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Search for Gravitational Waves from Intermediate Mass Black Hole Binaries

Permanent Link: http://ufdc.ufl.edu/UFE0042756/00001

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Title: Search for Gravitational Waves from Intermediate Mass Black Hole Binaries
Physical Description: 1 online resource (133 p.)
Language: english
Creator: PANKOW,CHRISTOPHER P
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

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Subjects / Keywords: Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: The radiation of gravitational waves (GW) produced by accelerating objects is predicted by the theory of general relativity. Several interferometric detectors, such as LIGO and Virgo, are used to detect gravitational-wave events from astrophysical sources. The coalescence of binary neutron stars and black holes is among the most promising sources for the first detection of gravitational waves. Also, intermediate mass black hole binaries (IMBHB) which have one or both of the component masses greater than $100\ms$ and total mass less than 450$\ms$ can be potentially detected by the LIGO and Virgo instruments I developed and conducted a search for gravitational waves from the coalescence of IMBHB in the total mass range of 100$\ms$ to 450$\ms$. The search used data collected by the LIGO and Virgo interferometers between May of 2005 and October of 2007. 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 event reconstruction was constrained to the elliptical polarization model which IMBHB sources are expected to conform to. As part of the search, both an accurate estimation of the background event distribution and the characterization of the search sensitivity to the target sources have been performed. The background was estimated using events from time-slides of the detector data beyond the expected maximum light-travel time, and the sensitive volume was measured using simulated waveforms from the Effective One Body Numerical Relativity (\emph{EOBNR}) family. No plausible gravitational-wave candidates were observed. 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 was measured to be 4\e{-7} per Mpc${}^3$ per year over the entire mass range.
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.
Statement of Responsibility: by CHRISTOPHER P PANKOW.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Klimenko, Sergey Grigoryevich.
Local: Co-adviser: Mitselmakher, Gena.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-04-30

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Classification: lcc - LD1780 2011
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Permanent Link: http://ufdc.ufl.edu/UFE0042756/00001

Material Information

Title: Search for Gravitational Waves from Intermediate Mass Black Hole Binaries
Physical Description: 1 online resource (133 p.)
Language: english
Creator: PANKOW,CHRISTOPHER P
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The radiation of gravitational waves (GW) produced by accelerating objects is predicted by the theory of general relativity. Several interferometric detectors, such as LIGO and Virgo, are used to detect gravitational-wave events from astrophysical sources. The coalescence of binary neutron stars and black holes is among the most promising sources for the first detection of gravitational waves. Also, intermediate mass black hole binaries (IMBHB) which have one or both of the component masses greater than $100\ms$ and total mass less than 450$\ms$ can be potentially detected by the LIGO and Virgo instruments I developed and conducted a search for gravitational waves from the coalescence of IMBHB in the total mass range of 100$\ms$ to 450$\ms$. The search used data collected by the LIGO and Virgo interferometers between May of 2005 and October of 2007. 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 event reconstruction was constrained to the elliptical polarization model which IMBHB sources are expected to conform to. As part of the search, both an accurate estimation of the background event distribution and the characterization of the search sensitivity to the target sources have been performed. The background was estimated using events from time-slides of the detector data beyond the expected maximum light-travel time, and the sensitive volume was measured using simulated waveforms from the Effective One Body Numerical Relativity (\emph{EOBNR}) family. No plausible gravitational-wave candidates were observed. 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 was measured to be 4\e{-7} per Mpc${}^3$ per year over the entire mass range.
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.
Statement of Responsibility: by CHRISTOPHER P PANKOW.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Klimenko, Sergey Grigoryevich.
Local: Co-adviser: Mitselmakher, Gena.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-04-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042756:00001


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SEARCHFORGRAVITATIONALWAVESFROMINTERMEDIATEMASSBLACKHOLE BINARIES By CHRISTOPHERPAULPANKOW ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011 1

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c 2011ChristopherPaulPankow 2

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Tomyparents,withoutwhothisworkcouldneverhavehappened,andtoDon,without whothiswork would neverhavehappened 3

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ACKNOWLEDGMENTS Firstandforemost,thisworkwouldnothavebeenpossiblewithouttheconstant assistance,guidance,andpatienceofmyadvisor,SergeyKlimenko.Iwouldalsolike toacknowledgethenumerouspiecesofadvicebothscienticandmotivational providedmebyIvanFuric.ThanksisalsoduetoNeilCornish,whohasprovided helpfulguidanceinpolishingthenalresults.DavidReitzehassupportedmewith bothencouragementaswellhelpfuldocumentsandmaterialsintheconstructionof thedocument.Myfriendsinthephysicsdepartmenthavedonemeagreatservicein keepingmesaneduringthewholeadventure.AspecialthanksgoestoFranciscoRojas, whonotonlyranthesanitarium,butalsoprovidedthemeansforothergures.Finally, I'dliketothankmycommitteefortakingthetimetohelpfurtherrenetheworkandllin thegapssometimesgapingholesinthenalproduct. IgratefullyacknowledgethesupportoftheUnitedStatesNationalScienceFoundationfortheconstructionandoperationoftheLIGOLaboratory,theScienceand TechnologyFacilitiesCounciloftheUnitedKingdom,theMax-Planck-Societyandthe StateofNiedersachsen/Germanyforsupportoftheconstructionandoperationofthe GEO600detector,andtheItalianIstitutoNazionalediFisicaNucleareandtheFrench CentreNationaldelaRechercheScientiquefortheconstructionandoperationof theVirgodetector.Ialsogratefullyacknowledgethesupportoftheresearchbythese agenciesandbytheAustralianResearchCouncil,theCouncilofScienticandIndustrialResearchofIndia,theIstitutoNazionalediFisicaNucleareofItaly,theSpanish MinisteriodeEducaci onyCiencia,theConselleriad'EconomiaHisendaiInnovaci oof theGoverndelesIllesBalears,theFoundationforFundamentalResearchonMatter supportedbytheNetherlandsOrganisationforScienticResearch,thePolishMinistry ofScienceandHigherEducation,theFOCUSProgrammeofFoundationforPolishScience,theRoyalSociety,theScottishFundingCouncil,theScottishUniversitiesPhysics Alliance,theNationalAeronauticsandSpaceAdministration,theCarnegieTrust,the 4

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LeverhulmeTrust,theDavidandLucilePackardFoundation,theResearchCorporation, andtheAlfredP.SloanFoundation. ThisworkwassupportedbytheUSNationalScienceFoundationgrantsPHY0855044andPHY-0855313totheUniversityofFlorida,Gainesville,Florida. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................9 LISTOFFIGURES.....................................10 ABSTRACT.........................................13 CHAPTER 1INTRODUCTIONTOTHEDETECTIONOFGRAVITATIONALWAVES.....15 1.1GravitationalWaves..............................15 1.2GravitationalWavesFromOrbitingSources.................18 1.3IntermediateMassBinaryBlackHoleBinaries................20 2GRAVITATIONAL-WAVEDETECTORSANDNETWORKS............26 2.1InterferometricGravitationalWaveDetection.................27 2.2NoiseSourcesinInterferometricGWDetectors...............30 2.2.1SeismicNoise..............................32 2.2.2ThermalNoise..............................32 2.2.3ShotNoise................................33 2.2.4RadiationPressure...........................33 2.3GravitationalWaveDetectorsDuringS5/VSR1...............34 2.3.1LIGO...................................34 2.3.2Virgo...................................35 2.3.3GEO600.................................36 2.4DetectorNetworks...............................37 2.4.1DominantPolarizationFrame.....................39 2.4.2NetworkSensitivity...........................40 3SEARCHALGORITHM...............................43 3.1Time-FrequencyAnalysis...........................44 3.2CoherentNetworkAnalysis..........................48 3.3RegulatorsandConstrainedLikelihood....................52 3.3.1HardRegulator.............................54 3.3.2WeakRegulator.............................54 3.3.3SoftRegulator..............................54 3.3.4NetworkResponseIndex........................55 3.3.5EnergyDisbalanceConstraint.....................55 3.4EllipticalConstraint...............................59 3.4.1DualStreamAnalysis..........................59 3.4.2MaximizedLikelihoodUndertheEllipticalConstraint........60 6

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4SEARCHFORGRAVITATIONALWAVESFROMINTERMEDIATEMASSBLACK HOLEBINARIES...................................63 4.1S5/VSR1....................................63 4.2TheSearchPipeline..............................64 4.2.1RunStaging...............................64 4.2.2RunningthePipeline..........................65 4.2.3Initialization,DataConditioning,Time-FrequencyDecomposition, andClustering..............................65 4.3CoherentTriggerProduction..........................66 4.4SelectionCriteria................................68 4.4.1NetworkCorrelationCoefcient....................68 4.4.2NetworkandDualStreamEnergyDisbalance............69 4.4.3CoherentNetworkAmplitude.....................70 4.4.4PenaltyFactor..............................71 4.5BackgroundEstimation.............................71 4.5.1Time-LagBackground.........................72 4.5.2UniqueLags...............................73 4.5.3SearchDataQualityandVetoes....................74 4.6Simulation....................................76 4.6.1NumericalRelativityInspiredWaveforms...............76 4.6.1.1EffectiveOneBodyNumericalRelativitywaveforms...77 4.6.1.2IMRPhenomfamily......................78 4.6.2EfciencyEstimation..........................78 4.7InjectionStrategy................................79 4.7.1MassDistribution............................80 4.7.2RadialPositionDistribution.......................81 4.8VisibleVolume.................................84 4.9UncertaintiesandEstimationofError.....................86 4.9.1CalibrationUncertainties........................87 4.9.2WaveformFamilyBiases........................88 4.10CombinedSearchUpperLimits........................89 4.10.1FalseAlarmRateStatistics.......................90 4.10.2FalseAlarmDensityandCombinedSearchStatistics........92 5RESULTSOFTHES5/VSR1IMBHBINARYSEARCH..............95 5.1ApplicationofCategory2EventVetoes...................95 5.2Post-productionTuning.............................97 5.3BackgroundSets................................101 5.3.1H1H2L1Network............................101 5.3.2H1H2L1V1Network...........................102 5.4LoudestEvents.................................102 5.4.1LoudestEventsintheH1H2L1Search................102 5.4.2LoudestEventsintheH1H2L1V1Search...............103 5.4.3LoudestEventsintheCombinedSearchbyIFAD..........105 7

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5.5ApplicationofCategory3Vetoes.......................106 5.6EffectiveRange.................................108 5.7StatisticalandSystematicUncertainties...................108 5.8CombinedUpperLimit.............................111 5.9ResultsSummary................................113 6FUTUREDIRECTIONS...............................114 6.1SpinEffects...................................114 6.1.1Spin-AlignedBinaries..........................115 6.1.2PhenomenologicalWaveformswithAlignedSpins..........115 6.1.3Results..................................116 6.1.4ArbitrarySpinDirections........................118 6.2EnhancedandSecondGenerationDetectors................120 6.2.1S6/VSR2/VSR3ScienceRun.....................122 6.2.2AdvancedLIGOandAdvancedVirgo.................123 LISTOFREFERENCES..................................125 BIOGRAPHICALSKETCH................................133 8

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LISTOFTABLES Table page 1-1Characteristicfrequenciesforbinariesofvariousmasscombinations......23 4-1ResolutionsatwhichthepipelinedecomposesT-Fdata.............66 4-2Livetimeaftervariousdataquality..........................75 4-3Overviewoftheparameterdistributionusedinthesearch............82 4-4FractionalerrorontheamplitudeofthecalibratedGWstraininthe40Hz 2kHzbandwidth...................................87 5-1Tableofcutsetstested...............................98 5-2Listingofthepost-productioncutthresholds....................100 5-3ThreeloudesteventsintheH1H2L1search....................103 5-4ThreeloudesteventsintheH1H2L1V1search..................104 5-5TriggersrankedbythecombinedsearchstatisticalongwiththeirFAPandnetworkinformation...................................106 6-1Comparisonofeffectiverangesforwaveformswithandwithoutspin......118 9

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LISTOFFIGURES Figure page 1-1Thelateinspiral,merger,andring-downofabinarycoalescence........21 1-2X-rayimagesofNGC4485andNGC4490andanULXintheFornaxCluster.22 2-1Theeffectofthe h + and h GWpolarizationonaquadrupoleantenna.....28 2-2AsimplieddiagramofaMichelsoninterferometer................29 2-3InitialLIGOnoisesources..............................31 2-4StrainsensitivityversusfrequencyfortheLIGOinterferometers.........36 2-5StrainsensitivityversusfrequencyfortheVirgointerferometer,andcomparisonwithLIGO.....................................37 2-6Angularsensitivityofinterferometerstogravitationalwaves...........38 2-7NetworksensitivitymagnitudeandsensitivityratiofortheLivingston-Hanford network........................................41 2-8NetworksensitivitymagnitudeandsensitivityratiofortheLivingston-HanfordVirgonetwork.....................................42 3-1AnillustrationofaMeyermotherwaveletfunction.................46 3-2Binaryversusdyadicwavelettransformations...................46 3-3AnexampleofbinaryversusdyadiconT-Fdata..................47 3-4Thegeometricalinterpretationofthelikelihoodanalysis..............51 3-5Themodiedbasisofforlikelihoodanalysisusingregulatorsandenergydisbalance........................................53 3-6TheNetworkResponseIndexasafunctionofskylocationforaneventinthe four-folddetectornetwork...............................56 4-1AnexampleoftimefrequencymapsintheH1H2L1V1networksearch......67 4-2ExamplelikelihoodTFmap.............................68 4-3Demonstrationoftime-shiftsdoneforbackgroundestimation..........73 4-4Injectiondistributioninthecomponentmassplane................80 4-5Injectedducialradius r max = R overthecomponentmassplaneintheS5/VSR1 IMBHsearch......................................83 4-6Distributionofinjections'radiusbeforeredandaftergreenrescaling....83 10

