. Removaloftheup-conversionnoisearound180Hzpowerlinewith16syntheticwitnesschannelsconstructedfrom16coilcurrentchannelsandthevoltagemonitor 3-6 59

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Often,thecarriertimeseriesarenotreadilymeasuredbyanyPEMchannels.Ifthecarrierfrequencyisquitestable,thecarrierwitnesschannelcanbesimulatedwiththemonochromaticsignal.Figure 3-9 showshowthismethodworksfortheregressionoftheup-conversionnoisearoundthecalibrationlines.Also,thecarrierwitnesschannelcanbeextractedfromthestrain(target)datawiththelinearpredictionerror(LPE)lter.Inthiscase,theLPElteristrainedontheWDMdatainthenarrow-band(1Hz)aroundthecarrierfrequency.TheoutputoftheLPElterisusedasthecarrierwitnesschannel.Figure 3-10 showsthesubtractionoftheseismicup-conversionnoisearoundthebeamjitterpeaks,whichwerenotmonitoredbyanywitnesschannelsduringtheLIGOscienceruns. 3.2.3MonitoringofGravitationalWaveDataLIGOhashundredsofauxiliarychannels.Someofthesechannels(likevoltagemonitors)areveryeffectiveinthemonitoringofthenoiseartifactspresentinthetargetdata.However,manychannelsdonotexhibitanysignicantcorrelationwiththestrainchannelorthiscorrelationvariessignicantlyoverthetime.Theproposedregressionanalysiscanbeusedtomonitorthecorrelationbetweenthetargetandthewitnesschannelsduringthedatatakingruns.ThemeasureofthecorrelationistheRMSvalueofthewhitenedpredictionsignal,whichshouldbebetween0(nocorrelation)and1(strongcorrelation).SinceitishardtovisualizeallRMSvaluesfromhundredsofLIGOchannels,themaximumRMSvalueamongalltheWDMlayersineachwitnesschannelcanbeusedtoidentifythestrongestcorrelationwiththestrainchannel.Figure 3-11 showsthemaximumRMSvalueforapproximatelysixhundredLHOchannels(allthechannelswhichhavesamplingrateof2048Hzorhigher).Toobtainthisgureofmerit,foreachwitnesschannelthe2048regressionlterswereconstructedcoveringa1Hzfrequencysub-bandeach.TheRMSvaluesofthelteroutputwerecalculatedandthemaximumvalueisdisplayedasafunctionofthechannelidenticationnumberandtheGPStimeofthetestdatasegment.Inthisplot,thecontributionofthepowerlines 60

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Figure3-9. Removalofup-conversionaround113Hzcalibrationlineusingcoilcurrentchannelsandasimulatednoise. hasbeenexcluded,astheyarecorrelatedwithmanychannels.Thereisnosignicantcorrelationformostofthetestedchannels:greenandblueareasintheplotwhereRMS0.5.Someauxiliarychannelshaveastrongcorrelationwiththestrainchannel:yellowandredareasintheplot.Thereareotherpossibleguresofmeritthatcanbeused.Forexample,theaverageRMSvalueofthepredictednoisefromseveralWDMlayersisagoodindicatorofthebroad-bandcorrelationwiththestraindata.Also,thewitnesschannelscanbemonitoredingroupsbyusingthemulti-channelregression.Thismonitoringmethodmay 61

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Figure3-10. Removalofup-conversionaroundjitterpeaksusingcoilcurrentchannelsandaself-predictedcarrier. helptoidentifychannels,whichareweaklycorrelatedwiththeGWchannelindividually,butmayrevealastrongcorrelationasagroup. 62

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Figure3-11. ThemaximumRMSvalueasafunctionofthechannelidenticationnumberandGPStime. 63

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CHAPTER4SEARCHALGORITHMTherearetwomethodswidelyusedtosearchfortheGW.TherstoneisamodeledsearchinwhichthetypeofsourcegeneratingGWisassumedapriori.TheGWproducedfromCompactCoalescingBinary(CBC)havebeenestimatedaccurately,and,withtheirexistenceindirectlyveried,thismethodmainlyfocusesonsearchingGWfromCBCsources.Thetemplatebank(ofwaveforms)isconstructedsoastooverlapandllapartoftheparameterspaceofCBCevents.Theothersearchmethodisanunmodeledsearch,generallycalledtheburstsearch,inwhichthereisnoprecedentedknowledgeofthesource.GWsources,suchassupernovaecollapse,lieinthiscategory.CBCsearch,beingmodeldependent,islessaffectedbythebackgroundnoisewhilesearchingonlyforlimitedparameterspace.Incontrast,burstsearch,beingmodelindependent,isaffectedmorebythebackgroundnoise,whilesearchinginwiderparameterspace.Bothsearchescanbeconductedincoherently,inwhichtriggersaregeneratedsingularlyforeachdetector,whichcanbecombinedlater[ 43 ].Thesearchcanalsobeconductedcoherentlywhendatafromthedetectorsareusedsimultaneouslytoproduceatrigger.Acoherentsearchismorenaturalfromthedataanalysisperspective,asasingleGWeventcanbevisibleinmultipledetectors.ItispossibletoreconstructtheGWwaveformandthesourcelocationinamorereliablewaycomparedtoincoherentmethods.Moreover,coherentmethodsallowimplementationofconsistentrelationshiponthedatastreamfromdifferentdetectornetworksandthereconstructedsignal.Theapproachforthepresentedworkiscoherent,whichconductslikelihoodanalysisontime-dependentquantities(suchas)inthetime-frequencyplane,asopposedtoonlythetimeorfrequencydomain.Tomitigatetheaffectofbackgroundnoise,analysisusesregulatorsandconstraints[ 44 ]. 64

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ThecollectionofGWsignal,fromeBBH,containsalargevarietyofpossiblewaveforms,whichcouldhavevaryingbandwidthinthefrequencydomain.Hence,ingeneral,analysismethodssearchingforsuchGWmustberobustwithrespecttotheexpectedwaveforms.Itisconvenientthatthealgorithmremainasrobustaspossibleagainstthepossiblevarietyofwaveforms.However,asthesearchinvolvessearchofGWfromparticulartypeofsources,certainsearchparametersneedstobeconstrained.eBBHsearchhasbeenconductedusingtheburstpipelinecoherentwaveburst(cWB).Inthepast,anolderversionofthispipelinehasalreadybeenusedtoconductallskyandintermediatebinaryblackholes(IMBH)search. 4.1Time-FrequencyAnalysisEachrunconsistsofreadingdatafromthedisk.Analysisisperformedonthedataleftaftertheapplicationofcategoryagsonthesciencedata.Originaldatahasasamplingrateof16384Hz;thisissampleddownto1024Hztoreducethecomputationalcost.Itdoesnotaffecttheresultsastheinformationabovethisfrequencyisnotrelevanttothissearch.Readdataarewhitenedtonormalizetheenergyacrossfrequencybandsandconditionedtoremoveperiodicinstrumentartifactssuchaspowerlinesandviolinmodes.Followingthis,itisdecomposedthroughsixresolutionswiththeWDMwavelettransformation(discussedinthissamesection)[ 42 ].Anexampleoftime-frequencymapsforindividualdetectorsisdisplayedinFigure 3-1 .Pixels(ontheT-Fmap)crossingadenedthresholdarecollected.Pixelssatisfyingaclusteringschemearecombinedanddenedasacluster.Coincidentclustersfromdetectorsinthenetworkarecombinedtogethertocreateasupercluster.CoincidenceisnotstrictlyimposedbutisgivensometolerancetoaccommodatethefactthatGWarrivesatdifferentdetectorsatdifferenttimes.Likelihoodanalysisisperformedonthesuperclusterdata(x). 65

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TheTFtransformationisobtainedbyapplyingltercoefcientonthetimeseriesofthedetectordata wnm=Xkfnm[k]x[k],(4)wherefnmaretheltercoefcients,denedas, wn0=Xk2Zx[2nM+k][k] wnm=p 2RCm+nXk2Zeikm=Mx[nM+k][k],0
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Figure4-1. Thetopplotsshow(!)andhighlightthedeningparameters:thewidthofthetopatregion(black)is2A,thetransitionregions(blue)aredenedbyB,andtheredlinespansthenominalbandwidth2A+B.Theshapeofthetransitionregiondependsontheparameterninn;n=4fortheleftplotandn=2fortherightplot.Thebottomplotsshowthecoefcientsoftheirdiscretetimedomainrepresentation.IntheseexamplesM=4. 4.2OverviewofthePipeline 4.2.1CoherentAnalysisStrain()isthedifferentialeffectofaGWonpathlengthinaninterferometeroverthetotalpathlength.Straindependsonthedetector'sresponse,whichisgivenas, (,,t)=F+(,)h+(t)+F(,)h(t),(4)whereF+andFarethedetectorantennapattern.Fromthispointonward,thenotationintheangulardependenceofF,duetotheinclinationofagravitationalplanewavewith 67

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respecttothedetector,willbeomitted.Furthermore,sincetheGWsignalsoccuroverarelativelyshorttimeperiod,theimplicitdependenceofangularbeampatternsontime,becauseofthemotionoftheEarth,isignored.Givenatimeseriestobeanalyzed,atime-frequencydecompositionofthisseriesintothewaveletdomain(w)isperformed.cWBdoesalmostallofitscalculationsinthisdomain.ThereasoningforthisisexplainedbelowinthesectiononcWB.Giventhisrepresentation,theremainingdenitionsfollowthecurrentcWBalgorithmdescription[ 45 ].Forasingledetectorwithanoisespectrumn,wewritethewaveletseriesasatime-frequencydecompositionwithindices(i,j)fortimeandfrequencyrespectively, w(i,j)=n(i,j)+(i,j).(4)LIGOconsistsofanetworkofmanydetectors,andtheformalismisextendedtoanarbitrarynumberofKdetectors,withindexdenotedbyk, w(i,j)=w1(i,j) 1(i,j),,wK(i,j) K(i,j),f+=f+ 1(i,j),,f+ K(i,j),f=f 1(i,j),,f K(i,j). (4) Therstisthedatavectorandthesecondistheantennapatternvector.Herekrepresentsthesquarerootofthevarianceofthedetectordataindicatedbytheindex.Thisactioncanbeunderstoodasthewhiteningprocessdonepriortotheselectionofenergeticpixels. 4.2.2DominantPolarizationFrameThedetectorresponsetoaGWdependsontheproductofboththeantennapatternsandthepolarizationofGW.Thewaveframe(abouttheaxesorthogonaltotheplanedenedbyh+andh)isfreetorotate,becausetherotationsdonotaffectthedetectorresponse.Theanglewhichsatisestheconditionf+f=0ischosenbecause 68

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itdisentanglesthelikelihoodfunctional.Thismakesitsformparticularlysimplebecausethecrosstermscancel.ThisframeiscalledtheDominantPolarizationFrame(DPF). 4.2.3Un-constrainedLikelihoodCoherentWaveBurst(cWB)usesaun-constrainedlikelihoodmethodforthedetectionandthereconstructionofwaveforms[ 45 ].Thismethodisbasedonacriterionknownasthelikelihoodratio.Itexpressestheprobabilitythatthesignalisagravitywaveovertheprobabilitythatitisdetector'sintrinsicnoise.Foratime-frequencyvolume()oftheeventanddetectorindexk,thisisexpressedas: \(w)=Ykexp)]TJ /F4 11.955 Tf 9.3 0 Td[((wk)]TJ /F3 11.955 Tf 11.96 0 Td[(k)2 2k=exp)]TJ /F4 11.955 Tf 10.49 8.09 Td[(w2k 2k.(4)Thenumeratoristobeunderstoodasthelikelihoodofthesignalhypothesisanddenominatoristhelikelihoodofthenullhypothesis.Thisratiobecomeslargeforlargeorsmallnoisen.Thenaturallog(L)ofthelikelihoodratiois, L=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2ln)-278(=Xw(f+h++fh)+(jf+j2h2++jfj2h2).(4)NotethatthenoisevariancehasbeenabsorbedintothedenitionoffasperthedenitionoftheantennapatternmatrixinEquation 4 .ThecalculationhasbeenperformedinDPF.Thevariationofthisfunctionalinthetraditionalmethodofthecalculusofvariationsyields: wf+=jf+j2h+,wf=jfj2h.(4)Thisismorecompactlystatedbynormalizingfsuchthate=f jfj,nowexpressedas jf+jh+=we+,jfjh=we.(4)Theseequationsarethensubstitutedbackintotheoriginallikelihoodfunctionaltoobtainthemaximumlikelihoodstatisticforagivenskylocation.Inordertomaximizeover 69