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4-7DeviationinSNRbetween EOBNR familiesandNRresults...........89 4-8PercentdifferenceintheuncorrectedEOBNRresultrelativetothebiascorrectedEOBNRresult.................................90 4-9FADversusrankingstatistic.............................93 5-1CoherentnetworkamplitudeversustimefortheH1H2L1V1network......96 5-2CoherentnetworkamplitudeversustimefortheH1H2L1network........97 5-3Comparisonofccand valuesfortheH1H2L1V1injectionandbackgroundset98 5-4Comparisonof and valuesfortheH1H2L1V1injectionandbackgroundset99 5-5ThecumulativeeffectiverangeasafunctionofdistancefortheH1H2L1V1 network........................................100 5-6Twodimensionalplotofeventrankingstatistic incolorversustimeandcentralfrequency F c fortheH1H2L1network.Thetimeisrelativetothestartof theS5run.......................................101 5-7CumulativerateofbackgroundeventsversustheirrankingstatisticfortheH1H2L1 network........................................102 5-8Twodimensionalplotofeventrankingstatistic incolorversustimeandcentralfrequency F c fortheH1H2L1V1network.Thetimeisrelativetothestart oftheS5/VSR1run..................................103 5-9CumulativerateofbackgroundeventsversustheirrankingstatisticfortheH1H2L1V1 network........................................104 5-10FalseAlarmDensityforbothnetworks,includingthebackgroundeventsand foregroundevents,asafunctionoftherankingstatistic..............105 5-11EffectofCategory3dataqualityontheH1H2L1V1backgrounddistribution..107 5-12EffectofCategory3dataqualityontheH1H2L1backgrounddistribution....107 5-13FalseAlarmDensityforbothnetworksafterCategory3dataquality......108 5-14TheeffectiverangemeasuredfortheH1H2L1V1searchoverthecomponent massplane.......................................109 5-15TheeffectiverangemeasuredfortheH1H2L1searchoverthecomponent massplane.......................................109 5-16Fractionaluncertaintyabsolutestatisticaluncertaintynormalizedbythetotal visiblevolumeonthevisiblevolumefortheH1H2L1V1search..........110 11

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5-17Fractionaluncertaintyabsolutestatisticaluncertaintynormalizedbythetotal visiblevolumeonthevisiblevolumefortheH1H2L1search...........111 5-18TheratedensityinMpc )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 yr )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 upperlimitonIMBHBcoalescenceoverthe componentmassplaneforthecombinedsearch..................112 5-19CombinedsearchproductivityasafunctionofFAD................112 6-1DiagramofBHbinarysystemwithspinningBH...................115 6-2Distributionofthemagnitudeofthecomponentspinvectors...........117 6-3Efciencyovercomponentspins..........................118 6-4Effectiverangeof IMRPhenomB familywithspineffects..............119 6-5Ratiooftheeffectiverangesforspinningovernon-spinningfamilies.......119 6-6LIGOstrainsensitivityduringpartofS6......................122 6-7ProjectedAdvancedLIGOstrainsensitivity....................124 6-8ProjectedAdvancedVirgostrainsensitivity....................124 12

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SEARCHFORGRAVITATIONALWAVESFROMINTERMEDIATEMASSBLACKHOLE BINARIES By ChristopherPaulPankow May2011 Chair:SergeyKlimenko Cochair:GuenakhMitselmakher Major:Physics TheradiationofgravitationalwavesGWproducedbyacceleratingobjectsis predictedbythetheoryofgeneralrelativity.Severalinterferometricdetectors,such asLIGOandVirgo,areusedtodetectgravitational-waveeventsfromastrophysical sources.Thecoalescenceofbinaryneutronstarsandblackholesisamongthemost promisingsourcesfortherstdetectionofgravitationalwaves.Also,intermediatemass blackholebinariesIMBHBwhichhaveoneorbothofthecomponentmassesgreater than100 M andtotalmasslessthan450 M canbepotentiallydetectedbytheLIGO andVirgoinstruments Idevelopedandconductedasearchforgravitationalwavesfromthecoalescence ofIMBHBinthetotalmassrangeof100 M to450 M .Thesearchuseddatacollected bytheLIGOandVirgointerferometersbetweenMayof2005andOctoberof2007. ThesearchwasperformedusingtheCoherentWaveBurstalgorithmwhichutilizes alikelihoodapproachtoreconstructGWeventsfromdatacollectedbymultipleGW detectors.Theeventreconstructionwasconstrainedtotheellipticalpolarizationmodel whichIMBHBsourcesareexpectedtoconformto.Aspartofthesearch,bothan accurateestimationofthebackgroundeventdistributionandthecharacterizationof thesearchsensitivitytothetargetsourceshavebeenperformed.Thebackground wasestimatedusingeventsfromtime-slidesofthedetectordatabeyondtheexpected 13

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maximumlight-traveltime,andthesensitivevolumewasmeasuredusingsimulated waveformsfromtheEffectiveOneBodyNumericalRelativity EOBNR family.No plausiblegravitational-wavecandidateswereobserved.Ipresenttheresultsofthe search,includingthesensitivevolumeandupperlimitsonthesourceratedensityasa functionofcomponentmass.Theupperlimitonthesourceratedensitywasmeasured tobe4 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(7 perMpc 3 peryearovertheentiremassrange. 14

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CHAPTER1 INTRODUCTIONTOTHEDETECTIONOFGRAVITATIONALWAVES GravitationalwavesarepredictedfromperturbativesolutionstoEinstein'sequations inaatvacuum.Intheweakeldlimit,theEinstein'sequationswiththeproper gaugeconditionstakeontheformofawaveequation.Thesolutiontothislinearized equationisafreelypropagatingtensorwavewithtwodegreesoffreedom.Theeffectin thevicinityofthepassingwaveistoalternativelystretchandcompressspace-time. Gravitationalwavesremaintheoneofthefewunveriedpiecesintheconrmation ofgeneralrelativity.PreviousexperimentssuchastestsoftherelativisticShapiro timedelay[1],theequivalenceprinciplethroughtheNordtvedteffect[2,3],gravitational redshiftviathePound-Rebkaexperiment[4],andmostrecentlythetestofframedraggingbyGravityProbeB[5]haveconstrainedpost-Newtonianparametersvery closelytothoseexpectedbygeneralrelativity.However,gravitationalwavesremain unidentiedbydirectobservations. IndirectobservationofgravitationalwaveswereperformedbyHulseandTaylor whocarriedoutexperimentswiththepulsarPSR1913+16andoveratimeperiodof aboutthirtyyearsmatchedtheexpectedlossofenergyviagravitationalwavestothe spindownoftheperiodofthepulsar[6].Thepulsaritselfpairedwithapartner,anddue totheirverycloseseparationdistancelessthanafewsolarradiiatmaximumthe binarycompletesafullrotationaboutonceeveryeighthours.Theenergyreleasedin thisprocessislargeenoughtomeasurethedecreaseinorbitalperiodofthebinaryover anextendedperiodoftime. 1.1GravitationalWaves Generalrelativitydescribesthebehavioranddynamicsofthespace-timemetric g representingthegeometryofspace-timeanditsinteractionwithmattersources. R )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(1 2 g R = 8 G c 4 T .1 15

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Equation1containsthecurvature,i.e.theRiccicurvaturetensor R andRicci scalar R thestress-energytensor T ,andthemetrictensor g Inthefollowingdiscussiononthederivationgravitationalwaves,themetricwillbe atspacee.g. g = whichisdescribedinEquation1. = 0 B B B B B B B @ )]TJ/F21 11.9552 Tf 9.298 0 Td [(1000 0100 0010 0001 1 C C C C C C C A TheEinsteinequationsareinherentlynonlinear,however,gravitationalwavesarise fromthelinearizationofthefullEinsteinequationsintheweakeldlimit.Inorderto linearizeit,aperturbation h ismadeonaatspacemetric suchthat g = + h where j h j 1.First,theRiemanncurvaturetensoriscalculatedandthencontractedto obtaintheRiccitensorandscalar.Inatspace,andkeepingonlytermsrstorderin h theRiemanntensorisdescribedinEquation1 R = 1 2 )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(@ @ h + @ @ h )]TJ/F26 11.9552 Tf 11.955 0 Td [(@ @ h )]TJ/F26 11.9552 Tf 11.955 0 Td [(@ @ h Twotermsoforder h 2 havebeendroppedinEquation1.TheRiccicurvature tensorisobtainedbyraisingtherstindexandcontractingwiththethird. R = 1 2 )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(@ @ h + @ @ h )]TJ/F26 11.9552 Tf 11.955 0 Td [(@ @ h )]TJ/F26 11.9552 Tf 11.955 0 Td [(@ @ h TheRicciscalaristhenformedfromthecontractionoftheremainingtwoindices. R = 1 2 )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(@ @ h + @ @ h )]TJ/F26 11.9552 Tf 11.955 0 Td [(@ @ h )]TJ/F26 11.9552 Tf 11.955 0 Td [(@ @ h Sincethemetricissymmetric,sois h .Withthissimplication,andtheidenticationofthetraceof h as h = h and @ as @ 2 ortheatspaceD'Alembertian )]TJ/F26 11.9552 Tf 9.298 0 Td [(@ 2 t + @ 2 x + @ 2 y + @ 2 z ,thenewformoftheequationisdisplayedinEquation1. 16

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@ @ h + @ @ h )]TJ/F26 11.9552 Tf 11.955 0 Td [(@ 2 h )]TJ/F26 11.9552 Tf 11.955 0 Td [(@ @ h )]TJ/F26 11.9552 Tf 11.955 0 Td [( )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(@ @ h )]TJ/F26 11.9552 Tf 11.955 0 Td [(@ 2 h = 16 G c 4 T The g termhasbeendroppedto because R wasalreadyoforder h and thereforemultiplyingbythefullmetricwouldhavegonebeyondthisorder.Equation 1canbereducedbyimposinggaugesconditionsonthedegreesoffreedomin theequation.First,adifferentwayofwritingthemetricperturbationcalledthetrace reversedmetricisintroducedinEquation1. h = h )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(1 2 h UsingthisdenitioninEquation1andsomesimplicationsprovidesEquation 1. @ @ h + @ @ h )]TJ/F26 11.9552 Tf 11.955 0 Td [(@ 2 h )]TJ/F26 11.9552 Tf 11.956 0 Td [( @ @ h = 16 G c 4 T Additionally,thedeDondergaugecondition,asinEquation1isapplied. @ h =0 Thisconditionisverysimilartothesameusedintheelectromagneticeldequationswhichgiverisetoelectromagneticwaves.Theconditionreducesthelefthandside toonlyoneterm: @ 2 h = 16 G c 4 T Equation1isasecondordertensorwaveequationwithmultipledegreesof freedom.Therepresentationofthewavecanbefurtherrestrictedbyxingthecondition inthewaveframesuchthatwavedisturbanceistransversetothepropagationdirection. Thisisknownasthetransverse-tracelessgaugeandreducestheoverallsolutionofthe 17

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equationtotwodegreesoffreedom,whicharelabeled h + and h .Inthespecicchoice ofthepropagationdirectioninthe z direction,thesolutionisrepresentedasinEquation 1. 0 B B B B B B B @ 0000 0 h + h 0 0 h )]TJ/F24 11.9552 Tf 9.299 0 Td [(h + 0 0000 1 C C C C C C C A Theamplitudes h + and h representthetwoindependentpolarizationstatesof thegravitationalwave.Theamplitudes h + and h aremeasuredinstrain.Strainis measuredinthedifferenceofperturbedpathlengthversusnonperturbedpathlength. Overall,theeffectofapassinggravitationalwaveistoalternatelystretchandcontract space-timeinorthogonaldirectionsproportionaltotheamplitude.Thetwopolarization statesproducethesameeffect,exceptwithonepolarizationstatebeingrotated45 degrees. OnecanbuildageneralsolutiontoEquation1withoutsourcesi.e.setting T =0.Thewaveequationadmitssolutionsintheformofplanewavesasshownin Equation1. h = H e k x WhereinEquation1, k isthewave4-vector, x isthecoordinate4-vector,and H isamatrixofyetundeterminedcoefcientsthatspecifytheamplitudeofthewave. 1.2GravitationalWavesFromOrbitingSources Uptothispoint,nosourcehasbeenspecied,andhencethesolutionpresentedin 1iscompletelygeneral.Theprototypicalexampleintheproductionofgravitational wavesfromastrophysicalsourcesaretwobodiesorbitingeachother.Classically,inan electromagneticanalogy,theelectronandprotonwouldorbiteachotherandbecause 18

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oftheacceleration,loseenergythroughelectromagneticradiation.Thelossofenergy shouldcausetheoverallorbittodecay,andwithoutothereffectsthetwoparticleswould spiralintowardseachother.Thecaseforgravitationisidenticalasitselectromagnetic counterpartwhengeneralrelativityisconsidered.However,whilequantumeffects preventthecollapseoftheatom,thereisnoanaloginthegravitationalcase,andtwo orbitingbodieswillemitgravitationalradiationandspiralintowardseachotherinarun awayeffect.Inasimpliedperturbativesolutiontothegravitationalwavesemittedfrom twobodycircularorbitproblem,thegravitational-waveamplitudesfromtwoorbiting objectsinthefar-eldapproximationaregiveninEquations1and1. h + t = 4 r Gm c c 2 5 = 3 f t )]TJ/F24 11.9552 Tf 11.955 0 Td [(r = c c 2 = 3 1+cos 2 2 cos t )]TJ/F24 11.9552 Tf 11.955 0 Td [(r = c h t = 4 r Gm c c 2 5 = 3 f t )]TJ/F24 11.9552 Tf 11.955 0 Td [(r = c c 2 = 3 cos sin t )]TJ/F24 11.9552 Tf 11.955 0 Td [(r = c Intheabovedescription,theamplitudeasafunctionoftimeisdependentonfour quantities.Therstbeingthechirpmass m c ,afunctionofthemassesofthetwo objects. m c = m 1 m 2 3 = 5 m 1 + m 2 1 = 5 Othersaretheinclination oftheorbitalplanewithrespecttotheearthandthe distance r totheobject.Finally, t describestheorbitalfrequency. t = 0 )]TJ/F21 11.9552 Tf 11.956 0 Td [(2 )]TJ/F21 11.9552 Tf 9.298 0 Td [(5 Gm c tc 3 )]TJ/F21 7.9701 Tf 6.586 0 Td [(5 = 8 Theotherquantitiesinvolvedaretheconstants G Newton'sconstant,andthe speedoflight c .ItshouldbenotedthattheexpressionsinEquations1,14,and 1arevalidfortimesbeforezero.Thetermsdivergeat t =0.Thisisindicativeofthe 19

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inabilityofthemodeltodescribethepointwherethetwoobjectsoverlapandmerge,so t =0couldbeconsideredanapproximationtothecoalescencetimee.g. t coalescence =0. Ofcourse,theamplitudeandphasingequationsareapproximationsbasedonpointmassesandlinearizedgeneralrelativity.Itispresentedasabasicdescriptionofthe dynamicsofthesystemandtheresultinggravitational-waveemission.Moreaccurate formulasforgravitational-waveemissionareobtainedthroughthepost-NewtonianPN expansionoftheEinsteinequations.Currently,theequationsofmotionsforbinary blackholeshavebeencalculatedtoseventhorderin v c or3.5PNorderforthephase evolution[7].Ultimately,higherordereffectsbecomedifculttocalculate,andtherefore non-analyticalmethodsarerequiredtodescribethestateofthesystemduringthe merger. Whilethepost-Newtonianexpansionsarevalid,theevolutionofthebinarybefore theplungeiscalledtheinspiralstage.Sincethepost-Newtonianapproachisanapproximation,itfailsintheregimewheregeneralrelativisticeffectsbecomestrongenoughto nolongeradmitregularorbits.Afterthelaststableorbit,thetwoobjectsinthebinary thenplungetogetherinamuchshortermergerstageproducingaburstofgravitational waves.Afterthemerger,theperturbedblackholeproducedwillring-downintoa symmetricalcongurationbyemittinggravitationalwaveslikeadampedsinusoid.The ring-downisaperturbedsolutiontoaKerrmetricandcanbedescribedanalytically[8]. Hence,theonlyportionoftheevolutionwhichhasnoanalyticdescriptionisthemerger. RecentadvanceshaveallowednumericalsolutionstotheevolutionproblemandSection 4.6.1willdescribehowtheinspiral,merger,andring-downstagesarettogethertoproducewaveformsfromtheentireevolutionprocess.Figure1-1showsthefullwaveform throughthemergerandring-downstages. 1.3IntermediateMassBinaryBlackHoleBinaries Whenrstproposed,blackholeswereonlyconceivedofasastellar-mass objects[911].Theoreticalandobservationaldevelopmentsovertheinterveningyears 20

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. Figure1-1.Thelateinspiral,mergernearthepeakamplitude,andring-downdamped sinusoidofabinarycoalescencerepresentedasatimeseriesintotalstrain onaninterferometricGWdetector.Thisisthewaveformthatwouldbe producedbya113 M +131 M binaryblackholesystem haveexpandedthemassrangeofblackholesfromminisculeprimordialblackholes allthewayuptothesupermassiveblackholeswhichexistwithingalacticcoreswith massesofupto10 6 M ormore.Themainfocusofgravitational-wavedetectionefforts uptothispointhasbeensolarmassblackholesblackholesformedintheendpointof stellarevolutionasthesignaturefrequenciesoftheirdynamicsfallwithinthesensitive regionofcurrentgroundbasedinterferometersseeTable1-1.Recentastronomical observationshaveindicatedevidenceforintermediatemassblackholesIMBH[12]. TheseIMBHwouldpopulatetheregionofmassspacebetweenstellarmassblackholes andsupermassiveblackholes[13,14].Withnoobservationalconrmationandlackof hardevidencefortheirformationprocesses,aformallimitoneithertheirsmallestor largestsizeisimpossible.However,generalagreementinliteratureplacesthemabove around100 M andbelowthelowerboundofobservedsupermassiveblackholesnear 10 5 M 21

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Figure1-2.Left:AChandraX-rayimageofNGC4486topandNGC4490bottom, twogalaxiesidentiedasULXsources[24].Intotaltherearevestrong X-rayemittingobjectsinthesetwoclusters,andassuchcouldbepowered byIMBH.Right:AcompositeX-ray/opticalimageofanULXsourceinthe Fornaxcluster[25].ThepotentialIMBHisthebrightX-raysourcelocated aboveandtotheleftofthecentersource. Interestinthesesourceshasincreasedinrecentyearsbecausetheyarepossiblesolutionstosomeunresolvedproblemswithdescribingenergeticastronomical phenomena.AnumberofUltraluminousX-RayULX[15,16]sourceshavebeenidentiedbyastronomersincludingX-raysourcesinLocalGroupmemberandstarburst galaxyIC10[17,18],themostluminousX-raysourceyetdiscovered,M82X-1[19,20], aULXinNGC1313[21]withablackholewithmassbetween50 M ,aswellasa suspectedX-raybinary[22]inthePinwheelgalaxyM101withapossiblesupergiant companion[23].ThesesourceshavebeensuggestedashostsforIMBHduetotheir super-Eddingtonluminosities.Iftheemissionfromthesesourcesisnotbeamed,then accretionmodelssuggestionthatULXsmaybepoweredbythelargeststellarmass blackholeseverobserved. Fromthestandpointofgravitational-waveastronomy,thesesourcesareofinterest becausethebinariesmightbeaccessibletogroundbasedinterferometersatthepresent time.Binarieswithtotalmassoflessthan450 M havering-downfrequencieswithin 22

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Table1-1.Characteristicfrequenciesfortheinner-moststablecircularorbitISCOand ring-downofvariousbinaryblackholesystems.Notethat,ingeneral,mass ratiosdeviatingawayfromunityandnon-zerospinwilladjustthevalues upward.Also,the f ring isonlyforthelowestemittedfundamentalmode,the l =2, m = 2mode. Componentmasses M f ISCO Hz f ring Hz 2,56301700 25,2588240 100,1002260 175,1751534 reachofcurrentgenerationdetectorsseeTable1-1andbinarieswith 450 M arepotentiallydetectableviatheirmergerandring-downattheedgeofthesensitive frequenciesoftheLIGOandVirgointerferometernetwork.Becauseasignicantamount ofpowerisradiatedduringthemergerandring-downphasesofthesemassivesystems, forafavorablecongurationthebinarycanbedetectedhundredsofmegaparsecsaway. NosearchofanykindhastargetedIMBHbinariesuptothispoint.Previous searchesfordifferentpartsofthefullcoalescencesignalcalledInspiralMerger Ring-downorIMRhavebeenperformedbymatched-ltersearchesusingdatafrom LIGOandVirgo.TheseincludeasearchforlowmassandprimordialbinariesinS3/S4 LIGOdata[26],severallowmasssearchesoverdifferentpartsofS5/VSR1data[2729], asearchforhighmass < M = M < 100binariesinallofS5[30],searchestriggered onelectromagneticsourcessuchasGRBs[31],andsearchesforring-downsignalsin S4[32].Thesearchforring-downsignalsiscurrentlybeingextendedintoS5data. WithlittleobservationaldataforIMBH,theformationofbinarysystemscanonly beinferredfrommodels.Mostofthesemodelsproposetheformationoftheseobjects throughtwopossiblynon-exclusiveprocesses.Onepossibilityisthenearlossless collapseofverymassivestars[33,34].Alternatively,IMBHcouldalsoformbythe runawaymergerofstellarmassblackholesinablackholerichcluster[3436]. Therstscenarioimpliestheexistenceofaverymetal-poorenvironment,suchas theearlyuniverse.PopulationIIIstarsareahypotheticalclassofstarswhicharisefrom 23

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thislowmetallicityenvironmentandformwithabundancesofhydrogen,helium,and othermetalsinproportionsinheriteddirectlyfromthepost-BigBangera[37].Then,this scenariopositsanendoflifemechanismwheretheseyoungstarsundergoafailed supernova:Insteadofsheddingmostoftheirmassinanexplosion,aswouldbenormal forasmallerstar,theshockwaveisunabletounbindtheoutermassshells.Hence,in thismodel,mostofthemassofthestarthencollapsesdirectlyintoablackhole[33]. PopulationIIIstarscouldhavemassesinthehundredsofsolarmasses;sothisprocess couldleaddirectlytoblackholesoftensorhundredsofsolarmasses.Themodelgains plausibilityintermsofIMBHformationasmostofthesestarswouldhavelifetimes whicharemuchsmallerthantheirmoderndaycounterparts,andwouldthereforehave quicklyevolvedintostellarremnants.Formationonthisearlytimescalecouldprovide therequiredtimeforthebinarytoevolveintoatightsystemwhichwillmergewithinthe lifetimeoftheuniverse. Thesecondscenarioismoreinterestingfromthepointofviewofgravitational-wave astronomy.Aglobularclustermayhostarelativelylargeblackholewhich,through gravitationalinteractionandtheassimilationofsmallermasses,sinkstothecenterof thecluster.Thiscentralblackholewillcontinuetoaccretemassintheformofsmaller blackholesorotherdensestellarremnants[35].Ithasbeensuggestedthattheseblack holescanaccreteupto10 3 M inadditionalmassfromthecluster.Therefore,thisnot onlylendscredencetotheformationofIMBH,butalsosuggestsapossibleavenuefor theformationofbinariesandcoalescenceswithmassivepartners.Otherauthorshave modeledtheformationofIMBHwithinglobularclustersandproducedevidenceofa plausiblepopulationof100 )]TJ/F21 11.9552 Tf 12.609 0 Td [(250 M ormoreIMBHs[34].Thisscenarioalsoimplies thatthoseclusters,beinglowmetallicityenvironmentsmayhaveconnectionstotheULX sources[36,38,39]iftheclustersinteractwiththelargerhostgalaxy[35]. Sinceglobularclustersarethemostlikelyhostsforthesesources,theirratesare generallyplacedintermsofthepopulationofglobularclustersGC.ForanIMBH-IMBH 24

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binaryarateof0.007 )]TJ/F21 11.9552 Tf 12.788 0 Td [(0.07GC )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 Gyr )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 isprojected[40].Thesamereferencealso projectsaratebetween3 )]TJ/F21 11.9552 Tf 12.553 0 Td [(20GC )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 Gyr )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 forintermediatemassratioinspiralsonto IMBH.Itisimportanttonotethattheseareoptimisticprojections,sinceasfewas10% ofglobularclustersmayhavethecoredensityrequiredtogenerateasuitablepopulation ofIMBH[41].Theastrophysicalratestranslatedintodetectionratesforinitialdetectors rangebetween0.001and10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 yr )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 [40]. DetectionrateestimatesforadvanceddetectorssuchasAdvancedLIGOand space-bornedetectorssuchasLISAarereadilyavailableandincreaseinordersof magnitudeaslowerfrequenciesbecomeaccessible.Prospectsfortheirratesare discussedinSection6.2. 25

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CHAPTER2 GRAVITATIONAL-WAVEDETECTORSANDNETWORKS ThedirectdetectionofgravitationalwavesGW,rstproposedbyWeberin 1959[42],isintrinsicallycoupledtomeasuringandamplifyingtheeffectofapassing gravitationalwaveonanelasticbody.Weber'sproposalandsubsequentimplementationwaswithanelytunedsetofaluminumcylinderscalledaWeberbar.The devicetranslatestheperturbativeeffectofagravitationalwaveintoaresonating acousticwaveandusesthiseffecttomeasurechangesinlengthoftheapparatus. ModernexperimentsofthistypeincludetheresonantbarsAURIGA[43],NAUTILUS[44], EXPLORER[45],andALLEGRO[46].Inrecenttimes,resonantspheretypedetectors suchasMiniGRAIL[47]havebeenconstructed. Theseexperimentshavehadsuccessinsettingupperlimitsontheenergyin gravitationalwavesemittedfromvarioussources[48],however,theyarelimitedin frequencysensitivitybytheirnarrowresonanceband.Thislimitationhasdriventhe developmentofwide-bandinterferometricdetectors.Thenovelideaofaninterferometer asaGWdetectorwasproposedinexploredintheearly1970's[49]andfurtherrenedin the1980's[50].Whilesimilarinprincipleofdetection,interferometersaresensitivetoa muchwiderclassofgravitationalwavesignals.Groundbasedinterferometerstypically havecomparableorbettersensitivity )]TJ/F21 7.9701 Tf 6.587 0 Td [(22 Hz )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = 2 toresonantmassdetectorsina widerfrequencybandHz10kHz. Sincethebeginningofthe21 st century,multiplehighlysensitiveinterferometric gravitational-wavedetectorshavecomeintoservice.Whenoperatingintandem,they formagravitational-wavedetectornetwork.Thesenetworkshavemanyadvantagesover singledetectors.Asingledetectorisconstrainedbythelackofcorrelatinginformation aboutanevent.Atleasttwodetectorsarerequiredtodisentanglethetwoindependent polarizationstates.ThreedetectorsarerequiredtolocalizeGWtoapointinthesky 26