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sourcecoordinates,thelikelihoodiscalculatedinthefollowingway, Lmax=maxX(we+)2+(we)2.(4)Thiswillberecognizedasthenormalizedantennapatternstakenintoaninnerproductwiththedatavector,maximizedoverskypositions.Equation 4 canberewrittenas, Lmax=XwPwT,(4)wherethematrixPistheprojectionconstructedfromthecomponentsoftheunitvectorse+ande: Pnm=en+em++enem.(4)Theindicesnandmrunoverallthedetectorsinthenetwork.Projectionresolves(inthedominantpolarizationframe)thedatavectorintoavectorinthesignalplanedenedbye+andeandtheorthogonalnullspace(tobeunderstoodasreconstructeddetectornoise).Figure 4-2 summarizesthisresolution. Figure4-2. Thegeometricalinterpretationofthelikelihoodanalysis.Thedatavectorxisprojectedintothesignalplane.Theprojectionofxontothisplaneis,andtheorthogonalprojectionisn,thenullvector. 70

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ThemaximumlikelihoodratiostatisticLmaxisaquadraticform,whichcanbesplitintotheincoherentEiandcoherentEcparts,denedas: Ei=XXnwnPnnwn,(4) Ec=XXn6=mwnPnmwm.(4)Thesecoherentstatistics,togetherwiththeenergyofthenullstreamEn,areusedintheburstsearchesfortheconstructionofselectioncuts.Thestatistic,coherentenergy(Ec),giveninEquation 4 ,dependsonthecross-correlationtermsbetweenthedetectorpairs.IntheeventofapresentGW,Ecisexpectedtobehighduetostrongcross-correlationinthedatastreamofthedetectorpairs.Thenetworkcorrelationcoffecientdenedascc=En=(En+Ec)wasusedasaselectioncutintheeBBHsearch.Thepipelineusespixelsofvariousresolution,ofwhichprincipalcomponentsareextractedfortheestimationoflikelihood.Whenallthepixelsareusedinthecalculationofthisratio,theratioistermedasnetccwhile,whenonlyprincipalcomponentsareuseditistermedassubcc. 4.2.4DualStreamAnalysis/PolarizationConstraintLikelihoodanalysiscanbeperformedonnoise-scaleddatastreamwaswellasonits90phaseshiftedquadratureew.ThiswillresultinthemaximumlikelihoodLmaxand~Lmax.Thequadraturedatastreamdoesnotcontainanynewinformation,neverthelessLmax6=~Lmax.Thisisbecause,foragiventime-frequencycluster,thequadraturecounterpartsmayhavedifferentcontributionsbothfromthesignalandthenoise.Thereforeinclusionofthequadraturestreamcanimprovethecollectionofthesignalenergyintheclusterand,respectively,improvethereconstruction.Also,thedualstreamisrequiredfortheinclusionofthesignalpolarizationmodelsintotheanalysis. 71

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Adualstreamdatasampleiischaracterizedbythedatavectorsw[i]and~w[i],whichcanbeusedtocalculatetheamplitudesforanarbitraryphaseshift: w[i]=w[i]cos+~w[i]sin, (4) ~w[i]=~w[i]cos)]TJ /F4 11.955 Tf 11.96 0 Td[(w[i]sin. (4) Inthelikelihoodfunctional,thesametransformationshouldbeappliedtothedetectorresponsesand~.ThequadraturelikelihoodfunctionalsLand~Lalsovaryasthephasetransformationisapplied,howeverthetotallikelihoodL=L+~Lisphaseinvariant,eveniftheindividualphaseshiftsiareappliedtoeachdatasample.Itispossibletoselecttheprincipledualstream(PDS)phase,suchthattheprojectionsofthedatavectorsonthenetworkplaneareorthogonaltoeachother.Withtheprojectionsunderstoodas, =F+h+(t)+Fh(t)~=F+~h+(t)+F~h(t),thecondition(~)=0ismetfor =jf+jh+g cos u( ), (4) ~=jfjh)]TJ /F4 11.955 Tf 11.95 0 Td[(g cos v( ). (4) Angle isestimatedassumingtheconditionv=0and~u=0.uandvarethetwounitandorthogonalvectorsinthenetworkplane,parametrizedwiththeangle .u,v,gandharedenedasfollows: u=e+cos +esin (4) v=ecos )]TJ /F10 11.955 Tf 11.96 0 Td[(e+sin (4) 72

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h=(jfj+~jf+j) 2jf+jjfjcos (4) g=(jfj)]TJ /F4 11.955 Tf 19.15 3.15 Td[(~jf+j) 2jf+jjfjcos (4) Theparametersand~arethen =(wu),~=(~wv). (4) Theyaretheamplitudesofthereconstructednetworkresponses, =u,~=v. (4) Thetransformationobtainedintheabovefewlines,toascertain(~)=0,isshowninFigure 4-3 Figure4-3. Phasetransformationofthedetectorresponsetoachievethecondition(~)=0.ThetotallikelihoodLmaxmax+~Lmaxisinvariantunderthistransformation.ThephasetransformedresponsesLmaxand~LmaxbeararelationshipbasedonthepolarizationstateoftheGW. Thephase-transformedprojectionsand~beararelationshipbasedonthepolarizationstateoftheGW.Figure 4-4 showstheirorientationbasedonthepolarization 73

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stateoftheGWforfourdifferentpixels(shownwithdifferentcolors).Amoreelaboratediscussionispresentin[ 46 ]. Figure4-4. Orientationofand~forfourdifferentpixelsbelongingtoaGWsignal,basedonitspolarization. 4.2.5RegulatorsForthecaseofaligneddetectors(sameantennaresponses),thenetworkisinsensitivetooneofthepolarizationstates.Eveninthecaseofmisaligneddetectors,thismayhappenfortheresponseatcertainpointsonthesky(intheDPFjfj=0).AsolutiontothisproblemisknownasconstrainedlikelihoodandwasproposedbyKlimenko,etal.[ 44 ].Theconstrainedlikelihoodanalysisintroducestheregulatorbydeningthedirectionoftheinitialdetectorresponsebasedonthenetworksensitivitytothesecondpolarization.Thetwomostwidelyusedregulatorsarethehardandsoftconstraints.Thedatavectorisprojectedontovectoru0,whichliesonthenetworkplane.Forexample,ahardregulatorusesf+asu0,therebyrestrictingsensitivityofthenetworktoonlyonepolarizationstate.Duetothis,itcompletelyremovesthesensitivitytoasmallclassofwaveforms.Figure 4-5 depictstheuseofregulators. 4.3ReconstructionofChirpMass(chirpcut)IntheeBBHsearch,reconstructedchirpmassconstrainthasbeenappliedforthereductionofbackground.ChirpmassplaysacentralroleindeningthefrequencyevolutionoftheCBCchirpingsignals.Althoughaneccentricbinaryisexpectedtohaveamulti-chirpsignatureonthetime-frequencymap,thechirpmasshasbeenconstructedassumingasinglechirpscenario.ThisworkswellfortheeBBHanalysisdoneonthe 74

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Figure4-5. ThemodiedbasisforlikelihoodanalysisofGWusingregulators.Thedatavectorxisprojectedontothevectoru,whichliesonthesignalplane.Thenewdetectorresponseisthisprojection. initialLIGO/VIRGOdata,asthefrequencythresholdishigh(around50Hz).IntheeBBHanalysis,reconstructedchirpmassconstraintreducedthebackgroundbytwoordersofmagnitude,whileloosingonlyasmallfraction(around5%)ofinjections. 4.3.1AlgorithmTheevolutionoffrequencyofachirpingCBCsignal,basedonthequadrupoleradiationformula,isgivenas, _f=96 58=3GMc c35=3f11=3,(4)whereMc=(m1m2)3=5=(m1+m2)1=5isthechirpmassofthebinarywithcomponentmassesm1andm2.IntegratingEquation 4 withrespecttotimeyields, 96 58=3GMc c35=3t+3 8f)]TJ /F5 7.97 Tf 6.58 0 Td[(8=3+C=0,(4)whereCistheconstantofintegration.Identifyingxtandy3 8f)]TJ /F5 7.97 Tf 6.59 0 Td[(8=3,thechirpmasscanbecalculatedbytheslopeofthelinettedthroughthedatapoints.Thettingisdoneonthecollectionofpixels(yi,xi)extractedfromtheT-FmapaftertheWDM 75

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transforms.Alinecanbettedbyminimizingthe2givenintheEquation 4 2Xi(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(b(xi)]TJ /F4 11.955 Tf 11.95 0 Td[(x0))2,(4)whereb=96 58=3)]TJ /F5 7.97 Tf 6.68 -4.98 Td[(GMc c35=3andx0=C=b.Thecenterofthepixelispromotedasthedatapointbyincludingthedimensionofthepixelin2,asgiveninEquation 4 2Xi2i,2i=(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(b(xi)]TJ /F4 11.955 Tf 11.95 0 Td[(x0))2 F2i+b2t2i,(4)whereFi=)]TJ /F4 11.955 Tf 9.3 0 Td[(f)]TJ /F5 7.97 Tf 6.58 0 Td[(11=3i(fandtarethedimensionsofpixelsonthefrequencyandthetimeaxis).Lineofbesttisidentiedbyscanningovervaluesofbandx0andthelinewhichminimizes2isselected.Figure 4-6 plotsreconstructedchirpmassvsinjectedchirpmassforrecoveredeventsinS6A(LHV)analysis.Asseenintheplot,anumberofeventshavebeenreconstructedwithverylowchirpmass.Thisisbecausethepixelsselectedforthereconstructionofchirpmass(andeventuallyforlikelihoodanalysis)includeoutliersduetobackgroundnoise.MoreoverthemergerstageofthechirpdoesnotfollowEquation 4 .Tocircumventthisproblem,thelinewhichintersectswiththemaximumnumberofpixels,whileexcludingthepointswhichhavehigh2ivaluewithrespecttoit,isselected.Ifthenumberofintersectingpixelsisthesameformultiplevaluesofslopeandintercept,thelinethatminimizes2isselected.Figure 4-7 plotsinjectedvsreconstructedchirpmasswiththisadjustmentandFigure 4-8 plotsreconstructionofchirpmassfortheBigDogevent. 4.3.2ErrorinReconstructedChirpMass2cannotbeusedtodeterminethegoodnessoftasitisnotnormalized.ErrorsinEquation 4 arenotstatisticalerrorsbuttheuncertaintyinthevaluesofFiandti.Toestimategoodnessoft,howcloselythecollectionofpixelsresemblesalineisestimated.Thetwoguresofmerittoquantifythisareenergyfractionandellipticity.Energyfractionistheratiooftheenergyofpixelsintersectingwiththelineandthetotalenergyofthecollectionofpixel.Ellipticityisthemeasureofhowellipticorcircular 76