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viatriangulationtwowouldprovideonlyaringcorrespondingtoidenticaltime-delays betweenthedetectors. Detectornetworksarealsodesirabletomakemorecondentdetections.Noise sourcesareinverygoodgeneralizationnotcorrelatedbetweennon-colocateddetectors.Therefore,coincidencebetweendetectorsbenetsdataanalysisbyrejecting non-correlatedsignalsinthedata. 2.1InterferometricGravitationalWaveDetection Themodesofgravitationalradiation,likeelectromagneticradiation,disallowthe monopolemodeviatheconservationofmass.Unlikeitselectromagneticcounterpart, dipolemodesarealsodisallowedbecauseoftheadditionalconstraintoftheconservationofmomentum.Therefore,thedominantmodeofmostprocessesemitting gravitationalradiationwillbeinthequadrupolemode,andhencemostmoderndetectors exploitthisexplicitlyintheirdesign.ThebasicdesignofaMichelsoninterferometer representsthesimplestformofaquadrupoleantenna,towit,twoorthogonaldipole antennasjoinedattheircentersuchasinFigure2-1. TheactualconstructionofaninterferometerofthistypeisrefertoFigure2-2an inputbeamarmwhichissplitatthecenterbyabeamsplitterandsentintoorthogonal armsofthedetector,denotedhereasthe x and y arms.Lightisthenreectedbackto thebeamsplitterandeitherreectedbackintothedeviceorrecombinedatthedarkport andfedintotheopticalreadoutsystem. Theoveralleffectofapassinggravitationalwaveistoincreasethelightpath differenceofthelighttravelingineitherarm.Thisisadifferentialeffect;thepathinone armisshortenedwhiletheotherarmisincreased.Consideralocallyatmetricaround theinterferometer,suchasinEquation2. ds 2 = dx 2 + dy 2 + c 2 dt 2 27

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Figure2-1.Theeffectofthe h + and h GWpolarizationonaquadrupoleantennablack crossattheorigin.Thepluspolarizationstateleftwilltendtostretchthe north-southarmandcompresstheeast-westarmalternatively.Thecross polarizationrighthasasimilareffectbutisrotatedby45degreesrelativeto theplus.The t valuesshowninthelegendsarerelativetotheperiodofthe wave.Itshouldbenotedthatthepolarizationlabelsarearbitrary,and labeledhereoutofconvention. Referringto2-2,they-axisisalignedwiththenorth-southdirectionandthex-axis alignedwiththeeast-westdirectioninFigure2-2.Concerningtheorientationofthe GWwaveframewiththedetectorarms,themaximumofthedisturbanceisalignedwith they-axis.Overlayingtheeffectofapassinggravitationalwaveontothismetricgives Equation2. ds 2 =+ h xx dx 2 ++ h yy dy 2 + c 2 dt 2 Forlight,thegeodesicisanullgeodesicand ds 2 =0.Thereforethetrajectoryof lightineitherarmisgivenbythesetofequationsin2. dx dt = c p 1+ h xx x-arm dy dt = c q 1+ h yy y-arm 28

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Figure2-2.Asimplieddiagramofapower-recycledMichelsoninterferometerequipped withFabry-Perotcavitiesattheendtestmasses.Lightisemittedbyalaser atthewestend,splitintotheeastandnorthchannelsatthecenterandread outinthesouthphotodiode.AdaptedwithpermissionfromFabien Cavalier[51]. Iftheincomingwaveislinearlypolarizedhavingonlyonepolarization h + andalignedasstatedearlier,thenthe h + straincomponentsaresimplyrelatedtothe interferometerarmsby h xx = h and h yy = )]TJ/F24 11.9552 Tf 9.299 0 Td [(h where h istheamplitudeofthewave. Furthermore,sincethewaveamplitudeismuchlessthan1,aspositedinSection1.1 thenthesquarerootisreplacedbyitsTaylorapproximation.Thenthelighttravelpathin eitherarmisgiveninEquation2. 29

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L x = c 2 + h t L y = c 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(h t Ingeneral, L x and L y aredependentonthefrequencyoftheGW.Forthefrequenciesconsideredinthissearch,thefrequencyisnothighenoughfor L x or L y tochangein alighttraveltime,andthisdependenceisneglected. Theinterferometerisdesignedtomeasurethedifferenceinpathlength,andso Equation2givesthetotaldifferentialpathlengthbetweenthetwoarms,andnormalizedbytheunperturbedpathlength L = ct .Thisisthestrainontheinterferometer. L x )]TJ/F24 11.9552 Tf 11.955 0 Td [(L y L = h 2.2NoiseSourcesinInterferometricGWDetectors Gravitational-wavedetectorsarelimitedintheirsensitivitybyseveralfundamental sourcesofnoisepresentinthestrainmeasurement.Somenoisesourcesarestochastic sourcesofnon-GWinducedstrainontheinterferometerwhicharedominantinvarious frequencybands.Others,suchascouplingtopowerlines,havenarrowfrequency resonances.Thesesourcesarecharacterizedinthefrequencydomainbytheirstrain powerspectraldensityPSDdenedinEquation2[52]. S f = 1 p 2 ZZ h t h t )]TJ/F26 11.9552 Tf 11.955 0 Td [( exp )]TJ/F21 11.9552 Tf 9.298 0 Td [(2 f dtd The S f ofvariousnoisesourcesisdisplayedinFigure2-3.Eachnoisesourceis expressedasadifferent S f whicharethenaddedtogethertoobtainthetotalnoise spectraldensity,anexampleofwhichisshownforbothLIGOinFigure2-4andVirgoin Figure2-5. 30

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Figure2-3.VariousnoisesourcesintheinitialLIGOinterferometers.Initial interferometersarelimitedmainlybyseismicnoise,thermalnoise,andshot noise.Thetotalsumofthenoisesourcesisoutlinedinred,whichformsthe baselinesensitivityoftheinterferometer.Adaptedfrom[53]. 31

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2.2.1SeismicNoise Seismicnoiseisthelimitingfactorinstrainsensitivitybelow100Hz.Thissource formstheseismicwallanareaofrapidlydecreasingsensitivityduetohuman activity,andvariousgeologicdisturbances.Itspowerspectraldensity S f isdenedin Equation2. S f / 8 > < > : const .1Hz < f < 10Hz 1 f 2 f > 10Hz Seismicnoiseispartiallyreducedbytheuseofseismicisolationstacks:setsof spring-loadedplatformsspecicallydesignedtodampouttheeffectsofseismicsources. 2.2.2ThermalNoise Intheregionofhighestsensitivity,theoorofthenoisepowerspectraldensity isacombinationofnoisyphenomenaarisingfromthethermalmotionexertedon andexhibitedbythetestmasses.Thesemassesareplacedattheendpointsofthe interferometerarms,andserveasthefreefallingmassthepointparticleonthe geodesic.Thermalnoisecanarisefrommultiplesources,forexample: Collisionswithresidualgasmoleculesremaininginthevacuumchamber. Motionsoftheatomsonthesurfaceofthetestmassmirrorsduetothermal energy.Onalargescale,theindexofrefractionofthemirroritselfcanchangewith temperature. Thermallyinducedmechanicaldisturbancesinthesuspensionsofthetest-masses ThesesourcescanbemodeledinfullgeneralityusingtheFluctuation-Dissipation theorem[54],butthereislittledoneinrstgenerationinterferometerstoameliorate thermaleffects.Bettervacuumsystems,cryogeniccooling,andbettermirrorcoatings willplayapartinthereductionofthesenoisesourcesinlatergenerationsofdetectors. 32

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2.2.3ShotNoise BeyondafewhundredHz,thedominantsourceofnoisecomesfromshotnoise. Thissourceisunderstoodasthediscretequantaoflightreectingfromthetestmasses atirregulartimeintervals.Thisirregularityinthearrivaltimeofthephotonscauses uctuationsinthemeasurementofpathlength.Theuctuationisfrequencydependent andisexpressedasinEquation2[55]where L isthepathlength, F isthecavity nesse, f p isthepolefrequencyoftheFabry-Perotcavity, isthewavelengthofthe laser,and P bs isthelaserpoweratthebeamsplitter. S f = 1 8 FL 2 4 ~ c P bs 1+ f f p 2 !! 2.2.4RadiationPressure UponexaminationofEquation2,themostapparentlysimplewaytoreduce shotnoisewouldbetoincreasethepowerpresentatthebeamsplitter P bs .However, theeffectofincreasingtheinputlaserpower,orrecyclingthepoweralreadyinthe interferometercomesatthecostofincreasedradiationpressure.Photonsexerta pressure p rad = 2 P rad Ac onthetestmasses,sobyincreasingthepower P rad ,thepressure onthetestmassesisincreased.Thisnoisesourcedominatesatfrequencieslowerthan 50Hz. Whenconsideredtogether,radiationpressureandshotnoisecanonlybereduced sofarbeforetheuncertaintyprinciplewillnotallowanyfurtherreduction.Thislimitis calledthequantumlimit.Whenoptimizingthecontributionsfromradiationpressure andshotnoise,thelimitinEquation2[55]isthesmallestthatcanbeobtainedfrom eithersource. S f = 1 2 fL 2 8 ~ M 33

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InEquation2, L isthepathlength,and M isthemassofthetestmass.Further reductionofnoisebeyondthequantumlimithasbeenexploredinsuchsituationsas lightsqueezingorquantumnon-demolition[56]. 2.3GravitationalWaveDetectorsDuringS5/VSR1 Severalinterferometershavebeenconstructedaroundtheworldandhavebeen inoperationsincethebeginningofthe2000s.Namely,theLaserInterferometer Gravitational-WaveObservatoryLIGOwhichconsistsoftwositeswiththreeoperatinginterferometers[57],German-BritishGEO600[58],French-ItalianVIRGO[59,60], andJapaneseTAMA300[61,62].Multiplefutureinterferometersarestillintheplanning orpre-constructionstagessuchasAIGO[63,64]andtheLCGT[65,66].Asofthiswriting,theLIGOdetectorshavecompletedsixscienceruns.Thesearchpresentedlateris performedusingdatacollectedduringLIGO'sfthsciencerun,itslongesttodate. LIGO'sfthsciencerunS5occurredbetweenNovember2005andDecember 2007.Duringthistime,veinterferometricGWdetectorswereactiveandtakingdata. LIGOproducedayears'worthofsciencedatainwhichallthreedetectorswereparticipatingduringthespanoftwoyears.LIGOwasjoinedbytheVirgodetectorinMayof 2007.ThejointruncompletedinDecember2007.ThissubsetisreferredtoasVirgo ScienceRun1VSR1,andthejointruniscalledS5/VSR1.Afthinterferometer, GEO600,wasalsorunningintheS5period,butitsdataisnotconsideredsensitive enoughtosignicantlyimprovetheoverallsensitivityofasearchforGW. 2.3.1LIGO TheLIGOdetectorsduringS5includedthreebroadbandinterferometricdetectors. Therst,locatedinLivingstonParish,LouisianaandcommonlyreferredtoasL1isa4 kminterferometer.Theothertwo,locatedinHanford,Washington,arelocatedwithinthe samevacuum,wheretherstH1is4km,andthesecondH2is2km. DuringtheS5run,alldetectorsinLIGOwereequippedwithNd:YAGlasersoperatingatafrequencyof1064nm.Thetwo4kminterferometerhadaninputlaserpowerof 34

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4.5Wandthe2kminterferometeratHanfordhadaninputlaserpowerof2W.Thelight intheinterferometeristhenpassedthroughaFabry-Perotcavityplacedinfrontofthe endmirrors.Thisactsasanamplierforthelaserpowerandalsothephaseshiftwhich isintroducedbyapassingGW.Apowerrecyclingmirrorrecirculateslightthatwould normallybereectedbackintotheinputlasersystem.Therecirculationactstoincrease thelightpowerintheinterferometer.Mostoftheopticalsystemsintheinterferometer areseismicallyisolatedbymass-springisolationstackstodampenmotionduetoground movementabove10Hz[57]. DuringS5,LIGOachieveditsinitialdesignsensitivitytargetof2 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(23 Hz )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = 2 at200Hzfora4kmdetector,itsmostsensitivepointinfrequency.Thisisnotedfrom Figure2-4[67].ThetwoLIGOsitesareseparatedbyabout3000kminstraight-line distance,buttheH1/H2andL1interferometershavesimilarresponsestotheincoming gravitationalwave.Thiswillbecomeimportantlaterinthecoherentanalysiswhich dependsontheabilitytodistinguishthepolarizationstates. 2.3.2Virgo TheVirgodetector,a3kmlonginterferometerislocatedinCascina,Italy[68].The VirgointerferometerhasmanysimilaritieswiththeLIGOdetectors.Majordifferences includeasuperattenuatorsystemwhichenhancesseismicisolationusingamulti-stage pendulumandactivefeedbacksystems.However,Virgohaslessinputlaserpower[60]. Overall,Virgo'ssensitivityiscomparabletotheLIGOinterferometer.Abroadband comparisonfromFigure2-5[69]showsVirgoisdesignedtobemoresensitiveatlower frequencies,butreversesthesituationatfrequenciesabovethemostsensitivepoint. RelativetotheseparationbetweenthetwoLIGOsites,Virgohasalongerbaseline approximately8000kminbothcases.Thereisonlya27degreeanglebetweentheL1 andH1arms.TheadditionofVirgowhichhasanorientationwhichisverydifferentfrom theLIGOdetectorstoaLIGOonlynetworkallowsforsensitivitytobothpolarization 35

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Figure2-4.Strainsensitivityin1 = p HzversusfrequencyforallthreeLIGOdetectors averagedovertheentireS5period.TheBinaryInspiralRangeshowninthe legendisthedistanceatwhichanoptimallyorientedbinaryneutronstar coalescencewouldproduceaneventwithasignal-to-noiseratioof8. statesinmanyskylocationsaswellasmuchimprovedsourceskylocalizationin general. 2.3.3GEO600 GEO600isa600mlonginterferometerinHannover,Germany[58,70,71].In additiontothepowerrecyclingmirrorasintheotherinterferometers,GEOalsoutilizes asignalrecyclingmirror,positionedinthesoutharm.Thisallowsthelaserpowerwhich isnotusedinthedifferentiallengthsignaltoberecirculatedbackintotheinterferometer. TheinputlaserpowerofGEO600isabout20timeslesspowerfulthanLIGOat0.5W. However,onenovelfeatureofGEO600isitsnear-automatedlockingprocesswhich 36