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Figure4-6. InjectedvsReconstructedchirpmassforS6A(LHV)unmodeledsearch(withoutpolarizationconstraint)usingEOBNRv2waveforms.OnlyinjectionsthathavecrossedcWBpost-productioncutshavebeenconsidered.Reconstructionschememinimizesthetotal2value,whileusingallthedatapoints. distributionofpixelsis.Anellipticityclosetoonemeansdistributionisclosetoaline,whileellipticityclosetozeromeansdistributionisclosetoacircle.Figure 4-9 plotsenergyfractionandellipticityforboththebackgroundandthesimulationfortheS6A(LHV)analysis. 4.3.3ApplicationofChirpMassCutIntheeventofachirpsignal,thereconstructedchirpmasswillbeclosetotheactualchirpmass(Figure 4-7 ).Ontheotherhand,reconstructedchirpmassforthebackgroundwillbeclosertozero.Figure 4-10 plotsthereconstructedchirpmassfortheS6(LHV)time-shiftanalysis.Byimposingconditionsofminimumreconstructedchirpmass(1solarmassormore)foraneventtoqualify,backgroundisreducedbyanorderofmagnitude 77

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Figure4-7. InjectedvsReconstructedchirpmassforS6A(LHV)unmodeledsearchusingEOBNRv2waveforms.Reconstructionschememaximizesthenumberofpixelsthroughwhichlineispassing,whileignoringpixelswithhigh2values. (Figure 4-10 ).Chirpmasscanbereconstructedwithasignicantvalueevenbybackgroundevents.Furtherreductionisdonebyusingenergyfractionandellipticity,asthesenumberswillusuallybehigherforachirpingsignalonatimefrequencymap.Anabsolutecutonellipticityisused.FortheEBBHanalysiseventswithellipticitylessthan.7arerejected.Anothercutisappliedonenergyfraction.Thevalueofenergyfractionhassomedependenceonthesizeofthecluster.Figure 4-11 plotsenergyfractionversusthesizeofthecluster.Apeakispresentatone;thishappenswhenmostofthepixelshavelargevalueoffi,whichmakesattedlinepassthroughmostofthepixels.Acutofenergyfractionlog10(sizeofcluster)>1.3removesasignicantportionofbackgroundeventswhilelosingasmallpercentageoftheinjectedevents. 78

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Figure4-8. ReconstructionofchirpmassfortheBigDogevent.Injectedchirpmassis4.962Mwhilethereconstructedchirpmassis4.800M Figure4-9. Energyfractionandellipticitydistributionforthe(leftplot)time-shiftanalysiswith1000lagsand(rightplot)simulation,fortheS6A(LHV)unmodeledsearch. 79

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Figure4-10. ReconstructedchirpmassforS6A(LHV)time-shiftanalysiswith1000lags. 4.3.4EccentricBinaryIftheorbitofbinaryiseccentric,thereispresenceofamulti-chirpsignatureontheT-Fmap.Amulti-chirpsignatureappearsbecauseofthepresenceofvariousmodes.Inaclassicalellipticorbit,orbitcanbedenedas r=Acos(!0t)+Bcos(2!0t)++Ccos(3!0t)+(4)where!0=Gm=a3.Gisthegravitationalconstant,misthetotalmassandaisthesemi-majoraxisoftheellipse.A,BandCareconstants.Theproblemofndingthechirpmassnowinvolvesttingmultiplelinesthroughthecollectedpixels.Althoughmultiplelinescanbettedthroughthedataset,itishelpfultoimposetheconditionthattheslopeofthettedlinesare(3=2))]TJ /F5 7.97 Tf 6.59 0 Td[(8.=3.,(4=2))]TJ /F5 7.97 Tf 6.58 0 Td[(8.=3.,etc.,timestheslopeofthesecondharmonic.Firstharmoniccanbeignoredasitscontributionismuchsmallerincomparisontoothermodes.Consideringonlyquadrupoleradiation,thedecayrateof 80

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Figure4-11. Redlinerepresentsthecut,energyfractionlog10(sizeofcluster)>1.3.Plotisproducedafterthereconstructedchirpmass,ellipticityandstandardcWBcutshavebeenapplied.Coloredpixelsareinjectionswhileblackdotsarebackgroundeventsobtainedfromtime-shiftanalysis.ResultisfortheunmodeledsearchperformedoverS6ALHVnetwork. eccentricityandsemi-majoraxisisgivenas, hda dti=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(64 5G3 c5m1m2(m1+m2) a3(1)]TJ /F4 11.955 Tf 11.96 0 Td[(e2)7=21+73 24e2(4) hde dti=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(304 15G3 c5em1m2(m1+m2) a4(1)]TJ /F4 11.955 Tf 11.96 0 Td[(e2)5=21+121 304e2.(4)Whenappliedtoeccentricbinarieswaveforms,thechirpmassreconstructionalgorithmdescribedintheprevioussectionisnotexpectedtobeoptimum.Figure 4-12 comparesreconstructedchirpmassfortheeccentricbinarywaveformwiththereconstructedchirpmassfortheEOBNRv2waveforms.Figure 4-13 reproducesFigure 4-11 foreccentricbinarywaveformsinjections,withtheeccentricitydistribution 81

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showninFigure 5-9 .ThelossinperformanceisnotsignicantfortheeBBHanalysisontheinitialdetectordata,primarilybecauseofthehighfrequencycut-off.BythetimeGWentersadetector'ssensitiveregion,binaryhasalreadylostasignicanteccentricity.Thereconstructedchirpmassvalueisstillusefulinreducingthebackgroundwhileremovingonlyafractionofinjections.Figure 4-14 showstheeffectofchirpmasserrorcutattwodifferentthresholds. Figure4-12. (Left)Distributionofreconstructed/injectedchirpmass.OverallratioissmallerfortheeBBHwaveforms.(Right)Distributionofellipticity.EllipticityislowerfortheeBBHwaveforms.GreencurveisfortheeBBHwaveformswhiletheredcurveisfortheEOBNRv2waveforms. 4.4RunStagingTheCoherentWaveBurst(orcWB)isthedataanalysispipeline,builtoverframeworkprovidedbyCERN'sroot[ 47 ].Rootisanobject-orientedframeworkforlarge-scaledataanalysis.Ithasbeenprimarilydesignedforparticlephysicsdataanalysis,butitisalsousedinotherapplicationssuchasGWdataanalysis.cWBisstagedrstbycreatingaprojectfolder;asetofscriptshavebeendevelopedtofacilitatethestagingprocess.Followingaresomerelevantscripts, 82

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Figure4-13. Redlinerepresentsthecut,energyfractionlog10(sizeofcluster)>1.3.Plotisproducedafterthechirpmasscutandstandardcwbcutshavebeenappliedonthe`g'searchresults.Coloredpixelsareinjectionswhileblackdotsarebackgroundeventsobtainedfromtime-shiftanalysisforthedetectornetworkLHV. cwb mkdir-createstheworkenvironmenttolaunchtheanalysis,forwhichfollowingdirectoriesarecreatedcong-directoryforcwbcongurationlesinput-directoryforinputdatales,suchaslocationofstraindatacondor-directoryforcondor.suband.daglestmp-directoryfortemporaryjobleslog-directoryforoutputanderrorloglesoutput-directoryforoutputjobles 83

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Figure4-14. Effectofchirpcutoneccentricbinarywaveforms.Bluecurveisthedistributionofeccentricityfortheinjections,andredandgreencurvehavebeenproducedwithtwodifferentthresholdsofreconstructedchirpmasscut.Theinjectionsareequallypenalizedfordifferentchirpcut,whiletheinjectionswithhighereccentricityarepenalizedmorebyaseverechirpcut.Thecurveshavebeennormalized,suchthatareaunderthecurveis1. merge-directoryformergedoutputjoblesreport-directoryfordatatobepublishedonwebmacro-macrosforuserdenedtaskcwb condor create-createscondorlesforsubmittingjobsonmultiplenodes.Analysisisbrokenintomanyjobsasthewholedatacannotbeanalyzedsimultaneously.Lengthofeachjobisfewhundredseconds.Additionally,eightsecondsontheendofeithersideofthesegmentareusedtocontainnumericalartifactscreatedbythedataprocessing. 84

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CHAPTER5SEARCHFORGRAVITATIONALWAVESFROMECCENTRICBINARYBLACKHOLESThischaptergivesanoverviewoftheeBBHsearchperformedontheLIGO-VirgodatacollectedduringtheS5/S6-VSR1/2/3sciencerun.ThesearchwasperformedfortheL1H1H2andL1H1H2V1networkontheS5/VSR1runandfortheL1H1andL1H1V1networkontheS6runs.FortheS6Crun,onlytheL1H1networkwasanalyzedasduringthistimetheVirgodetectorwasnotoperating.TheH2detectorwasnotoperatingduringtheS6sciencerun.Thefrequencybandbetween48and1024HzwassearchedfortheeBBHsignals.Searchtuningstudieswereperformed,whichincludedtuningofthenetworkcorrelationandthereconstructedchirpmassparameters.OnlyforegroundcandidatespassingpostproductioncutsandCAT1,CAT2andHVETOagswereconsidered.Backgroundsetwasalsoappliedwithpost-productioncutsandcategoryagstocreatethenalbackgroundset.ThesignicanceofaneventwasestimatedusingFADstatistics.Fortheestimationofratelimitdensity,visiblevolumewasestimatedbyusingtheeBBHwaveforms[ 27 ].Simulationwasperfomedforthecomponentmassrange,from5Mto25M.Eccentricitydistributionisderivedbysimulatingtheencountersinthegalacticnuclei,inthepresenceofasupermassiveblackhole(Section 1.3 ).Table 5-1 summarizestheparameterspacechosenfortheeBBHsearch.TocorrectlyassessthesensitivityofthenetworktotheGWevents,theinjectionsshouldlltheparameterspace.Moreover,thereshouldbesufcientinjectionstoavoidlargestatisticalexperimentalerror.Thiscanbeachievedbyincreasingthenumberofinjectionsmadetothedatastream.However,thishaspracticallimits.Forexample,theS6Alive-timefortheL1H1V1networkallowsamaximumof13,000injections.Thislimitationarises,astheinjectionsshouldnotoverlapandbesufcientlyseparatedintime.Toovercomethislimitation,statisticsisincreasedbyrepeatingmultiplerunsonthesamedataset. 85

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Table5-1. Summaryofnetworks,live-timeandparameterspaceforeBBHsearch. S5-VSR1S6-VSR2/3 NetworksH1H2L1V1andH1H2L1H1L1V1andH1L1Observationtime(days)60.0+238.9=298.942.1+79.2=121.3Total-massrange(M)[5-25][5-25]MassRatio[.2-1][.2-1]Eccentricity[0,1][0,1] 5.1DataQualityFlagsGWchannelisaffectedbyvarietyofnoises.Theseincludeinstrumentalnoisetransients(glitches)suchasoverowofthefeedbackcontrolsignalsusedtocontroltheinterferometerarmlengthsandmirroralignments,droppingoutofsinglefrequencycalibrationsignalswhicharecontinuouslyinjectedintothefeedbackcontrolsystem,scatteringoflightfromH1toH2orviceversawhenoneoftheinterferometergoesoutofphase,etc.Therearealsonoisetransientscausedbyenvironmentalnoisesuchaselectromagneticdisturbanceduetopower-lines,weather-relatedtransients,seismicdisturbances,etc.Theapplicationoftheseagsiscrucialinremovingtheeventswhicharenotofastrophysicalorigin.Thefollowingtablesummarizesvariouscategoriesofsuchvetoes. Table5-2. CategorydenitionandtheirapplicationintheeBBHanalysis. CategoryDenitionApplicationoneBBHanalyses CAT1ReectseveremalfunctionsofthedetectorAppliedattheproductionstageCAT2FlagsnoisyperiodswherethecouplingbetweengravitationalwavechannelandthenoisesourceiswellunderstoodAppliedattheproductionstageCAT3FlagsnoisyperiodswherethecouplingbetweengravitationalwavechannelandthenoisesourceisnotwellunderstoodAppliedatthepost-productionstage Otherthanthedataqualityags,hierarchicalvetoes(HVETO)arealsoappliedatthepost-productionstage.Hundredsofauxiliarychannelsaresearchedforanycouplingpresentbetweenthemandthegravitationalwavechannel.ItisdifferentfromCAT2 86

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andCAT3agswherenoisecouplingisalreadyknownoraplausiblesourcehasbeenidentied. 5.1.1ApplicationofVetoesFortheeBBHanalysisCAT1andCAT2vetoeswereappliedattheproductionstageswhileCAT3andHVETOvetoeswereappliedatthepost-productionstage.Vetoeswereappliedontheforegroundaswellasthebackgroundset.Thereisaverysmalllossinlive-timebecauseoftheapplicationofHVETOvetoes;onthecontrary,live-timelossissignicantbecauseoftheapplicationofCAT3vetoes.Aneventwasrejectedifthetimeatwhichtheeventisdetectedmatcheswiththetimeanyofthedetectorinthenetworkwasagged.Figure 5-1 and 5-2 showstheeffectoftheapplicationofvetoesonthestatisticsobtainedfromthetime-shiftanalysis. Table5-3. Live-time(days)fordifferentrunsaftertheapplicationofCAT2,CAT3andHVETOvetoes. RunAfterCAT2CAT2andCAT3CAT2,CAT3andHVETO S5L1H1H2347.94308.48305.51S5L1H1H2V167.7059.2157.79S6ALH12.0610.6410.23S6ALHV10.589.078.76S6BLH22.7222.5722.13S6BLHV16.4815.2114.90S6CLH51.2848.4647.78S6DLH51.6640.5440.20S6DLHV24.7518.1717.99 5.2Post-ProductionTuningThegoalofthepost-productiontuningistoestimatethevaluesofthecutswhichpenalizethebackgroundeventsmost,whilemaximizingtherecoveredinjections.Thresholdonthecoherentnetworkamplitudewasxedbasedontheloudestlagzeroeventinalltheruns.Parametersthatrequiredtuningwerenetcc,subccandreconstructedchirpmass.Figure 5-3 and 5-4 plotsnetccandsubccwithrespecttothecoherentnetworkamplitude.Anetccvalueof.6rejectsmorethan60%ofthebackgroundwhilerecoveringmostoftheinjections. 87

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Figure5-1. EffectofCategory3andHVETOvetoesontheeventsidentiedinthetime-shiftanaysis,forthedatafromtheL1H1H2networkduringS5-VSR1sciencerun.ThisplotshowstheeventssurvivingtheapplicationofthestandardcWBcuts(blue,redandgreen),theeventssurvivingaftertheapplicationofthestandardcWBcutsandCAT3vetoes(redandgreen)andtheeventssurvivingaftertheapplicationofthestandardcWBcuts,CAT3andHVETOags(green). Duringtheproductionstage,chirpmassoftheeventwasreconstructedonthetimefrequencyplane.Reconstructionwasachievedbyttingachirpingsignalswhileignoringtheoutliers.Theerrorinthereconstructionwasquantiedusingenergyfractionandellipticity.Theenergyfractionisdenedastheenergyofpixelsusedinthereconstructiondividedbythetotalenergyofallthepixels.Ellipticityisameasureofhowcloseisdistributionofthecollectionofpixelsistoaline(afterappropriatetransformationachirpingsignalcanbemappedtoastraightline).ThechirpmassreconstructionisdiscussedinmoredetailintheSection 4.3 .Thefollowingchirpcutswereapplied 88

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Figure5-2. EffectofCategory3andHVETOvetoesontheeventsidentiedinthetime-shiftanaysis,forthedatafromtheL1H1H2networkduringS6D-VSR3sciencerun.ThisplotshowstheeventssurvivingtheapplicationofthestandardcWBcuts(blue,redandgreen),theeventssurvivingaftertheapplicationofthestandardcWBcutsandCAT3vetoes(redandgreen)andtheeventssurvivingaftertheapplicationofthestandardcWBcuts,CAT3andHVETOags(green). a)reconstructedchirpmass>1solarMassb)ellipticity>.7,andc)energyfractionlog10(numberofpixelsintheevent)>1.3.Applicationofchirpcutreducedthebackgroundbymorethananorderofmagnitude.Achirpingsignalfromacoalescingbinaryhasaknownsignatureonthetime-frequencyplane.Iftheaveragefrequencyofthepixelsonthetimefrequencymapiscalculated,itisexpectedtolieaboveacertaincertaincut-offfrequency.Thiscut-offfrequencywilldependonthesensitivityofthedetectordataandtheparametersofthebinary.FortheeBBHbinarieswithcomponentmass5)]TJ /F4 11.955 Tf 12.39 0 Td[(25Mandeccentricity0)]TJ /F4 11.955 Tf 12.39 0 Td[(1.,theaverage 89

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Figure5-3. Distributionofnetworkcorrelationcoefcientversuscoherentnetworkamplitude.Thisplotincludesdatafromalltheruns.Theplothasbeenproducedbeforetheapplicationofchirpcut.Greendotscorrespondtothesimulatedeventswhilereddotscorrespondtothebackgroundevents. frequency,fortheS5/6-VSR1/2/3data,wasfoundhigherthan100Hz.Onthecontrary,manytime,backgroundeventswerefoundhavingaveragefrequencylessthan100Hz.Tofurtherreducethebackground,ahardcutof100Hzfrequencywasappliedonthereconstructedevent(timeshiftanalysisandinjections).Thefrequencycutmaybeunderstoodasamodel-basedconstraint.Figure 5-6 and 5-5 plotsaveragefrequencydistributionofeventsfromtime-shift. 5.3CalibrationUncertaintiesThechangeinthedifferentiallengthbetweenthearmsofaGWdetectorisreadoutdirectly.Toconvertitintothecorrespondingstrain,atransferfunctionisapplied.Thetransferfunctionhasassociatederrorswhichpropagateintheamplitudeand 90

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Figure5-4. Distributionofsubnetworkcorrelationcoefcientversuscoherentnetworkamplitude.Plotincludesdatafromalltheruns.Plothasbeenproducedbeforetheapplicationofchirpcut.Greendotscorrespondtothesimulatedeventswhileredpixelscorrespondtothebackgroundevents. phasereconstructionoftheGWstraininasearch[ 48 ].ThecalibrationuncertaintiesfortheLIGO[ 49 50 ]andVirgo[ 51 52 ]instrumentsarelistedinTable 5-4 fortheS5/S6-VSR1/2/3runs. Table5-4. MaximumfractionalerrorontheamplitudeofthecalibratedGWstraininthe40Hz)]TJ /F1 11.955 Tf 12.62 0 Td[(2kHzbandwidth. S5-VSR1S6-VSR2/3 L110.4%19%H110.1%16%H214.4%-V16.1%4.5% Althoughthecalibrationuncertaintiesaffectphaseandamplitude,theeffectonamplitudeismorerelevantfortheburstsearches.Theeffectofamplitudeisdirectly 91

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Figure5-5. Thedistributionofthecentralfrequencyofthepixelsusedtoreconstructtheeventsforthetime-shiftanalysis.PlotisforS6CLHdata. translatedintoanuncertaintyontheeffectiveradius.Thus,anuncertaintyofRe,cal=14.4%wasconservativelyappliedontheeffectiveradiusfortheS5-VSR1runandanuncertaintyofRe,cal=19%wasconservativelyappliedfortherunsS6-VSR2/3run. 5.4BackgroundEstimationPipelineconsidersenergeticpixelsaspossibleevents.Itisalsopossibleforevents,suchasglitches,tobeconsideredasaGWevent.AGWevent(foregroundevent)isdistinguishablefromaneventproducedfromthenoisebyitscoincidentappearanceinmostofthedetectorsinthenetwork.Tocalculatethefalsealarmprobability(FAP)ofaforegroundevent,informationaboutthebackgroundisrequired.Anidentiedeventneedstobecomparedbytheeventscausedsolelybythebackground.Signicanceofaforegroundeventisthenquotedastheprobabilityofitbeingproducedfromthenoise. 92

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Figure5-6. Thedistributionofcentralfrequencyofthepixelsusedtoreconstructtheevents,whenthesimulationwasperformedusingtheeBBHinjections.PlotisforS6CLHdata. AsthereisnowaytodistinguishbackgroundeventsfromtrueGWeventsbyjustusingsearchresults,itisimpossibletoestimatethesignicanceofanevent.Moreover,theamountofdatatoestimatethebackgroundislimitedinarun.ToincreasethestatisticsandefcientlyseparatebackgroundeventsfromtrueGWevents,time-shiftanalysisisperformed. 5.4.1Time-ShiftAnalysisThetime-shiftanalysisisusedtogeneratethestatisticsofeventsproducedsolelyduetothebackground.Thetriggersaregeneratedaftershiftingoneormoreofthedetectordatawithrespecttoareferencedetector.Thedata,whichgoesbeyondtheendofjobsegment,arewrappedbacktothebeginning(Figure 5-7 ).Dataareshiftedby 93

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adurationwhichismuchlongerthanthemaximumtraveltimeofagravitationalwavebetweenthedetectors,therebyremovingexistenceofanycoincidentGW.Becauseofthetime-shifts,thepossibilityofaGWeventisremovedandtheonlyeventsidentiedbythepipelineareduetothebackground.Manytime-shiftsareperformed. Figure5-7. Datafromreferencedetectorareleftastheyare,whiledatafromotherdetectorsareshiftedtoremovepresenceofanycoincidentGW.Thetime-shiftTkisappliedtothedata.Thedatathatgobeyondarewrappedbacktothebeginning.Anyinitialreferencetimets,becomests+Tkintheshifteddata. 5.4.2BackgroundSetsTheeBBHsourcesweresearchedinfournetworksandtwoeachforfthandsixthscienticruns.NetworkL1H1H2andL1H1H2V1weresearchedforthefthscienticrun,H2wasdismantledforthesixthrunsoLHandLHVnetworkwereanalyzedforthesixthscienticrun.Thelive-timeforS5isaroundanyearwhile,forS6,itishalfayear.Runsweremadeexclusive;thereforeeventsappearinginH1H2L1wereremovedifH1H2L1V1wasalsorunning.SimilarlyforS6,theeventsinLHnetworkwerenotconsideredifLHVnetworkwasrunning.Live-timewasalsocalculatedaccordinglyfordifferentruns.Allresultsshownareaftertheapplicationofdataqualityags(uptoCategory2forforegroundcandidatesandCategory3forsimulatedinjections)andstandardpost-productionthresholdsestimatedafterpost-productiontuning.Foraparticular,runeventsarerankedintermoftheirFAR(Equation 5 ).Figure 5-8 plotsthedistributionofallthebackgroundevents,fordifferentdetector 94

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networksusedintheanalysis,beforeandaftertheapplicationpostproduction,frequencyandchirpcut.AdjacenttothatistheplotshowingFARforthebackgroundevents. 95