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Figure2-5.Strainsensitivityin1 = p HzversusfrequencyfortheVirgodetectorin magenta.Itissuperimposedupontheotherinterferometersoperating duringthattime. allowsforaveryhighdutycycle,typicallyhigherthan80%ascomparedto50%to60% fortheotherinterferometers.ThestrainsensitivityofGEOisalsoshowninFigure2-5. 2.4DetectorNetworks Gravitational-wavedetectorscanbeformedintonetworks.Detectornetworksare characterizedbyafewfundamentalpropertiesrelatingtheirgeometrytotheresponse ofanincomingGW.Interferometers,essentiallybeingquadrupoleantennas,willhave differentresponsestoapassingGW,dependingontheorientationbetweentheGW waveframeandtheinterferometerarms.Theangularsensitivityofthedetectorsis 37

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Figure2-6.Angularsensitivityofaninterferometertogravitationalwaves.Theleft representsthesensitivity F + tothe h + polarizationstate,themiddleisthe sensitivity F tothe h polarizationstate,andtherightisaveragedover both.Adaptedfrom[53]. encodedinthedetectorantennapatterns[72] F + and F andtheirangulardependence ismadeexplicitinEquation2.AdiagrammaticalrepresentationisshowninFigure 2-6. F + = 1 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(1+cos 2 cos F =cos sin Therelativeorientationisexpressedintheanglesbetweenthegravitationalwave frameandtheinterferometerframeviatheangles and .Theangles and are thesameasthoseusedtodeneavectorinnormalsphericalcoordinates,exceptthe vectorrepresentingthedirectionofanincomingwavepointstowardstheorigin,and thecoordinateaxesarexedtoadetectoranditsarms.Thesourcelocation and is relativetothedetector'slocation,withtheanglesbeingmeasuredrelativetothearmsof theinterferometerandthe z -axisorthogonaltothem. Consideringonespecicdetectorandusingthecoordinatesystemsetupfor Equation1,theeffectofapassingGWisdetailedbythedetectorresponse h in Equation2. 38

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h t = F + h + t + F h t Equation2incorporatestheassumptionthatthedurationof h t isshortenough thattheantennapatternswillnotchangemuchduetotheEarth'srotation.Theroleof thepolarizationangle iselaboratedfurtherinSection2.4.1. Foranetworkofdetectors,therearemultipleantennapatternstwoforeach detectorwhichhavethesameformasEquation2,butdifferentorientationsrelative totheframeofthewave.EachdetectorframecanberotatedintothetheEarth'sxed frame,asetofcoordinateswhichrelateseachdetector'ssourceanglepair and to axedrightascensionanddeclinationwhichisinvariantwithregardstoanarbitrary positionontheEarth.Therefore,eachdetector'santennapatternsetwillhavethesame inputsourcelocationinrightascensionanddeclinationbutwilltakeondifferentvalues basedonthelocationofthedetector.Fromnowon, and shouldbeconsideredtobe intheEarth'sxedframe. Inordertoefcientlyhandlethemultiplegeometriesintroduced,theantennapatternsofallthedetectorsareorganizedintoavectorspacecomposedofthetwovectors F + and F .Thesetwovectors,withadimensionequaltothenumberofdetectorsinthe networkandwithdetectorsubscript k formageometricbasisforthelikelihoodanalysis, asinFigure3-4. F + = F +,1 F +, k F = F ,1 F k 2.4.1DominantPolarizationFrame Thepolarizationangle istheparameterinarotation R whichalignsthe transverseaxesplaneofthepolarizationoftheGWwiththeplaneoftheinterferometer arms.Becausetherotationrelatesthecoordinateframeofthewavetothecoordinate 39

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frameoftheinterferometer,anyrotationwhichaffectstheorientationof h + h will alsoaffectthedetectorantennapatterns F + F .Theoveralleffectleavesthedetector response h invariantto andthereforeoneisfreetochoosewhicheverangleismost convenient. TheconventionchosenhereistheDominantPolarizationFrameDPF[73].This framepicks toalwaysleavethevectors F + F orthogonaltoeachother.Thischoice willbecomeimportantwhenconsideringtheoptimalwaytodetectGWinanun-modeled senseinSection3.2. 2.4.2NetworkSensitivity Theoverallsensitivityofanetworkshouldimproveasmoredetectorsareadded. Thenetworksensitivitytothepluspolarizationisencodedin F + andthecrossin F OneconventionalmeasureoftheoverallnetworksensitivitytoaGWisthemagnitude oftheantennapatternvectorsaddedtogetherintheDPF p j F + j 2 + j F j 2 .Aswas notedinSection2,onedetectorisinsufcienttoobservethesecondpolarizationstate. However,simplyhavingtwodetectorsisnotenough.Thetwodetectorsinthenetwork musthavetheirarmsrotatedrelativetoeachother.Ifthetwodetectorsarealigned,then thereisnotenoughinformationtodiscernthetwopolarizations.TheLIGOnetworkisa nearlyalignednetworkandFigure2-7showsittobeblindtothesecondpolarizationin mostskylocations,wheretheratioofantennapatternvectors F F + andhencetheratioof polarizationsensitivitiesissmall.ThesituationimprovesgreatlyinFigure2-8whenthe Virgodetectorisaddedtothenetwork.However,forasignicantfractionofthesky,the networkhaslowsensitivitytothesecondpolarization. Thediscussionofsensitivity,uptothispoint,hasonlyconsideredtherelative geometrybetweendetectorsandtheGW.However,asoutlinedinSection2.2,detectors arelimitedintheirsensitivitybythecombinationofnoisesources.Eachdetectorhas noisevariance 2 k relatedtoitsnoisespectraldensitybytheNyquistfrequency f N by 40

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Figure2-7.Top:Magnitude p j F + j 2 + j F j 2 oftheantennapatternvectorsintheDPF fortheLivingston-Hanfordnetwork.Bottom:Ratio j F j = j F + j oftheantenna patternvectormagnitudesintheDPFfortheLivingston-Hanfordnetwork. S f = f 2 k = f N ,thenetworknoisepowerspectraldensity S net isdenedinEquation 2where f N istheNyquistfrequency. S net = f N X k )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 Therefore,fromEquation2itcanbeseenthat,if K detectorswithsimilarnoise variancesareinthenetwork,thentheoverallnetworknoisespectraldensityisreduced byafactorof p K 41

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Figure2-8.Top:Magnitude p j F + j 2 + j F j 2 oftheantennapatternvectorsintheDPF fortheLivingston-Hanford-Virgonetwork.Bottom:Ratio j F j = j F + j ofthe antennapatternvectormagnitudesintheDPFforthe Livingston-Hanford-Virgonetwork. Thedetectornoisecanbeaddedtothegeometricsensitivitybyweightingthevector componentsof F + and F by k ,thenthenetwork'ssensitivityiscompletelydescribedby twovectors f + and f asinEquation2. f + = F +,1 k F +, k k f = F ,1 k F k k 42

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CHAPTER3 SEARCHALGORITHM Thefundamentalproblemofpurelyun-modeledGWreconstructionistodetermine theresponseofadetectortoapassingGWusingonlytheinformationcontainedinone ormoredetectors'datastreamswithoutrecoursetouseofsourcespecicinformation. Atrst,methodsfordetectionofGWviaexcesspowerinsingledetectorswasused[74]. Thesemethodswouldidentifyeventsbycomparingtheirsignalenergytothenoise energyinadetector.Itisthenpossibletocombinetheeventsfromdifferentdetectors usingatimeandfrequencycoincidenceanalysis. Coherentnetworkanalysis,ontheotherhand,usesthecombinationofsimultaneousdatafrommultipledetectorstreamstoformnetworkevents.Insteadofcreating eventsinindividualdetectors,theeventisconstructedforalldetectorssimultaneously. Thisismorenaturalfromthedataanalysisperspective,becauseitisasingleGW creatingeventsinmultipledetectors,andhencetheconsistencybetweendetectors shouldbesatisedintheprocessofconstructingthenetworkevent.Itisbelievedthat usingthedirectcross-correlationbetweenmultipledetectors'datasteamsismorerobustagainstinstrumentalnoise[75].Coherentnetworkanalysisisbetterabletoreject environmentallyinducedeventsandenforcestheconsistencyofreconstructedGW events. Theincorporationofcoherentdataanalysistechniquesdidnotariseuntilmultiple highlysensitivebroadbandGWdetectorsliketheLIGOnetworkcameintooperation.TheformulationoftheproblemwasproposedbyG urselandTintoin1989[76]. Itsolvestheproblemofcombiningmultiplestreamsofdetectordatabyminimizingat ofafunctionthedataamplitudesgiventheirtime-delaysbetweenmultipledetectors.A moresolidstatisticalapproachbasedonlikelihoodanalysiswassuggestedbyFlanagan andHughes[77].Thelikelihoodapproachutilizesconsistencybetweenallthedetectors participatingthenetwork.Thisconsistencyisimposedbyrequiringthethewaveforms 43

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arereconstructedcorrespondingtothedetectorresponseoriginatingfromconsistent skylocation.Thelikelihoodanalysispresentedin[73]representstime-dependent quantitiessuchas x and inthetime-frequencyplaneasopposedtoonlythetimeor frequencydomain.Theuseofregulatorsandconstraintsonthelikelihoodanalysiswill alsobeconsidered. 3.1Time-FrequencyAnalysis MatchedltersearchesdonebytheLIGO-VirgoCollaborationinthepast[26 28]usefrequencydomaintemplatestospanthesourceparameterspace.Insteadof dependingonknownfeaturesoftheexpectedwaveforms,un-modeledorweaklymodeledGWsearchesgenerallylookforcorrelatedexcesspoweronthetime-frequency T-Fplane[74].TheT-FplaneisconstructedbyFast-FourierTransformingFFTmultipleblocksofbandlimiteddata.InthecaseofGaussiannoise,thedistributionofthe powerinanyT-Fblockshouldbe 2 distributed.Therefore,aGWsignalshouldproduce powerinoneormoreblockswithasignicantexcessthanwhatisexpectedfromthe 2 distribution. Clearly,theconstructionofT-Frepresentationscanhaveanimpactontheability toreconstructsignals,especiallyinthecasewherenomodelinformationisavailable tovalidatetheresult.Astechniqueshaveadvancedandmoredetectorsbecomeavailable,methodshavebeendevelopedtocalculateexcesspoweroveratime-frequency mappingofdetectordata.Intermsofun-modeledtransientsearches,havingmultiple resolutionsisadvantageousinallowingonetochoosetheoptimalresolutionforthe reconstructionofthesignalparametersandenergy. Short-timeFourierTransformsSTFTcanbeusedtoperformtime-frequency decomposition,however,theyhavesomedisadvantages.Thewindowingrequired toperformSTFTcanleadtoaliasingenergylossandthereisnowaytoinvertthe transform.DissatisfactionwithSTFTledtothedevelopmentofwaveletsintheuseof spectralanalysis.WaveletsusedinT-Fanalysisrespectallthesamerelationshipsas 44

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atraditionalFouriertransform,includingorthogonalityandinvertibility[78].Ageneral wavelettransformisdenedbyitsscale a andoffset b F a b = 1 p j a j Z t )]TJ/F24 11.9552 Tf 11.955 0 Td [(b a f t dt InEquation3, F a b representsthewavelettransformedfunctionof f t with wavelet a b t .Inamanneranalogoustothetransitionfromcontinuoustodiscrete Fouriertransforms,onecanalsodenethediscretewavelettransform.Inthiscase, F a b isreplacedbyasetofdiscretecoefcients c a b .ADiscreteWaveletTransform DWTisthendenedin3. w a b = X n x i a b n Withappropriatechoicesof a b correspondingtoapropertime-frequency decompositionandasumover n samples,theDWTthenactsasaSTFTbutpossessing moredesirableproperties.Waveletsareadvantageousinbeingnaturalbasisfor transientanalysis,andwiththechoiceofasuitablewaveletfamilyhavelittle energylossthroughaliasing. Thepipelineusedforthesearchdoesalmostallofitscalculationsinthewavelet domainusingaMeyerseeFigure3-1waveletbasis.Thewavelettransformused bythesearchpipelineisabinarysplittingprocess:oneapplicationofthetransform amountstoabandpasslterforfrequenciesaboveandbelowthemidpointoftheinput frequencybandwidth.Foranexampleofthisbinarydecomposition,versusadyadic transformation,seeFigure3-2.However,intheprocessofthetransform,theenergyin successivetimesamplesisaddedtogether.Hence,eachsuccessiveapplicationofthe lterincreasesthefrequencyresolutionbyafactoroftwobutdecreasesthetiming resolution.Thisisincomparisontoadyadictransformationwhichwouldsuccessively dividethelowestlayerinthesametypeofhalvingtransformation.Foracomparison betweenthetwomethods,seeFigure3-3. 45

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Figure3-1.AnillustrationofaMeyermotherwaveletfunction. Figure3-2.Thesearchpipelineusesbinarywavelettransformationtrees.Therefore,all pixelsontheT-Fmaphaveidenticalareaandshape.Thisisopposedtoa dyadictransformationwhichwouldhavedifferingpixelsshapesbetween frequencybandwidths. 46

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Figure3-3.Left:UnitvarianceGaussiandistributeddatadecomposedbyadyadic transformation.Right:Thesamedatadecomposedviaabinary decomposition. Gravitational-wavedataexistsinitiallyasadiscretetimeseries.Thetimeindexis denotedby i .AT-Fdecompositione.g.awavelettransformationisatransformation whichtakesatimeseriesandproducesaT-Fmap.Usuallythisdataispresentedasa mapofsignalenergydistributedintoT-Fpixelscorrespondingtoblocksoftimeduration andfrequencybandwidth.InthepresenceofasignalfromaGW,astraindatastream fromadetectorcanbedividedintosignal h andnoise n denotedwithtimeindex i andfrequencyindex j asinEquation3. x i j = n i j + h i j SincethePSDofthenoiseindetectordatavariesthroughdecadesoverdifferent frequencyregions,thedataiswhitenedbynormalizingtheinputdatastreambythe squarerootofitsnoisevariance i j 2 asinEquation3.Furthermore,anetwork consistsofseveraldetectors,withindexdenoted k ,andsothedenitionsofthedata vector x nowlabeled x andantennapatterns F + F stillintheDPFareextended intotheT-FplaneinEquation3. 47