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96

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Figure5-8. DistributionofbackgroundeventsandtheirFAR,fordifferentdetectornetworksusedintheanalysis,aftertheapplicationofpost-production,chirp,frequencycuts,CAT1,CAT2,CAT3andHVETOags.Intheleftplot,theblackhistogramincludesalltheevents,whilethegreenhistogramincludesonlytheeventssurvivingthecutsandtheapplicationofags.OntherightplottheblackgraphisFARforalltheevents,whiletheredgraphisfortheeventsaftertheapplicationofCAT3ags.Fromtoptobottom,plotsareforthenetworkS5L1H1H2,S5L1H1H2V1,S6ALH,S6ALHV,S6BLH,S6BLHV,S6CLH,S6DLHandS6DLHV.Theresultsarefortheinclusiveruns. 97

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5.5SimulationSimulationstudieswereperformedfortheestimationofthevisiblevolume.Visiblevolumeiscalculatedtoestimatethesensitivityofadetectornetwork,i.e.,howfaranetworkcandetectGWwithcondence.Thedistanceatwhichabinarycanbedetecteddependsonthecomponentmassesofthebinaryalongwithotherparametersandalsoonthelocationofthebinarywithrespecttothedetectornetwork.Visiblevolumecanbeconsideredasanaveragereachforthedetectornetwork,forthegivenparameterspace,whenbinariesaredistributeduniformlyinthespace.VisiblevolumeisalsousedincalculationoftheFAD,becauseofwhichtheeventsfromdifferentrunscanbecomparedfortheirsignicance.Section 5.5.4 describesthevisiblevolumeindetail. 5.5.1EventGeneratorFirst,theparameterspaceneedstobedenedfortheestimationofvisiblevolume.Thisisdonebytheeventgenerator.TheeventgeneratorsimulateseventsbasedontheastrophysicalmodelasdescribedinSection 1.3 .ReferringtoFigure 1-5 ,blackholedensityfallsoffveryrapidlyoutside1pc.Togenerateanevent,arandomdistancebetween.001and1pcischosenfortheoccurenceoftheevent.Thisdenestherelativevelocityoftheeventandrp,max.Arandomrpischosenbetweenrpcorrespondingtolaststableorbit(7.5M)andrp,max.Thisdenestheeccentricityoftheformedbinary.Eventsareevolveduntiltheyreachburstfrequency(r)]TJ /F5 7.97 Tf 6.59 0 Td[(1p(rp=M))]TJ /F5 7.97 Tf 6.58 0 Td[(1=2[ 27 ])orrpreaches10M(Kutta-MersonmethodisusedtoevolveEquation 1 ).Eventsformedforrpvaluesbetween7.5Mand10MenterLIGOsensitivitybandthemomenttheyareformedandhencearenotevolvedusingPeter-Matthewformalism.Figure 5-9 showstheeccentricitydistributionof100,000eventsfortwoSMBHmasses.MWreferstoMilkyWaylikeSMBH(3.5106M),while10MWreferstoaSMBH,tentimesthemassofMWSMBH(3.5107M).Distributionofmassesdependsonvariable.Figure 5-10 plotschirpmassdistributionfordifferentvalues. 98

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Figure5-9. EccentricityofthebinarieswhenGWgeneratedbythemhaveanburstfrequencyof32Hz.Someoftheevents(10%forMilkyway-likegalaxy)haveaburstfrequencyof32Hzormorethemomenttheyareproduced. AsseeninFigure 5-10 ,thenumberofbinariesfallsoffforloweraswellashigherchirpmass.Iftheinjectionsaremadefollowingmassdistributiondictatedbytheeventgenerator,largestatisticaluncertaintieswillfollowintheestimationofthevisiblevolumesforsuchbinaries.Toavoidthis,theinjectionsaremadeuniformlyonthecomponentmassplane,asshowninFigure 5-11 .Thisadjustmentdoesnotbreakthemodelastheeccentricitydistributiondoesnotchange,asshowninFigure 5-12 .Thisstrategyisgenericintermsofthemassdistributionanddoesnotdependonthevalueof.Toobtainresultsspecictoacertainvalueof,genericresultsareconvolutedwiththemassdistributionoftheastrophysicalmodel. 5.5.2MassRangeTheeBBHanalysiswasconductedinthemassrange5Mto25M.Multipleworkshaveproposedtheformationofhighermassbinaries(componentmassof25 99

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Figure5-10. Distributionofchirpmassfordifferentvalues.ThedistributionisfortheMWmodel,althoughtheeffectoftheSMBHmassisnotsignicantonthisdistribution. Mormore)inthegalacticcores[ 53 ].GWfromhighermassbinarieshavelowISCOfrequencyandshouldentertheLIGOsensitivitybandathigheccentricity.Althoughthecontributionfromthehighermodes(modulationofGWamplitude)isnotsignicant,themostdominantstageisamergerforsuchbinaries.Allskyanalysesforthebinarieswithcomponentmassof25MandhigherhavealreadybeenconductedintheIMBHsearchontheS5-VSR1aswellasS6-VSR2/3data[ 54 55 ].Theupperlimitestimatedforthebincenteredat(67.5M,37.5M),intheIMBHsearchis.18Mpc)]TJ /F5 7.97 Tf 6.58 0 Td[(3Myr)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Whiletheupperlimitestimatedforthebincenteredat(23M,23M),intheeBBHsearch,is.6Mpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3Myr)]TJ /F5 7.97 Tf 6.59 0 Td[(1.ExtensionofeBBHsearchtohighmassesshouldhaveresultedinanupperlimitclosetoIMBHsearch,therebynotgaininganythingnew.Moreover,thereconstructedchirpmasscutandloosecircularpolarizationconstraintworkbestforbinariesinthecomponentmassrangeof5to25 100

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Figure5-11. Injectionsmadeonthemasscomponentplane.Theinjections,ineachmassbin,arecloseinnumber. M.Figure 5-13 plotsthereconstructedchirpmassforthebinariesinthecomponentmassrange25to50M. 5.5.3RadialPlacementTheinjectionsareplaceduniformlyovertheskylocation.Theradialplacementischosendependingonthecomponentmassesofthebinary.Ifinjectionscorrespondingtoheavybinariesareplacedatasmalldistance,thevisiblevolumewillbeunderestimated.Ontheotherhand,ifalightbinaryisplacedatalargedistance,itwillnotbedetectedbythealgorithmanditwillnotcontributetotheestimationofthevisiblevolume.Aheavystellarmassbinariesisexpectedtobedetectedatahigherdistancecomparedtoalightstellarmassbinary.Thisisaccountedforbyscalingtheinjectionvolumeasafunctionofthechirpmass.Ifa25M-25Mbinary(Mc0=21.764M)isinjectedina 101

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Figure5-12. EffectonthedistributionofeccentricitywheninjectionsaremadefollowingdistributionshowninFigure 5-11 .Thechangeinthedistributionofeccentricityisinsignicant. ducialradiusofRd(Mc0),andtheducialradiusforotherbinariesisscaledaccordingto Rd(Mc)=(Mc=21.764M)5=6Rd(Mc0).(5)Figure 5-14 plotsducialdistancefor=2andRd(21.764)=200Mpc. 5.5.4VisibleVolumeThevisiblevolumeistheeffectivevolumeinthespaceinwhichanoccurredGWwillbecondentlyseenbythedetectors.Itisdenedas Vvis=ZVinj(r,,)dV,(5) 102

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Figure5-13. Reconstructedversusinjectedchirpmassforhighermassbinaries(componentmassof5to25M.Becauseofashortenedinspiralstage,thereconstructedchirpmasshasbeenunderestimated.ResultisforS6ALHVrunperformedusingEOBNRv2injections. whereistheefciencyofdetectionatco-ordinate(r,,)andVinjistheducialvolume.Theangulardependencecanbeaveragedtorewritetheintegralonlyintermsoftheradius, Vvis=4ZRinj0(r)r2dr(5)whereRinjistheducialradius.Efciencyisnotknowapriori,but,foraparticularshell(r,r+dr),itcanbeestimatedbytheratioofthenumberofdetectedeventsandthenumberoftheinjectedevents(Ndet=Ninj).Thedensityofinjectionsisthendenedas, 1 i=4r2i dNinj=dr,(5) 103

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Figure5-14. Injectiondistributionfor=2andRd(21.764)=200Mpc.Fiducialradiusiscalculatedforeachinjectionaccordingtoitschirpmass;injectiondistanceisthenrandomlychosensoastoproduceainjectiondensityof/r2. wheredNinj=dristheradialinjectiondensity.Withthisdenition,Equation 5 becomes, Vvis=NdetXi1 i.(5)ThevisiblevolumeisestimateddirectlyfromEquation 5 .Theinjectiondensityattheoccurrenceofeachdetectedevent(ridistancefromEarth)isestimatedbycountingthenumberofinjections(Ni)madebetweentwoconcentricshellswithradiusri)]TJ /F4 11.955 Tf 11.45 0 Td[(.5Mpcandri+.5Mpc.Visiblevolumeisthengivenas, Vvis=NdetXi4((ri+.5)3)]TJ /F4 11.955 Tf 11.95 0 Td[((ri)]TJ /F4 11.955 Tf 11.96 0 Td[(.5)3) Ni.(5) 104

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Theeffectiverangeistheradiuscorrespondingtothevisiblevolume, Rvis=3Vvis 41=3.(5)Figure 5-15 plotsvisiblevolumeasafunctionofcoherentnetworkamplitudeforvariousdetectornetworksusedintheeBBHsearch. Figure5-15. VisiblevolumeasafunctionofcoherentnetworkSNR(p (NIFO)]TJ /F4 11.955 Tf 11.95 0 Td[(1)),forvariousdetectornetworksusedintheeBBHsearch.Plothasbeenproducedwithauniformnumberofinjectionsoverthemassbins.Visiblevolumeatacertaincoherentnetworkisnotrepresentativeofthesensitivityofthedetectornetwork. Thevisiblevolumesuffersfromvariousuncertainties.ThefollowingsectiondiscussesuncertaintiesconsideredintheeBBHsearch,namelycalibration,systematicandstatisticaluncertainties. 105

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5.5.5StatisticalUncertaintyThelikelihoodofthedetectionofaninjectiondependsonitslocation.Thefarthertheinjection,thelesseristhelikelihoodofdetection.Similarly,aninjection'sskylocationaffectsitsdetectability,asdetectornetworkshaveanon-uniformsensitivity.Angulardistributionofinjectionsintheskyisuniform,andastheinjectionsareindependent,theefciencyofdetectionwithdistanceisabinomialdistribution.ThisarisesfromthefactthatestimationofefciencyisthesumofBernoullitrials.Thestatisticaluncertaintyiscalculatedbypropagatingtheerrorstoobtainthequadraturesumoftheuncertaintiesoneachtrial.Thevarianceofabinomialdistributionisgivenas, Var=Np(1)]TJ /F4 11.955 Tf 11.96 0 Td[(p),(5)usingEquation 5 ,thevarianceinthevisiblevolume,foraparticularshell(r,r+dr),canbewrittenas, Var(V)=1 2iNinj0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(0).(5)Also,asNdet=0Ninj,thequadraturesumcanbewrittenas, Var(V)=NdetXi1 2i.(5)Thissumismoreconservativebutitsimpliesthecalculationbydroppinginfavorof1.Therelativeerrorinthevolumeisthengivenas V Vvis=vuut NdetXi4((ri+.5)3)]TJ /F4 11.955 Tf 11.96 0 Td[((ri)]TJ /F4 11.955 Tf 11.95 0 Td[(.5)3) Ni2=Vvis.(5)Thestatisticaluncertaintyintheeffectiveradius,estimatedfromEquation 5 ,isgiveninEquation 5 R=V 43Vvis 4)]TJ /F5 7.97 Tf 6.59 0 Td[((2=3).(5) 106