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x i j = x 1 i j 1 i j x K i j K i j f i j = F +,1 1 i j F +, K K i j f i j = F ,1 1 i j F K K i j Therefore,Equation3isrewritteninthenetworkvectorformulationinEquation 3. h i j = f + i j h + + f i j h x i j = n i j + h i j 3 3.2CoherentNetworkAnalysis CoherentWaveBurstcWBusesamaximumlikelihoodestimatorMLEfordetectionandreconstructionofwaveforms[79].TheMLEchecksthedetectorresponses againsttheGWwaveforms.Thefundamentalquantityofinterestisthedetectorresponse h ,whichisameasurementoftheresponsetoaGWinadetectornetwork. InordertosolvetheproblemofconsistentreconstructionacrossallGWdetectors inanetwork,alikelihoodratiotestispositedfromthejointprobability P x t j h +, t hypothesisthataGWwithasetofparameters h + h ispresentinadetectortothe jointprobabilitythatnoGWispresent P x t j h =0.Intheseprobabilities, x t isthe detector'sdatastreamorthestrainwhichthedetectorsmeasure.Thelikelihoodratioof thetwohypothesesisformedby P x i j j h = P x i j j 0.Therefore,itispositedthatthe maximumvalueofthisratioshouldbeobtainedforthebestmatchof h + and h against theknownmeasurements x i j .This,inturn,speciesthereconstructedwaveform 48

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amplitudes h .Inthecaseofun-modeledwaveformreconstruction,thetparametersare theamplitudes h themselves,sincetheunderlyingsourceparametersarenotknown. InthecaseofGaussiannoise,theprobabilitythatadatapointamplitudeisdue tonoiseisproportionaltoexp )]TJ/F50 11.9552 Tf 9.298 0 Td [(x i j 2 ,wherethedatapointhasbeenweighted bythenoisevariance,andtheprobabilityofthesignalhypothesisisproportionalto exp )]TJ/F21 11.9552 Tf 9.298 0 Td [( x i j )]TJ/F55 11.9552 Tf 11.956 0 Td [( i j 2 L = Y i j exp )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [()]TJ/F21 11.9552 Tf 9.299 0 Td [( x i j )]TJ/F55 11.9552 Tf 11.955 0 Td [( h i j 2 = exp )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [()]TJ/F50 11.9552 Tf 9.298 0 Td [(x i j 2 Thefunctionalistintermsoftheunknownwaveformamplitudes h fortheselected region intheT-Fplane.Sincetheexponentialfunctionismonotonic,themaximumof theargumentcorrespondstothemaximumofthefunction.Therefore,thelogarithmof theratioasinEquation3ismaximizedinstead. log L = 1 2 X )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [()]TJ/F50 11.9552 Tf 9.298 0 Td [(x h )-222(j h j 2 TheT-Findicesarenowsuppressedforbrevity.Thevariationoflog L overtheGW amplitudesresultsinEquation3. h + j f + j 2 + h f f + = x f + h j f j 2 + h + f + f = x f Toeliminatethecrossterminthesecondtermonthelefthandsidethecalculation isperformedintheDPFfromSection2.4.1.Thevalueofthetotallikelihoodisindependentof andcanbeselectedsuchthattheplusandcrossantennapatternvectorsare orthogonali.e. f + f =0.ThissimpliestheformofEquation3andthesolutions foreitherpolarizationstatebecomeindependentofeachotherasinEquations3. 49

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h + = x f + j f + j 2 h = x f j f j 2 Substitutionofthemaximizedvaluesof h backintoEquation3providesthe maximizedfunctionalintermsoftheantennapatternvectorswhicharenownormalized tounity.TheseunitvectorsaregiveninEquation3 e + = f + j f + j e = f j f j L = X x e + 2 + x e 2 Equation3istheMLEforanun-modeledGWwaveform.Thelikelihoodis thenfurthermaximizedbyselectingthesourcecoordinates whichmaximizes thevalueofthelikelihood.Thismaximizationoversourcecoordinatesisperformed simultaneouslyforalldetectorsinthenetworkandisthereforeequivalenttodemanding maximumconsistencybetweenthereconstructedwaveformovertheentirenetworkby thehypothesisthatwaspositedinEquation3. Forageometricpicture,thedetectordata,representedasthevector x inFigure3-4 isdecomposedintoitsprojectionontotheantennapatternplanedenedby f + and f theexpectedGWresponse = x f + f + + x f f andtheresidualenergyfromnoise theprojectionontothe z -axis,calledthenullstream n ThecriticalportionoftheMLEhypothesisisthecorrelationandconsistencyofthe expectedwaveformenergyacrossallthedetectorsparticipatingintheanalysisnetwork. Iftheexpectedenergyineachdetectorisnotconsistentwiththesignalhypothesis, thenthedatastreamsarenotcoherent.Thelikelihoodcalculatedoverthenetworkis adirectmeasurementofthecombinedincoherentandcoherentpowerinthenetwork. 50

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Figure3-4.Thegeometricalinterpretationofthelikelihoodanalysis.Thedatavector x is projectedintothedetectorplane.Theprojectionof x ontothisplaneis andtheorthogonalprojectionis n ,thenullvector. ThelikelihoodmatrixasdenedinEquation3expressesthequadraticformof thelikelihoodasthecross-correlationofindividualdetector k l datasampleswhen projectedintothesignalplane. L kl = x k e + k e + l + e k e l x l Thetraceofthelikelihoodmatrixformstheincoherentenergy E I andthesumofthe remainingtermsformscoherentenergy E C asdenedinEquation3. 51

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E I = X Tr L E C = X X k l L kl )]TJ/F24 11.9552 Tf 11.955 0 Td [(E I Thetotalenergy E T inthedatastreamsisrelatedtothelikelihoodbyEquation 3,where N istheenergyinthenullstream,orestimatedresidualnoiseenergy. N = E T )]TJ/F24 11.9552 Tf 11.955 0 Td [(L 3.3RegulatorsandConstrainedLikelihood Oneofthedrawbacksofmodelindependentalgorithmsisthatthespanofsolution spacedoesnotalwayscontainphysicallyallowedsolutions.Anumberoftheseunphysicalsolutionscanbetracedbacktothefactthateveninthemodelindependentcase,the na velikelihoodalgorithmpresentedbeforewillnotusealloftheinformationavailable aboutthedetectordata.Forexample,inthecaseofaligneddetectorssameantenna responsesthesensitivitytoasecondpolarizationstateofGWisexactlyzerointhe DPFbecause j f j =0. Asolutiontothisproblemisknownasconstrainedlikelihood[73],whichintroduces regulators apriori constraintsontheexpecteddetectorresponse.Theconstrained likelihoodanalysisintroducestheregulatorbydeningthedirectionoftheinitialdetector responsebasedonthenetworksensitivitytothesecondpolarization.Theproblemof networksensitivityisespeciallyapparentintheLIGOonlynetwork,whosepolarization sensitivitiesaredisplayedinFigure2-7.Theregulatorinitialdirectionisdenedtobe alongtheunitvector u 0 seeFigure3-5. Theimplementationofregulatorsismostconvenientlyexpressedusingthebasis vectors u 0 and v 0 .Therstbasisvector v 0 isselectedtobeinthesignalplane,but 52

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Figure3-5.Themodiedbasisforlikelihoodanalysisofgravitationalwavesusing regulatorsandenergydisbalance.Thedatavector x isprojectedontothe signalplaneswith u 0 .Thenewdetectorresponsevector isdenedfor theminimumenergydisbalancebytherotationof u 0 into u by orthogonaltothedatavector x .Thevector u 0 isconstructedasitsorthogonalpartnerin thesignalplane.Theprojectionvectorofthedatavectorontothisbasisistheresponse vector 0 ,withitsredenitioninEquation3. 0 = x u 0 u 0 Dependingonthenumberandorientationofthedetectorsinthenetwork,various regulatorscanbeappliedusingthisbasis.Thestrictestregulatorswillforcereconstructiontoberestrictedonlytoasinglepolarization.Theselectionofregulatorcanbemade adaptivelybythealgorithmviatheNetworkResponseIndexasdescribedinSection 3.3.4. 53

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3.3.1HardRegulator Thisregulatorforcesthedirectionof u 0 toliealong f + intheDPF.Thiseffectively limitsthesearchtoasinglepolarization,anddisregardstheotherpolarizationstate. Oneexampleofwhenthisregulatorisrequiredifallthedetectorsinthenetworkare aligned,suchaswouldariseintheHanford-onlycase. 3.3.2WeakRegulator Theweakregulatormeasurestherelativesensitivityofthenetworktobothpolarizationsandemploysthehardregulatorinthecasethatthe j f j = j f + j ratioistoosmall.The directionforeithercaseisoutlinedinEquation3. u 0 8 > < > : f + if < j f j = j f + j hardreg. otherwise ConsiderFigure2-7:becausethetwoHanforddetectorsareco-located,the Livingston-HanfordLIGOdetectorsareconsideredtobeaneffectivetwodetector network.Inthiscase,thesensitivitytothesecondpolarizationstate j f j ismuch smallerthantherststateinlargepartsoftheskysuggestingthisregulator'susewitha conservativevalueof 3.3.3SoftRegulator Thevector u 0 forthesoftregulatorisdenedinEquation3. u 0 = x f + f + + x f f Thisregulatorcanbeusedinnetworkswiththreeormorenon-aligneddetectors, suchastheLivingston-Hanford-VirgoLIGO-Virgonetworkcombinations.Inthiscase, theareaoftheskywherethesensitivityratioissmalliscomparativelylessthanthe LIGOonlynetworkseeFigure2-8. 54

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3.3.4NetworkResponseIndex TheNetworkResponseIndexNRIisanindicatoroftheregulatorwhichshould beappliedinagivenskylocationwithregardstotherelativemagnitudes f + f .The NRIistypicallycalculatedforaspeciedskylocation,takingonintegervalueswhichare mappedtohowseverelytheskylocationshouldberegulated. Forthe k thdetectorthefollowingconditionsarechecked: 1. u 2 k j u j 2 > )]TJ/F26 11.9552 Tf 11.955 0 Td [( and 2. u 2 k j u j 2 + v 2 k j v j 2 > Where isapre-speciedthresholdbetweenzeroandunity.Conditionone incrementstheNRIandconditiontwodecrementsit.IftheNRIislessthanthenumber ofdetectorsinthenetworkminusone,thenthisskylocationisconsideredtobeonly marginallysensitivetothesecondpolarizationstate,andthehardregulatorisapplied. Otherwise,thesoftregulatorisused.Itiseasilynotedthatsetting =0willcausethe hardregulatortobeusedand =1causesthesoftregulatortobeusedregardlessof thevaluesof u and v 3.3.5EnergyDisbalanceConstraint Whiletheregulatorsalreadyintroducedalleviatesomeoftheproblemswiththe totallyun-modeledMLE,theyarealsoad-hocandamoredesirablesolutionisneeded. Sincethereisnooptimalruletoapplyregulators,theselectionoftheinitialprojection maynotbecorrectandhencetheuseofregulatorscancauseincorrectwaveform reconstruction.Theenergydisbalancecanbeusedasamodelindependentmethodof ndingthecorrectbasisvectortouseintheconstrainedlikelihoodanalysis. Energydisbalancearisesfromthenullenergywhichisreconstructedfromthe constrainedlikelihoodanalysis.FromFigure3-4itisclearthatthetotalenergy E T in thedatavectorcanbedividedbetweenthelikelihood L asdenedin3andthenull energy N asinEquation3.ThenullvectorcanbeeasilyfoundfromFigure3-4, 55

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Figure3-6.TheNetworkResponseIndexasafunctionofskylocationforaneventinthe four-folddetectornetwork.Theredpixelsindicatingavalueof3willhave thesoftregulatorapplied.Allotherlocationswillhavethehardregulator applied. whichistheresultantvector n = x )]TJ/F55 11.9552 Tf 12.678 0 Td [( ofnoiseinEquation3afterthelikelihood analysis. N = X j x j 2 + j j 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(2 x N = X j x j 2 )-222(j j 2 +2 j j 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( x ThethirdtermontherighthandsideofEquation3istheenergydisbalance term.Thisterminthenetworkanalysisshouldalwayssumtobezero. = X x )]TJ/F55 11.9552 Tf 11.955 0 Td [( =0 Equation3denesthenetworkenergydisbalance .Thesolutiontothe incorrectselectionoftheinitialbasisvectoristoredirectthebasisvectortobeatthe 56

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minimumoftheenergydisbalance.BymakingreferencetoFigure3-5,theenergy disbalancequadraticform isintroducedinEquation3. = X X k h 2 p k p k Theenergydisbalancevector p k isdenedwithdetectorweightingcoefcients c k = u k = u 2 k +2inEquation3. p k = c k u k x k )]TJ/F21 11.9552 Tf 11.955 0 Td [( x u u k Theenergydisbalanceconstraintisequivalenttominimizingthemagnitudeofthe energydisbalancevectorbyrotatingitinthesignalplanebytheangle seeFigure 3-5: d p p d =0 d p d p =0 Thisformcontainsthedetectorresponse h inthedetectorplane.Thedetector responseissimplytheprojectionofthedatavector x ontothevector u 0 .The g termis alsodenedforuselater. h = x u h = x u 0 cos + v 0 sin h = h 0 cos + g 0 sin g = @ h @ = )]TJ/F24 11.9552 Tf 9.298 0 Td [(h 0 sin + g 0 cos = x v Where h 0 = x u 0 and g 0 = x v 0 .Rotationinthisplanebytheangle is equivalenttodeningnewvectors u and v 57