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5.5.6StatisticalandSystematicUncertaintiesintheeBBHAnalysisTheparameterspaceissampledforthecalculationofthevisiblevolumeand,asthenumberofsamplesisnite,thiscalculationisaffectedbystatisticaluncertainty.Table 5-5 summarizesthenumberofinjectionsmadeineachrun. Table5-5. Numberofinjectionsmadeineachrun. RunNumberofinjectionsLiveTime,inclusiverunsafterCAT2(days) S5L1H1H23750869347.94S5L1H1H2V1146192267.70S6ALH63669012.0S6ALHV56798210.6S6BLH121940022.7S6BLHV88674716.5S6CLH276281851.3S6DLH278403151.7S6DLHV133315024.7 CalculationofstatisticalerrorsisdiscussedinSection 5.5.4 .Figures 5-16 and 5-17 plotthefractionalstatisticaluncertaintiesontheeffectiveradiusoverthecomponentmassplane. 5.5.7SystematicErrorsTheestimateofthesearchsensitivitydependsonthewaveformfamilyusedtoconductthesimulationstudies.Thedifferenceinthewaveformsusedintheestimationofthevisiblevolumefromtheoneproducedfromtheastrophysicalsourcesresultsinthesystematicerrors.Thereisnowaytoknowthewaveformsgeneratedbyastrophysicalsourcesotherthanbydirectmeasurement.However,numericalrelativitywaveformsareexpectedtobeclosetothetruewaveforms(assumingEinstein'stheoryofgeneralrelativityiscorrect).FortheeBBHanalysis,theestimationofthesystematicerrorsencounteredproblemsbecauseoftheabsenceofviabletoolsneededtoproducethenumericalrelativitywaveformsorcompleteIMRpost-Newtonianwaveforms.ThesystematicerrorwascrudelyestimatedbycomparingtheeBBHwaveformsateccentricityequaltozerowiththeEOBNRv2waveforms.Thisisnotasevere 107

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Figure5-16. Percentageuncertainty(absolutestatisticaluncertaintynormalizedbythetotalvisiblevolumeandmultipliedby100)onthevisiblevolumefortheS5L1H1H2run. shortcoming,asitwasfoundthattheeBBHanalysiswasonlyweaklydependentontheeccentricity.Figure 5-18 plotsthepercentagedifference100(1)]TJ /F4 11.955 Tf 12.44 0 Td[(REOBNRv2=ReBBH)betweentheeBBHandtheEOBNRv2waveforms.Thesystematicerrorsweretakentobethemaximumnumberonthecomponentmassplane.Aconservativeestimatederrorof8.3%wasappliedtotheeffectiveradius. 5.6SignicanceofanEventCondentacceptanceofatruegravitationaleventandrejectionofabackgroundeventdependonthesignicanceoftheevent.Awell-denedstatisticalprocedureisrequiredtoachieverankingstatistics.Roughlyspeaking,thesignicanceofaforeground 108

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Figure5-17. Percentageuncertainty(absolutestatisticaluncertaintynormalizedbythetotalvisiblevolumeandmultipliedby100)onthevisiblevolumeforthefortheS6Drun. eventisestimatedbycomparingitwiththebackgroundstatisticobtainedfromthetime-shiftanalysis.Rankingstatisticsareusedto estimatetheastrophysicalratesincaseofdetections setupperlimitsonastrophysicalratesincaseofnodetection 5.6.1FalseAlarmRateDensityThebackgroundstatisticisobtainedbythetime-shiftanalysis.Themethodensuresthatnotrueeventsareincluded.Thecountingrates,bothforthesignalandthenoise,dependonthechosenselectioncuts.Thefalsealarmrateforaforegroundeventis 109

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Figure5-18. ThepercentagedifferencebetweentheestimatedeffectiveradiusfortheeBBHandtheEOBNRv2(100(1-REOBNRv2=ReBBH))waveform,fortheS6Arun. denedas FAR()=N() T,(5)whereN()isthenumberofbackgroundeventscrossingthethresholdandTisthesumoftheobservationtimeforallthetimelagintheanalyses.Thefalsealarmrate(FAR)canbeusedtodeterminethesignicanceofagravitationalwaveeventrelativetothebackgrounddistributionofasearch.ThesignicancecanbeexpressedintermsoftheFalseAlarmProbability(FAP).TheFAPforthebackgroundtoproduceNevents, 110

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crossingthreshold,isgivenas FAP()=1)]TJ /F5 7.97 Tf 12 14.94 Td[(N)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xn=0n n!exp()]TJ /F3 11.955 Tf 9.3 0 Td[(),(5)wherethebackgroundeventshavebeenassumedtobefollowingaPoissondistribution.Iftherearenoforegroundeventsofsignicance,aratelimitdensitywith90%condencecanbeplacedbyusingloudesteventstatistics[ 56 57 ],denedas R90%=2.303 Tobs(^).(5)InEquation 5 ,theefciency,(^),isevaluatedfromnumberofrecoveredinjectionswithatleastasgreatastheloudestevent()intheforegroundandTobsisthecombinedobservationtimeforzerolag,aftertheapplicationofdataqualityags. 5.6.2FalseAlarmDensityForaparticulardetectornetworksensitivity,FARcanbeusedtoestimatesignicanceofanevent.Asthenumberofbackgroundeventscrossingacertainthresholdchangeswiththechangeindetectornetworkorchangeinthesensitivityofthenetwork,variousrunscannotbecomparedwitheachother.Otherthanthebackgroundinformation,anotherpieceofinformationrequiredisthesensitivityofthenetworktothesearchedastrophysicalsources.AstraightforwardapproachtoextendFAR,whichincludessensitivityofthedetector,iscontainedintheFalseAlarmDensity(FAD)statistics.Thisisdescribedinthefollowingequation, FAD()classical=FAR(j) Vvis().(5)Equation 5 isnotmonotonicin,asthedecreaseinthenumberofeventscrossingfromthebackgroundandthesimulationwithincreasedmaynotbelinear.Forathresholdi,aFADvaluecanbedenedbyassigningthelowestvalueofEquation 5 111

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amongallthevaluesfori,asgiveninEquation 5 FAD(i+1)=min(FADclassical(i)]TJ /F5 7.97 Tf 6.58 0 Td[(1),FADclassical(i)),(5)therebymakingtheFADmonotonicallydecreasing.TheFADisconstructedforeachsearchnetworkandtherankingsaredonerelativetotheindividualsearches.ThesignicanceofaneventisestimatedbycomparingitsFADwithproductivityofthesearchandproductivityisdenedinthefollowingequation, (FAD)=XkTobs,kVvis(FAD),(5)whereindexkrunsoveralltherunsinvolvedinthesearch.Fromtheproductivity,meanoccurrenceoftheeventfromthebackgroundiscalculatedbyusingEquation 5 (FAD)=FAD(FAD),(5)followingwhichEquation 5 isusedtocalculatethesignicance. 112

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CHAPTER6RESULTSOFTHECWBS5-S6/VSR1-VSR2-VSR3EBBHSEARCHInthischapter,theeBBHsearchresultsarepresented.ThesearchwasperformedoverscienticrunsS5-S6/VSR1-VSR2-VSR3.Noeventwasfoundsignicantenoughtobeconsideredaneventofgravitationalwaveorigin.Intheabsenceofadetection,smallestFADvalueforthezerolageventswasusedtoxthethresholdsontherankingstatisticforthedifferentnetworksandtocalculatethevisiblevolume.TheastrophysicalstatementsaremadeintermsofupperlimittotheeBBHcoalescerate. 6.1FalseAlarmDensityandLoudestEventsTheFADplotsarecreatedforvariousbackgroundsets.ThesignicanceoftheforegroundeventsisthencalculatedusingEquation 5 .Figure 6-1 showstheFADdistributionofthebackgroundsetsderivedfromtheeBBHsearch.Theplotalsoincludesforegroundeventsshowninthereddots.Thereareatotalof8foregroundeventsthathavecrossedthepost-productioncutsandCAT2ags.ThethreeloudesteventsaresummarizedinTable 6-1 Table6-1. Listoftherstthreeloudestforeground(lagzero)events. GPStimeNetworknetccMchirp(M)ellipticityFAD(Myr)]TJ /F5 7.97 Tf 6.58 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3)FAP 970785943S6DLH5.690.771.70.81.35.96950142837S6CLH6.310.673.20.841.71.92947044429S6CLH5.890.6212.80.93.84.99 6.2EffectiveRadiusOncetheloudestforegroundeventhasbeenidentied,theeffectiveradiuswascalculatedattheFADthresholdsetbytheloudesteventforeachrun.Followingthis,effectiveradiusandupperratelimitarecalculated.TheeffectiveradiusisameasureofthesensitivityofthedetectornetworkfortheeBBHsources,i.e.,howfaritcanseeeBBHsourceswithcertaincondence.Figure 6-2 plotseffectiveradiusfortheS5L1H1H2runandFigure 6-3 plotseffectiveradiusfortheS6DLHVrun. 113

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Figure6-1. TheFADversuscoherentnetworkamplitudeplot.TheplotincludesallthebackgroundsetsusedintheeBBHsearchandforegroundeventswhichareshownbythereddots.Theplothasbeenproducedwiththeuniformnumberofinjectionsinthecomponentmassbins.Onlyeventscrossingthepost-productioncutsandtheCAT2agshavebeenconsidered. Asexpected,thereisanincreaseintheeffectiveradiuswiththechirpmass.Theeffectiveradiusremainedunchangedwiththeconstantchirpmass,ascanbeundertoodfromthefactthattheonlydissipativeterminthewaveformgenerationmodelisquadrupolar. 6.3UpperLimittoRateDensityIntheeventofnodetections,upperlimitsarecalculated.Astherearenosignicanteventstoclaimdetection,therate(inunitsofMyr)]TJ /F5 7.97 Tf 6.58 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3)ofcoalesceforeccentricbinaryblackholesduringthetimeitwasobserved,cannotexceedacertainvalue. 114

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Figure6-2. TheeffectiverangemeasuredfortheS5L1H1H2searchoverthecomponentmassplane.ThisplothasbeencalculatedfromtheeventscrossingthestandardcWBcuts,havingtheFADvaluelessthantheloudestzerolagevent. Theloudesteventstatisticsisusedtocalculatetheratelimitdensitywith90%condence.ThisisdiscussedindetailinSection 5.6 .TheratedensityupperlimitoverthecomponentmassplaneisshowninFigure 6-4 .Theratelimitissensitivetothedistributionofcomponentmassesonthemassplane.Theaverageratelimitoverthemassplaneforuniformnumberofinjectionswascalculatedtobe3.6Myr)]TJ /F5 7.97 Tf 6.59 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.58 0 Td[(3.Table 6-2 liststheupperratelimitdensityforvariouseBBHmodels.UpperlimitoftheratedensityhasbeenestimatedaftertheapplicationofCAT1,2,3,4andHVETOags. 115

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Figure6-3. TheeffectiverangemeasuredfortheS6DL1H1V1searchoverthecomponentmassplane.ThisplothasbeencalculatedfromtheeventscrossingthestandardcWBcuts,havingtheFADvaluelessthantheloudestzerolagevent. Becauseoftheincreaseinthevisiblevolume,ratedensitydecreasedwiththetotalmass,asexpected.Forthesametotalmass,ratedensitydecreasedwiththeincreaseinthemassratio(massratioisbetween.2and1). Table6-2. Upperlimittotheratedensityforvarioussourcemodels.ResulthasbeenobtainedbytheconvolutionofresultsfromFigure 6-4 withthecomponentmassdistributionobtainedforaparticularsourcemodel. SMBHMass(M)RateLimit(Myr)]TJ /F5 7.97 Tf 6.58 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3) MW24.7MW37.310MW24.710MW37.3 116