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u = u 0 cos + v 0 sin v = )]TJ/F50 11.9552 Tf 9.298 0 Td [(u 0 sin + v 0 cos Theremainderof isdenedbythefollowingequation. p k = u k x k )]TJ/F24 11.9552 Tf 11.956 0 Td [(hu k Thegoalistominimizetheoverallenergydisbalance withrespecttotheangle d h 2 p k p k d =0 Thisisequivalenttolocatingthecoordinateframeinwhichtheprojectionof x isalongthenewvector u andtheenergydisbalanceisminimum.First,however,it isapparentthatthederivativeof p k withrespectto q k inEquation3willbe required. q k = dp k d = v k x k )]TJ/F24 11.9552 Tf 11.955 0 Td [(hu k + u k v k h + u k g Passingthederivativethroughgivesthefollowingequation,wheretheprime denotesthederivativewithrespectto 2 h @ h @ p 2 k +2 h 2 p k q k =0 Therstsolutiontothisequationisimmediatelyapparent.Itisthetrivialsolution h =0andisdiscardedasnotphysicallymeaningfulinthecurrentcontext.Equation 3isthenwritteninamoresuggestiveforminEquation3. p k gp k + hq k =0 58

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Thisequationisofthirdorderincos andanalyticallyintractable.However,the useoftheMLEimpliesthatenergydisbalancebeingnegativelycorrelatedwiththe detectorresponseshouldalreadybenearminimumifthedetectorresponseischosen nearmaximum.Hence, shouldbesmall,andthetrigonometricfunctionsarereplaced withtheirrstorderapproximants.Theapproximatedsolutionisthenfounditeratively. 3.4EllipticalConstraint Giventhatmanyoftheexpecteddetectablesignalsforgroundbasedinterferometersaretheproductofobjectsinapproximatelyperiodicmotion,itisexpectedthat gravitationalwavesfromthesesourceswillbeellipticallypolarized.Theintendedtarget ofthissearchisforasetofsourcesinthiscategory,namelybinaryblackholes.Hence, itisdesirabletoincorporateinformationaboutthesourcemodelintoanun-modeled searchtomakeitmoresensitivetothesourcewhileretainingwidecoverageofthe parameterspace.InformationaboutthepolarizationstatecanbefoldedintotheMLEvia theuseofconstraints.Thisconstraintisexpressedbyredeningthedetectorresponse h h + h h h h = f + h + f h Equation3denesthedetectornetworkresponsetoanellipticalpolarizedwave. Inthefollowingdiscussion,aquantitywithatildewillindicatethe90degreesphase shiftedversionofaquantity. 3.4.1DualStreamAnalysis Thisimplementationoftheellipticalconstraintiscalledthedual-streamapproach. Thesecondstreamisconstructedbyatime-delaylterbankwhichformsthe90degree phaseshiftedversionoftheoriginaldatastream.Inthisway,theMLEisafunctiononly 59

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ofasinglegravitationalwaveamplitude h .Thehypothesisistestedbydemandingthat itisconsistentwithitsphaseshiftedversion h .Thisreducestheun-modeledparameter spacebytwo-foldbecauseonepolarizationstateisjustaphaseshiftawayfromthe otherandaddsonlytheadditionalparametersofpolarizationangle andellipticity parameter Thedual-streamapproachisincorporatedintotheMLEbeforetheenergydisbalancestage,providingtwomeasuresofenergydisbalance,oneforeachstream.The energydisbalanceformedfromthedual-streamisusedasanadditionalconsistencycut oncandidateevents. 3.4.2MaximizedLikelihoodUndertheEllipticalConstraint Thenewdenitionsof h h and aresubstitutedintothelikelihoodfunctionalin equation3.Intheun-modeledcase,therearetwoindependentlikelihoodfunctionals, oneforeachindependentpolarizationstate.Intheellipticallyconstrainedcase,two likelihoodfunctionalscanbedened,onefortheun-phaseshifteddata L ,andonefor thephaseshifteddata L .ThesumoftheselikelihoodsisexpressedinEquation3. L + L = X i j x f + )]TJ/F26 11.9552 Tf 11.955 0 Td [( x f h + x f + + x f h )]TJ/F21 11.9552 Tf -58.147 -32.191 Td [( j f + j 2 + 2 j f + j 2 h 2 + h 2 Maximizationoverthesumofthedatasamples i j for h and h asopposedto h + and h aswasdoneinSection3.2isstraightforwardandthesolutionsarelistedin Equation3,where x + = x f + and x = x f correspondtothescalar innerproductsofthedatavectorandthecorrespondingantennapatternvector. h = x + )]TJ/F26 11.9552 Tf 11.955 0 Td [( x j f + j 2 + 2 j f + j 2 h = x + )]TJ/F26 11.9552 Tf 11.955 0 Td [( x j f + j 2 + 2 j f + j 2 60

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Oncethesolutionsfor h and h areobtainedandreplacedbackintothelikelihood ratio,allthatremainsistomaximizethelikelihoodratiointermsofthepolarization angle andtheratioparameter .Ofcourse,thelikelihoodmustbemaximizedoverall samples.Giventhecomplexityoftheexpression,afewshort-handdenitionsaremade in3.Notethatthefunctional dependenceisomittedin F and G ,becausetheyare invarianttothevalueof chosen. F = x + 2 + x 2 + x + 2 + x 2 + 2 x + x )]TJ/F21 11.9552 Tf 13.516 0 Td [( x + x U = x 2 + x 2 V = j f + j 2 G = j f + j 2 + j f + j 2 Likelihoodasafunctionof and isshowninEquation3,wherethesumis overallT-Fsamples. L 0 = X i j L i = X i j F + 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 U G + 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 V ThemaximizationofEquation3requiressolvingEquation3. X i j dU i d = X i j L 0 dV i d Theexplicitdependenceon inEquation3isshowninEquation3. )]TJ/F24 11.9552 Tf 5.48 -9.684 Td [(x 2 + + x 2 + x 2 + + x 2 sin2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(2 x + x + x + x cos2 = L 0 )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(j f + j 2 )-222(j f j 2 sin2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(2 L 0 f + f cos2 61

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Clearly,inEquation3,thesolutionto requires L 0 .Toreducethecomplexity ofthesolution,instead,therelation E T = L + N isused,withtheapproximationin Equation3thatthereconstructednoiseenergy N isequaltothenumberofsamples. L 0 = E T )]TJ/F24 11.9552 Tf 11.955 0 Td [(N Thesolutionfor isshowninEquation3. S =2 X i j x + x + x + x )]TJ/F24 11.9552 Tf 11.955 0 Td [(L 0 X i j f + f C = X i j j x + j 2 )-222(j x j 2 + j x + j 2 )-222(j x j 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(L 0 X i j j f + j 2 )-222(j f j 2 tan2 = S C = 2 P i j x + x + x + x )]TJ/F24 11.9552 Tf 11.955 0 Td [(L 0 P i j f + f P i j j x + j 2 )-222(j x j 2 + j x + j 2 )-222(j x j 2 )]TJ/F24 11.9552 Tf 11.956 0 Td [(L 0 P i j j f + j 2 )-222(j f j 2 Maximizationover ismadesimplerbythefactthatofthefourshort-handterms, only F hasdependenceon .SolvingEquation3providesthesolutionfor givenin Equation3. X i j x + x )]TJ/F21 11.9552 Tf 13.515 0 Td [( x + x + U i )]TJ/F24 11.9552 Tf 11.955 0 Td [(L 0 V i =0 = 2 P i j x + x )]TJ/F21 11.9552 Tf 13.516 0 Td [( x + x P i j x 2 + + x 2 + x 2 + + x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(L 0 P i j j f + j 2 + j f j 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(C 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(S 2 Where,inEquation3,theshort-handfor C and S havebeenreusedfrom3. 62

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CHAPTER4 SEARCHFORGRAVITATIONALWAVESFROMINTERMEDIATEMASSBLACKHOLE BINARIES 4.1S5/VSR1 InLIGO'sfthsciencerunJune2005October2007,oneyearofsciencedata wascollectedinwhichallinstrumentswererunningincoincidence.Virgo'sdesign sensitivitywasrealizedin2007,andtheVirgointerferometerjoinedtheS5runinthe sameyear,designatingthepartoftherunwhereVirgoparticipatedasVSR1. Timeinwhichadetectorwasproducingsciencequalitydataalsoknownas ScienceModeisconsideredtobelivetime.Livetimeisanysegmentofdatawhich hassufcientqualitytobeanalyzedforGWsignals.Sincethedetectorsdonotfunction synchronouslywitheachother,therearemultipledisjointsegmentsoftimeinwhich anygivensetofdetectorsisproducingsciencedata.Intermsofacoherentsearch, analyzablelivetimeofagivennetworkofdetectorsisanintersectionofsciencemode timesegmentsinwhichdatawasproducedbythedetectorsinthespeciednetwork. ThoughthealgorithmusedintheIMBHsearchiscapableofanalyzingtwo-folddetector congurationsanddidsoforearliersearches[31,81]notwo-folddetectornetworks areconsideredinthissearch.WiththefourdetectorsdescribedinSection2.3,there isonefour-foldcongurationpossible,andfourthree-foldcongurationsavailable.The combinationsarenotmutuallyexclusiveintime,however,andtimewhichanalyzedas asupersetsuchasthefour-foldnetworkissubsequently not analyzedinthesubset suchaseitherofthethree-foldcongurations. Thissearchusesonlythefour-foldnetworkalsocalledH1H2L1V1andone three-foldH1H2L1conguration.Allofthethreeunusedcongurationsalsohave arelativelysmallavailablelivetimebecauseofsignicantoverlapwiththefour-fold detectornetwork.Oftheotherthree,theVirgo-Hanfordcombinationisaneffectivetwo detectornetworkwithlowersensitivitythanthesimilarH1H2L1.Removingthesecond HanforddetectorH1V1L1leavesanalyzablelivetimebecauseofitssignicantoverlap 63

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withH1H2L1V1andhenceisnotused.TheremainingcongurationH2L1V1wouldnot beofmuchvaluebecauseofthelackofusablelivetimeanddegradedsensitivitydueto thesubstitutionofthetwicelesssensitive2kminterferometerforits4kmcounterpart. LivetimesforthetwonetworksinthesearchareindicatedinTable4-2. 4.2TheSearchPipeline TheCoherentWaveBurstorcWBpipelineisafullend-to-enddataanalysis pipelineforthereconstructionofcoherentnetworkeventsfrominterferometricdata.It iswritteninC++andlinkedagainstCERN'sROOTlibrary[82].Itsworkowbeginswith readingdataforalldetectorsaswellassimulatedinjectionsifthatmodeisengaged. Then,vetosegmentsareappliedandthewaveletltersareinitialized.Afterthis,timefrequencymapsofthedetectordataareformedandclusteredacrossdetectors,the dataiswhitenedandconditioned,andtheMLEisappliedtotheseclusters.ThoseclusterspassingtheMLEareconsideredtobecoherentevents,andfurtherdownselection isappliedtothecoherentstatisticscalculatedfromthelikelihoodalgorithm.Finally, coherenteventinformationiswrittenouttodisk. 4.2.1RunStaging GiventhesizeandsamplingfrequencyoftheS5/VSR1dataset,itwouldbe impossibletorunthepipelineonallofitsimultaneously.Additionally,becausethe segmentsofanalyzablelivetimearedisjoint,itissensibletodividethedatasetintojob segments.Thesejobsegmentsareformedfromthedivisionofthelongersciencemode segmentsintoblocksofnolongerthan600seconds.Additionally,eightsecondsonthe endofeithersideofthesegmentareusedtocontainnumericalartifactscreatedbythe dataprocessing.Wherepossible,thiseightsecondsareoverlappedintoapreviousor subsequentsegmenttominimizetheanalyzabletimelost.Thisisnotpossibleatthe beginningorendofasciencemodesegment.Moreover,sciencemodesegmentswhich are128secondsorlessarediscarded,asdataconditioningwouldnolongerbereliable onadatasegmentthisshort.Thejoblististhenanumberedlistofsegmentsmeeting 64

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theabovecriteria.Eachentryinthislistisatimeintervalwhichanindividualinstanceof thepipelinewillanalyze. 4.2.2RunningthePipeline ThepipelineconsistsoftwoC++scripts.Therstisashortscriptwhichdescribes theparametersofthesearchaswellasthelocationoftheGWstraindataandother dataproductsrequiredforthesearch.Thisparametersleiscreatedbeforeruntimeand usedeachtimethepipelineisinvokedforajob. Thesecondscriptencapsulatesthesearchpipelineitself.Thisscriptisusedfor allnetworkcongurations,asthenetworkdependentdetailsarehandledbytherst script.ThesescriptsareinvokedthroughROOTviaabashscriptwhichistheactual executable.Forexample: #/bin/bash $inputdir=/path/to/init/files/ cat${inputdir}/inputfile.$1|root-b-q-lparameters.Cnet.C Theonlyparametergiventothebashscriptisthejobnumber,thereforeinvoking thisscriptwiththeparameter1willrunthepipelineonthejobsegmentdesignatedas job1inthejoblistwiththeappropriateruntimejobparametersreadinfromtheinputle. 4.2.3Initialization,DataConditioning,Time-FrequencyDecomposition,and Clustering Afterthepipelineisinitialized,datafromdiskisread.ThoughtheGWstrainchannel inthedetectordatalesissampledat16kHz,thedataisdownsampledto4kHzto reducecomputationalcost,andinformationabovethisfrequencyisnotrelevanttothis search. Oncetheinitializationstepsarecompleted,thecoherentsearchbegins.Datafor eachdetectorinthenetworkissequentiallydecomposedthrougheightresolutionssee Table4-1withtheMeyer'swavelettransformationfromSection3.1.Followingthis,the 65