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Figure6-4. Theratelimitforthemassbinsonthecomponentmassplane.Tocalculatetheaverageratelimitofanastrophysicalmodel,aweightedarithmeticmeanofthesenumbersiscalculated.Weightsarethefractionofbinariesthatlieinabinonthecomponentmassplane,asdictatedbytheastrophysicalmodel. Theratelimitcanbeusedtosetupperlimitonthe1GNrateusingEquation 6 R90%TV=TVSMBHdensity)]TJ /F16 7.97 Tf 7.31 -1.8 Td[(1GN(MMW),(6)whereTistheobservationtimeandVisthevisiblevolumeofthesearch.TheSMBHnumberdensitycanbecalculatedbyusingthefollowingEquations[ 58 ] dNSMBH dMSMBH=c0MSMBH M)]TJ /F5 7.97 Tf 6.59 0 Td[(1.25e)]TJ /F20 7.97 Tf 6.58 8.81 Td[(MSMBH M(6) SMBHMassDensity=ZMSMBHdNSMBH dMSMBHdMSMBH,(6) 117

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wherec0=3.210)]TJ /F5 7.97 Tf 6.59 0 Td[(11M)]TJ /F5 7.97 Tf 6.59 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.58 0 Td[(3andM=1.3108M.Figure 6-5 plotsthenumberdensityasafunctionoftheSMBHmass.ThemassdensityisplottedinFigure 6-6 Figure6-5. TheSMBHnumberdensityasafunctionoftheSMBHmass.Whenintegratedover104Mto3109M,theareaunderthecurveis0.351Mpc)]TJ /F5 7.97 Tf 6.58 0 Td[(3. Theoveralldetectionrateiswrittenas R=Z)]TJ /F16 7.97 Tf 7.31 -1.8 Td[(1GN(MSMBH)dNSMBH dMSMBHdMSMBH(6)thiscanbewrittenas R=Z)]TJ /F16 7.97 Tf 7.31 -1.79 Td[(1GN(MMW)MSMBH MMW9=28dNSMBH dMSMBHdMSMBH.(6) 118

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Figure6-6. TheSMBHmassdensityasafunctionoftheSMBHmass.Whenintegratedover104Mto3109M,theareaunderthecurveis6.39105MMpc)]TJ /F5 7.97 Tf 6.58 0 Td[(3. FollowingEquation 6 ZMSMBH MMW9=28dNSMBH dMSMBHdMSMBH.05=Mpc3,(6)theupperlimiton)]TJ /F16 7.97 Tf 7.31 -1.79 Td[(1GN(MMW)isR90%=.0510)]TJ /F5 7.97 Tf 6.59 0 Td[(4=yr.Thisisatleastthreeordersofmagnitudelargerthanthetheoreticalestimate[ 4 ].However,thetwolimitsareexpectedtobecomparable,iftheeBBHsearchisconductedinfutureonthedatacollectedfromtheadvanceddetectors. 6.4ComparisonwiththeCBCIMRsearchTemplatesearchfortheCBCsourceshasbeenconductedinthetotalmassrangeof2to100M(twoseparatesearchesinthemassrange2Mto25Mand25Mto 119

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100M).ThesearcheswereconductedfortheS5-VSR1aswellastheS6-VSR1/2/3scienceruns[ 59 63 ].Aseparateburstsearch,usingcWB,inthetotalmassrangeof50Mto450MhasbeenconductedfortheS5-VSR1andtheS6-VSR1/2/3aswell[ 54 55 ].Thesearchwasscienticallyjustiedbecauseoftheabsenceofreliablewaveformsforthebinarieswiththetotalmassesgreaterthan100M.Moreover,massivebinarieshaveshortersignalswhichcanbeadequatelycapturedbyaburstsearch.Incontrast,theeBBHsearchhasbeenconductedforthestellarmassbinaries.AslistedinTable 6-3 ,theeBBHsearchhasasensitiverangewithin20%oftheCBCsearch.ThisresultsuggeststhatcWBwillrecoveraround50%oftheCBCeventsrecoveredbyatemplatesearch.TheratelimitdensityintheeBBHsearchwasfoundtobearoundtwicethetemplatesearch.Asimilarcomparisoncanbeseeningure 6-7 whererecovered/injectedSNRhasbeenplottedforthestellarmassblackholebinaries.Around90%oftheSNRisrecoveredbythepipeline.ThepercentageoftherecoveredSNRshouldresultinalowerratelimitandahigherrangecomparedtothereportedvalues.Thelargerdifferenceinthesensitiverangeisduetotheuseofaconservativethresholdonthecoherentnetworkampitudeforthethreedetector-network.Onlyeventswith5.7weresavedand,asseeninFigure 6-1 ,thecoherentnetworkamplitudecorrespondingtotheFADofloudestforegroundeventisbelow=5.7.ThelargerdifferenceintheratelimitisattributedtothelongerlivetimeintheCBCsearch(moredetectornetworkswereanalyzed),smallersensitiverangeintheburstsearch,differenceinmethodologyfortheapplicationofcalibration,andsystematicerrorsanddifferenceintheloudestforegroundevent.TheonlymodelconstraintsusedintheeBBHsearcharereconstructedchirpmassconstraintandpolarizationconstraint.Thisleavesparameterspaceopenforeccentricity,spin,andotherparameters.Whiletemplate-basedsearchcanhaveattingfactorofaround.8whensearchingspinningbinariesusingnon-spinningwaveformsandttingfactoraslowas.2whensearchingeccentricbinarieswithcircularwaveforms(and 120

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Table6-3. ComparisonoftemplatebasedCBCsearchandeBBHsearchperfomedusingcWB,ontheS5/S6-VSR1/2/3scienceruns.Comparisonismadebasedonsomesampledvaluesofchirpmass,whiledisregardingotherbinaryparameters(massratioetc.).Binarymassesarethecentreofthebinsfor9MwidecomponentmassbinsforCBCsearchwhile4MfortheeBBHsearch.TheCBCrangesareaveragedovertheS6-VSR2/3scienceruns,whilecWBresultsarefortheS6Dsciencerun.Althoughthewaveformsusedinthetwosearchesdifferbecauseofthepresenceofeccentricity(intheeBBHsearch),thetwosearchescanberoughlycomparedbecauseoftheweakdependenceofcWBsearchoneccentricity. WaveformEOBNReBBHEOBNReBBHSearchDataS6-VSR2/3S6D(LHV)S5-S6/VSR1/2/3S5-S6/VSR1/2/3PipelineTemplateBurstTemplateBurstChirpMassReReRateLimitRateLimit(M)(Mpc)(Mpc)(Myr)]TJ /F5 7.97 Tf 6.58 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3)(Myr)]TJ /F5 7.97 Tf 6.59 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3) 1310283.91.71611696.61.118140108.4.8 therebylosssignicantnumberofevents),presentedeBBHsearchandpastIMBHsearchesshowthatthereisnoorveryweakdependenceofcollected/injectedSNRoneccentricityandspin(collected/injectednetworkSNRforMWmodelis.95comparedto.92for10MWmodel.Thecollected/injectedSNRcertainlybecomessmallerasthetotalmassofthebinarybecomessmaller(around.65forNS-NSbinaries);fora5-5Mbinary,asmuchas90%oftheSNRiscollected,asshowninFigure 6-7 6.5EffectofEccentricityontheResultsTheeBBH,waveformshavebeendiscussedindetailinSection 5.5 .Thepresenceofmulti-chirpstructureintheeBBHwaveformsdifferentiatesitfromcircularBBHwaveforms,whichhaveasinglechirpstructuresontheTF-plane.ForcircularBBHwaveforms,themodewhichcarriesmostoftheenergyisn=2whilefortheeBBHwaveformsdistributionoftheenergyamongvariousmodesdependsontheeccentricityofthebinary.Thisresultsin,(a)adecreaseinthedurationofthewaveform,(b)amodulationofthewaveformamplitude,andc)anoverallincreaseintheamplitude.Itis 121

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Figure6-7. RecoveredvsinjectedSNRforthecWBsearchperformedusingEOBNRv2injectionsoverS6A-VSR2runfortheLHVdetectornetwork. expectedthattheparameterspacepopulatedwithhighlyeccentricmergerswillhaveahighereffectiveradiuscomparedtocircularBBH.IntheeBBHsearchasignicantincreaseintheeffectiveradiuswasnotobserved.ThevisiblevolumeforbothMWand10MWmodelswasalmostthesame;Figure 6-8 plotsvariationofthevisiblevolumewithrespecttothecoherentnetworkamplitude,ThesamepatternisobservedinFigure 6-9 ,whichplotsefciency(ofdetectedevents/ofinjectedevents)asafunctionoftheeccentricity. 6.6SummaryofResultsThispaperpresentstheresultsofthesearchforGWfromeBBHinthetotalmassrangeof10Mto50M,withmassratiobetween.2and1.ThesearchwasperformedoverS5/S6-VSR1/2/3data.Noforegroundeventwasfoundthatcanbeclaimedtohave 122

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Figure6-8. Plotshowsthevisiblevolumeversusthecoherentnetworkamplitude.Thereisnovisualdifference.Theonlyvisibledifferenceappearsforlargevaluesofcoherentnetworkdifference.ResultisfortheS6D-VSR3analysisforthemodel=2andMSMBH=MW. anastrophysicalorigin.Intheabsenceofdetection,ratelimitswerecalculated.Theratelimit,averagedoverthecomponentmassplane,forequalnumberofinjectionsinthecomponentmassbins,isestimatedtobe3.6Myr)]TJ /F5 7.97 Tf 6.58 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.58 0 Td[(3.Theminimumratelimitwasestimatedforthemassbincorrespondingtothelargestcomponentmasses(23M-23M)andisequalto.6Myr)]TJ /F5 7.97 Tf 6.59 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3.Themaximumratelimitwasofthemassbincorrespondingtotheminimumcomponentmasses(7M-7M)andisequalto15.3Myr)]TJ /F5 7.97 Tf 6.58 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3.Estimatedratelimitvalueswereusedtocalculatethelimiton)]TJ /F16 7.97 Tf 7.32 -1.79 Td[(1GN(MMW). 123

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Figure6-9. Theefciencyversuseccentricityplot.Thereisnovisibledependenceoftheefciencyontheeccentricity.Thereisadropinefciencywhenchirpcutareused;thisisbecausechirpcutpenalizehighlyeccentricevents.Thisisexpectedbecauseofthepresenceofmulti-chirpstructuresineBBHwaveforms.ResultisfortheS6D-VSR3analysisforthemodel=2andMSMBH=MW Thiswasestimatedtobe10)]TJ /F5 7.97 Tf 6.58 0 Td[(5,whenaveragedoverthemassplane.ThemaximumeffectiverangeofthesearchwasfoundfortheS6DLHrunandwasequalto145Mpc.TheeBBHsearchincorporatedanastrophysicalmodel,forthepopulationoftheparameterspace,withtheaimoftestingthemodel.Thesearchthoughhasalimitedreach.Theestimated1GNMWratewereatleastthreeordersofmagnitudehigherthanthetheoreticalestimates.Theyareexpectedtobecomparablewiththetheoreticalestimates,foranyeBBHsearchtobeperformedovertheadvanceddetectordata. 124

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TwomodelbasedconstraintswereusedintheeBBHsearch,namelyloosecircularpolarizationconstraintandreconstructedchirpmassconstraint.Applicationoftheseconstraintsreducedthebackgroundbyaroundthreeordersofmagnitude.Thesearchdidnotapplyanyconstraintsontheeccentricityorspinofthebinary.TheMWand10MWmodelswerefoundtohavethesimilareffectiverange,suggestingweakdependenceoftheanalysisontheecentricityofthebinaries.Althoughspinwasnotintroduced,searchisexpectedtobenotaffectedbythespinofthebinaries,ashasbeenobservedintheIMBHsearches.ThisisanimportantpointwhencomparingCBCIMRsearchconductedusingcWBandtemplatematchedsearch.Templatebasedsearchcanhaveattingfactorofaround.8whensearchingspinningbinariesusingnon-spinningwaveformsandttingfactoraslowas.2whensearchingeccentricbinarieswithcircularwaveformsandtherebyloosesignicantnumberofevents.Ontheotherhand,theeffectiverangeofbursteBBHsearchwasfoundtobeapproximately80%oftheCBCIMRsearch.ItisexpectedcWBwillbeabletorecover50%oftheCBCeventsrecoveredbyanidealtemplatematchedsearch. 125

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CHAPTER7FUTUREPROSPECTSNoeventswerefoundinthecurrenteBBHsearchthatcanbeclaimedasdetections.UpperlimitsoneBBHcoalesceratewereestimatedtobe3.6Myr)]TJ /F5 7.97 Tf 6.58 0 Td[(1Mpc)]TJ /F5 7.97 Tf 6.59 0 Td[(3,whenaveragedoverthecomponentmassplane,withequalnumberofinjectionsinthebins.ByusingSMBHnumberdensityandtheratelimit,the)]TJ /F16 7.97 Tf 7.31 -1.8 Td[(1GN(MMW)ratewasestimatedtobe10)]TJ /F5 7.97 Tf 6.59 0 Td[(4.Thisvalueisatleastthreeordersofmagnitudehigherthanthetheoreticalestimates.eBBHsearchisexpectedtobeconductedintheadvanceddetectorera(ADE),whenanewgenerationofGWdetectorswillstartcollectingdata.Thesedetectorsareexpectedtobesensitivebymorethananorderofmagnitudecomparedtothepreviousdetectors.Thiswillassureanincreaseinthevisiblevolumebythreeordersofmagnitudeanddecreaseinupperlimitsof)]TJ /F16 7.97 Tf 7.31 -1.79 Td[(1GN(MMW)bythesameamount.Extensiveworkhasbeendonetoshieldthedetectorfromseismicnoise.Duetothis,thedetectorswillhavealowercutofffrequency.Thiswillimprovethedetectabilityofheavierbinaries.ThefollowingsectiondiscusssomeconsiderationsfortheeBBHsearchintheADE. 7.1SecondGenerationDetectorsAdvancedLIGOandVirgodetectorsareexpectedtobeoperatingsoon(2015-2016).Theyhaveundergonemajorupgradestoachieveimprovementinsensitivity.Theearlyadvanceddetectorsareexpectedtobe3-4timesmoresensitivethantheinitialdetectors,whilethedesignsensitivityisexpectedtobeachievedby2019.Figure 7-1 showsthetime-linefortheimprovementintheadvanceddetector'ssensitivity.Majorupgradesarebeingdonetoachievethissensitivity.Atlowfrequency,initialdetectorswereprimarilycontaminatedwiththeseismicnoise.Anentirelynewseismicisolationsystemhasbeenbuilt.Itwillbringdowntheseismiccutofffrequencyfrom40Hzto10Hz.OthermajornoiseswhichaffectsGWdetectorarethermalandquantumnoise.Amorepowerfullaserwillbeusedtoreducethequantumnoise(increasedfrom 126

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Figure7-1. Time-linefortheprogressiveimprovementinthesensitivityoftheadvanceddetectors.Thedesignsensitivityisexpectedtobereachedin2019,whichwillbearoundanorderofmagnitudelargerthantheinitialdetector'ssensitivity.ThedistancesinMpcdenotetheaveragedistanceatwhichNSbinariescouldbeobserved.Imagereference[ 64 ]. 10Wto200W),intheadvanceddetectors.Aheaviertestmasswillbeusedtohandletheradiationpressurecausedbytheincreasedpowerofthelaserbeams(increasedfrom11kgto40kg).Largertestmasses,madeoffused-silica,willbelargerindiametertoreducethethermalnoisecontributions. 7.2eBBHSearchintheADEEraWiththeimprovementinsensitivity,thevisiblevolumefortheeBBHsourcesisalsoexpectedtoincrease.ForthecurrenteBBHsearch,theeffectiveradiuswasestimatedtobearound100Mpc.TheeffectiveradiusisexpectedtoincreasebyanorderofmagnitudeduringADE.AlthoughDopplershiftcorrectionswerenotappliedinthepresenteBBHsearch,theywillneedtobeappliedtoeBBHsearchduringADE(whentheeffectiveradiusformostmassiveeBBHisexpectedtoextendupto1-3Gpc).DuetotheDopplershift,theGWarered-shifted,makingthesourceappearatalargerdistance.AstheGWfrequencyalsodecreasesduetotheDopplershift,thesystemappearsmoremassive.Theco-movingdistance(distancewhichdoesnotget 127

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affectedbytheexpansionoftheuniverse)isrelatedtoluminositydistance(distanceatwhichanastrophysicaleventisobserved)byEquation 7 DL=DC(1+z),(7)wherezistheredshift. 7.2.1DecreaseinCut-OffFrequencyAdvanceddetectorswillhavelowerfrequencycut-offwhichisexpectedtoreducefrom40Hzto10Hz.Duetothis,thedetactibilityoftheburssignal-to-noisetsofGWwhichoccuratperiastronpassagewillincrease.Theseburstsarelocalizedintimebutbroadbandinfrequency.Theimprovement,though,willbemorevisibleinmatchedlteranalysisthantheburstanalysis.Aburstanalysiswillusuallymisstheinitialburstsastheyaresignicantlyseparatedintimeforthemtogetclusteredwiththeinspiralandmergerstage.TheeffectoflowerfrequencyisnotcompletelyabsentintheburstanalysisasshowninFigure 7-2 .Figure 7-3 showsaffectoflowerfrequencycut-offonthedetectabilityofeBBHsources.ThecurrenteBBHsearchhasbeenperformedinthe5)]TJ /F4 11.955 Tf 12.6 0 Td[(25Mmassrange,butithasbeenproposedthatheavierbinariescanbeformedinthegalacticnuclei[ 53 ].Withlowerfrequencycut-off,thedetectibilityoftheheavierbinarieswillincreaseduetotheinspiralandthemergerofsuchbinariesoccuringatalowerfrequency. 7.2.2IncreaseinSensitivityoftheDetectorsFigure 7-4 showstheeffectiveradiusforeBBHsourcesforsimulatedearlyADEstrain.Thereisanincreaseofeffectiveradiusbyafactorofthree,whencomparedwiththeS6Dsensitivity.Thedesignsensitivityoftheadvanceddetectorsisexpectedtobeanorderofmagnitudelarger.Thiscorrespondstoanincreaseindetectionratesbythreeordersofmagnitudeand,intheeventofnodetection,decreaseinratelimitbythesameamount. 128

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Figure7-2. Centralfrequencydistributionoftherecoveredinjections.ThedistributionhasalowermeanforinjectionmadeoversimulatedadvanceddetectorsnoisecomparedtoinjectionsmadeoverS6Ddata,suggestingincreasedcontributionfromlowerfrequencypixelsinthereconstructionoftheevent. 7.2.3RequiredImprovementsintheAnalysisTheeBBHanalysisduringtheADEcanbeimprovedinvariousdepartments.cWBusesWDMtransformwithmultipleresolutiontomaximizetherecoveredSNR.Inspiteofthat,itwasobservedthat,evenwheninjectionwasstrongenoughforitsinspiralandmergerstagetobedetectable,individualburstoccurringduringperiastronpassagewereoftenmissedbytheanalysis.Tworeasonscanbeattributedtothat:eithertheburstsareseparatedintimeanddidnotgetclusteredwiththeremainingwaveformorthepixelswerenotstrongenoughtocrossthethreshold.Clusteringtechniquesaredenedtosavorgenericburstsignatures;theymaybespecializedfortheeBBHwaveformsforbetterclustering.ItwasalsonotclearwhichandhowmanyWDMtransformshouldbeusedintheanalysis.ThelikelihoodisbasedontheprincipalcomponentsandtheideathatincreasingthenumberofresolutionsdoesnotguaranteeanincreaseintherecoveredSNR.Onthecontrary,backgroundnoisemayresultinthedecreasedperformancewiththeincreaseinthenumberofWDMtransform.Figure 7-5 129

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Figure7-3. Spectrumofaninjectedsignal(black)andreconstructedsignal(red).Leftplotisthereconstructioninthefrequencyregion16-1024Hzwhilerightplotin64-1024Hz.Toptwoplotsareinjectionsmadeonsimulatedadvanceddetectorstrain;thecoherentnetworkamplitudeincreasesfrom11.2to11.4fromlefttorightplot.BottomtwoplotsareforinjectionmadeonS6Dstrain;thecoherentnetworkamplitudestaysat11.2.thereisasuddendropinredcurveataround60Hzplotfortherightcurve. comparesthereconstructionofaninjectedwaveformusingdifferentnumbersofWDMtransforms.Reconstructedchirpmasscutwasappliedatthepost-productionstageforthereductionofthebackground.Chirpmassreconstructionwasbasedonthesinglechirpsignature,madebythecircularbinary,ontheTF-map.Incontrast,theeccentricbinarieswaveformsappearasmultiplechirpsontheTF-map.Inspiteofthat,itwasefcientenoughinreducingthebackgroundbyaroundtwoordersofmagnitudewhilelosingonlyasmallportionofinjectedevents.ThesensitivitycurveforS5/S6datahashighfrequencythresholdwhichmadetheinspiralandmergerstagescontribute 130

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Figure7-4. ExpectedeffectiveradiusfortheeBBHsourcesduringearlyADE.TheanalysishasbeenperformedfortheLHVnetworkusingsimulatedstrain. mosttotheSNR.DuringtheADE,whenthefrequencythresholdisgoingtodecrease,performanceofthereconstructedchirpmasscutisexpectedtodecrease.Amodiedreconstructedchirpmasscutwillbeneeded,whichwilltakeintoaccountthemultiplechirps.Theproblemisexpectedtochangefromttingalinetottingmultiplelines,eachcorrespondingtodifferentmodesofeBBHwaveforms.AnothertopicthatneedsconsiderationarethewaveformsfortheeBBHandtheestimationofthesystematicerrorsassociatedwiththem.TheeBBHwaveformsusedinthepresentanalysisareaccurateonlyuptothequadrupoleterm.Themergerstageisgenericandhasbeenpatchedupwiththeinspiralwaveforms.DuetotheabsenceofviablecompetingtoolsthatgenerateeBBHwaveformstohigherdegreeaccuracyinPN 131

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Figure7-5. ReconstructionofaninjectedsignalusingWDMtransformwithfrequencyresolution.(Top)1Hzto128Hz,and(Bottom)2Hzto64Hz. 132

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Figure7-6. ChirpmassreconstructionalgorithmforeBBHanalysisduringADEwillnotbeoptimum.Plotshowsdistributionofellipticityforvariouscases.Adecreasedellipticitysigniesdistributionofthepixels,notclosetoaline. orders,estimationofthesystematicerrorswasdonebycomparisonwiththeEOBNRv2waveformsatzeroeccentricity.ThereisneedfortheeBBHwaveformswithhigherPNaccuracy. 133

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BIOGRAPHICALSKETCH VaibhavTiwaricompletedhisundergraduatedegreeinaerospaceengineeringfromtheIndianInstituteofTechnology,Kharagpur(India).Followingthis,hechangedhismajortophysicsandcompletedhisMasterofSciencedegreeinphysicsfromtheUniversityofSouthCarolina,Columbia.InSpring2015,VaibhavdefendedhisdissertationandcompletedhisDoctorofPhilosophydegreeinphysicsattheUniversityofFlorida. 139