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Table4-1.Time-frequencyresolutionsatwhichcWBanalyzesGWstraindata.Each pixelontheT-Fmapisbandwidth durationinareaforagivenresolution. NotethattheT-Fproductofeachoftheseresolutionsis1/2,sincethebinary wavelettransformpreservestheareaofthepixelthroughdecomposition. PixelbandwidthHzPixeltimedurations 81/16 161/32 321/64 641/128 1281/256 2561/512 dataiswhitenedtonormalizetheenergyacrossfrequencybands,andalinearpredictor lterisappliedtoremoveperiodicinstrumentartifacts,e.g.,powerlineresonances coupledintotheinterferometer.IfatimelaganalysisisrequestedexplainedinSection 4.5thenuniquelagcongurationsforthatnetworkarecalculatedandthepropersubset areinitialized. Anexamplesetoftime-frequencymapsforindividualdetectorsisdisplayedin Figure4-1.Ateachresolution,setsofcoincidentpixelswhichpassathresholdtest betweenanycombinationoftwodetector'spixelsaremarkedasclusteredinthe network.Pixelclustersthatarecloseenoughintimeandfrequencyasdenedin theinputparametersaremergedtogether.Theclusteringisperformedoneach resolution,andclusterinformationforeachresolutionisrecorded.Ifatimelaganalysis isrequested,thedetectorsaretimelaggedrelativetoeachotherandthisprocessis repeatedforeachtimelagconguration.Additionally,theclustersareconnectedacross resolutionsinaprocesscalledsuperclustering. 4.3CoherentTriggerProduction ThelistofclustersispassedthroughtheMLEasdescribedinSection3.2.Inorder toutilizetheMLE,rst,theproperregulatormustbeapplied.TheNetworkResponse IndexNRIfromSection3.3.4iscalculatedforeachlocationinthesky.Thenthe calculationoftheMLEforthatlocationwillfollowtheNRIcondition.IntheH1H2L1 66

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Figure4-1.AnexampleoftimefrequencymapsintheH1H2L1V1networksearch.Top leftisL1,toprightisV1,bottomleftandrightareH1andH2respectively. SeeFigure4-2fortheeventisolatedinlikelihood.Thisisaveryclearglitch intheLivingstondetectortimelaggedwhichisconnectedtoadisturbance inthesamefrequencybandinVirgo.Theeventidentiedinthesemapis isolatedinFigure4-2. network,therearerelativelyfewlocationswheretheNRIindicatesanythinglesssevere thanthehardregulator,therefore,thatnetworkisforcedtothehardregulatorinalmost allskylocations. Then,theMLEisappliedtoeachcluster,andtheadditionalcoherentstatistics fromSection3.2arecalculated.Theun-modeledlikelihoodiscalculatedrst,andthen thedataisdecomposedintothedualstreamlikelihood,andtheellipticallyconstrained MLEisapplied.Clusterswhichdonotpassloosethresholdsontheselectioncriteria asdescribedinSection4.4areremovedfromtheanalysis.Thesethresholdsare weakerthanenforcedinthenalanalysis,andremoveeventswhichareverypoorly reconstructedorveryweakandthereforeareexcludedfromtheanalysis 67

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Figure4-2.Theaboveistheeventafterclusteringacrossdetectorsandthemaximized likelihoodhasbeencalculatedforeachpixel. Followingthis,allinformationabouttheeventsarewrittentodiskandthejobends. Eventswhichareloudenougharealsopassedtoanindependenteventprocessor calledtheCoherentEventDisplay[83]forvisualizationandfollowup. 4.4SelectionCriteria ThedatacollectedinS5/VSR1isnon-stationaryandpollutedwithenvironmental andinstrumentalartifacts[84,85].Inconjunctionwiththeuseofdataqualityags, additionalpost-productionselectioncriteriaareusedtodistinguishbetweenpotential gravitationalwavesignalsandtransientenvironmentalphenomenainthedata.In theIMBHsearchsimilartoothercWBsearchesseveralstatisticsareemployedfor selectionofcandidateevents. 4.4.1NetworkCorrelationCoefcient First,thenetworkcorrelationcoefcient cc isdenedinEquation4. cc = E C E N + E C 68

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Thecoherentsignalenergyisaweightedsumoftheoffdiagonaltermsofthe likelihoodquadraticformseeSection3.2fromallpairsofdetectorsinthenetwork,and theresidualnoiseenergyrepresentstheenergythatremainsafterthereconstructed signalenergyissubtractedfromthetotalenergyoftheevent. Transientenvironmentalnoisewhichisuncorrelatedacrossdetectorswilloften bereconstructedwithlargeresidualnoiseenergyascomparedtoitscoherentenergy. Whileameasureofthenullenergycouldbeusedasacriteria,itishinderedbythefact thatGWsignalscanproducesignicantnullenergy.Thenetworkcorrelationcoefcient variesbetween-1and1,wheretruegravitationalwavesshouldhaveavalueof cc near unity,andglitcheswillhavevaluesof cc signicantlylessthanunity. 4.4.2NetworkandDualStreamEnergyDisbalance Thenetworkenergydisbalance isderivedfromtheenergydisbalanceconstraint inSection3.3.5,anddenedinEquation4. = X k jh x k k i)-222(h 2 k ij characterizesbythemismatchbetweenthereconstructedenergyoftheevent andtheenergyofthedata.Aglitchinonedetectorcanleadtolargedetectorresponses relativetothedataamplitudesbeingreconstructedinotherdetectors.Forexample,the h 2 k i termwillexceedthe h x k k i termandproducelargevaluesof .Ideally,thisstatistic shouldbezero. Because isscaledbythesignalenergy x k ,itcanhavearbitrarilylargevalues forveryloudevents.Therefore,themeasurewhichisactuallyusedasacriteriaisthe normalizedenergydisbalance : = E C 69

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Thiscriteriaisanun-modeledstatistic,andcanbeusedinanyun-modeledcoherentsearch.Sincethissearchhastheaddeddual-streamellipticityconstraintfrom Section3.4,eachstreamhasitsownenergydisbalance.Therefore,inthissearch,the normalizedenergydisbalanceisreplacedbythedualstreamenergydisbalance = max E C Equation4denesthedualstreamenergydisbalancetorequirethatboththe zerophase streamandthe90degreephaseshifted streampassthethreshold. Becausethisincorporatesmodeldependentquantities,thiscriteriaismorerestrictiveto glitcheventswhichwouldsurviveinanun-modeledsearchwhilepreservingefciencyin selectionoftrueellipticallypolarizedgravitationalwaveevents. 4.4.3CoherentNetworkAmplitude TheCorrelatedNetworkAmplitudestatisticisthemaindetectionstatisticinthe S5/VSR1un-modeledsearchandcharacterizesthestrengthofthecandidateevents. = r E C cc K InEquation4, K isthenumberofdetectorsinthenetworkand E C andccarethe coherentenergyandnetworkcorrelationcoefcient,respectively.Thecoherentnetwork amplitudecanbethoughtofasanaveragecoherentSNRperdetector.Mostnoise inducedeventsarecharacterizedbysmallcoherentenergyandlargenoiseenergy,so theywilltendtohaveasmallnetworkcorrelationcoefcient.Therefore,thecoherent networkamplitudeisusedinconjunctionwiththenetworkcorrelationcoefcienttoreject varioustypesofglitchesattheeventproductionstage.Thecombinationofthetwo statisticsallowstherejectionofbothconsistentbutweakglitches,andstrongbutless consistentglitches. 70

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4.4.4PenaltyFactor Thepenaltyfactor P f isameasureoftheconsistencybetweenreconstructed detectorresponses.Thisisaselectioncriteriameanttorejectcandidateeventswhich haveunphysicallyreconstructedresponses.Suchcandidateeventshaveareconstructedsignalenergywhichisgreaterthanthetotalenergyinagivendetector,here indexedby k .ThisstatisticisdenedinEquation4. P f =min k s h x 2 k i h 2 k i Thequantities h x k i and h k i arethesummeddatasampleamplitudesandreconstructeddetectorresponsesforaselectedtime-frequencyclusterinanindividual detectork.Ifaneventisinconsistent,thedetectorwhichismaximallyinconsistentis chosen.Iftheeventisconsistent h k i < h x k i forall k ,thenthisparameterisdenedto beunity. Thiscriteriawasusedinprevioussearches.Thepenaltyfactorisusedinsomeof thetestcutsetsin5.2whereitwasfoundtohavesignicantcorrelationwiththedual streamenergydisbalance.Therefore,itisnotusedinthenalsetofcutsforthesearch. 4.5BackgroundEstimation ItispossibleforenvironmentalnoisetomimictheexpectedresponseofaGW signal.Theprobabilitythatacandidatearisesfromanon-GWoriginisitsfalsealarm probabilityFAP.Acalculationofthefalsealarmprobabilityrequiresacomparison ofaneventcandidatetoeventscausedbythebackground.Therefore,thedetection signicanceofacandidatemusttakeintoaccountthefalsealarmprobability,andis requiredforanaccuratestatementofdetection. AsampleofbackgroundeventsinwhichthereisdenitelynoGWmustbecollected todenetherateoffalsealarms.ConstructionofabackgroundsampleisnotstraightforwardbecauseitisnotpossibletoturnoffallpotentialGWsources.Ifadetector isfunctioning,thenitalwayshasthepotentialtodetectaGWsource,andthedatato 71

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beanalyzedwillcontainanunknownnumberanddistributionofpotentialGWsignals. AnymeasureofFAPusingthesearchresultswithoutsomehowremovingtheGW a priori wouldstillhavethepossibilityofincorrectlyidentifyingaGWsignalasasignal originatingfromthebackground.Thiswouldtaintthebackgroundsample.Therefore, itisimpossibletousetheresultsofthesearchtocharacterizethebackgroundagainst whichGWcanbeidentied. Moreover,anymeasureofthebackgroundusingonlythesearchdatawouldbe limitedbytheamountofdatatakenintherun.Hence,toincreasestatisticsandremove thepossibilityofGWcontaminationinthebackground,amethodanalyzingthedatawith multipletime-lagcongurationsisused. 4.5.1Time-LagBackground ControldatauntaintedbypotentialGWmustbeobtainedtocollectstatistics aboutthebackgroundpopulation.Inordertocreateabackgroundsample,thedata istransformedwithamethodwhichdestroysanypossibilitythatatrueGWsignal couldbematchedbytheMLE.Thismethodistoshiftthedetectordatastreamsintime byadurationwhichismuchlongerthanthemaximumtraveltimeofagravitational wavebetweendetectors.Whenrecombinedcoherently,noskylocationcanproducea potentialtimeshiftlongenoughforagivendetectortoreconstructaGWsignalinthe network,becausetheMLEisdesignedonlytosamplethepotentiallighttraveltimes. Thus,theMLEhasessentiallynopossibilityofreconstructingatruenetworkeventfrom aputativeGWsignal. Theadvantagesofthismethodarethatdifferenttimelagcongurationsproduce differentbackgroundpopulations,andtheycanbecombinedtoproduceamuchlarger statisticalbackgroundsample.Theprimarylimitationofthismethodisthatthenumber oftimeshiftswhichcanbedoneislimitedbythetotallengthofdatainaspecicjob. Typically,theminimumshiftcanbenoshorterthanonesecond,andthetimelags aredoneonacircularbuffer,suchthatdatawhichisshiftedbeyondtheendofthe 72

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Figure4-3.Thetopgureshowstheunshifteddetectordatasegments,theyare9 secondsinlength.Thebottomgureshowsthesegmentsafterthetimelag hasbeenapplied:H1remainsunshiftedthereferencedetector,whileL1 andV1hasbeenshiftedby2secondsand7secondsrespectively.Alsonote thewrappingbehavior,sincetheportionwhichwouldbeshiftedoffthe segmentisinsteadplacedatthebeginning. jobsegmentiswrappedaroundtothebeginningofthejobsegmentseeFigure4-3. Therefore,forapairofdetectors,thenumberoflagsthatcanbedoneisequivalentto thetotaltimeofthejobdividedbythetimelagstep. 4.5.2UniqueLags Ofcourse,thefullbackgrounddistributionwouldrequireallnon-degeneratetimelagstobeperformedforalldetectors.Thisiscomputationallyinfeasible.Amuchsmaller fractionofthissetcanbeusedtoobtainareasonablestatisticalsampling.Theselection ofthissetmustbedonecarefully,however,orcorrelationswithinthebackgroundsample canarise. Thereshouldnotbemuchcorrelationinnoisesourcesbetweendetectors,and thereforetheoccurrenceofbackgroundeventsisconsideredtobeaPoissonianprocess 73

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withameanthatisdependentoneventsstrength.IntheS5/VSR1search,itwasfound thatcertainlagcombinationsweredegenerate,andthereforewouldproduceevents withacommonsource.Asanexample,astrongglitchintheHanforddetectorcouldbe draggedthroughtheLivingstondata,reconstructedessentiallythesameeventwith fewdifferencesinanumberoftimelags.ThesemultiplicitiesofeventsintroduceanonPoissoniancomponentintothebackgrounddistribution.Obviously,thisrevealsabias inthebackgrounddistributiongenerationprocedure.Toamelioratethisbias,thesetsof lagsgeneratedbythealgorithmhavetwoimportantproperties.First,notwoindividual detectorlagsaredegenerate,andsecondly,arerandomlysampledfromtheoverallset ofavailablelags. Theendeffectofthisontheanalysisisthatatwositenetworkregardlessof thenumberofdetectorsinvolvedinthenetworksuchastheLivingston-Hanford congurationislimitedinthenumberoflagsthatareindependentratherthanthe numberoflagsthatcanbeperformedforasatisfactorystatisticalanalysis.Inthesearch presented,theH1H2L1networkhasatmost600lags.Theothernetworkhaving threesitespossessthreenon-degeneratepairswhichallowsforavastincreasein thenumberofstatisticallyindependentlags.ForcomputationalreasonstheH1H2L1V1 networkwaslimitedto1000lags. 4.5.3SearchDataQualityandVetoes Thelivetimeofthenetworkissubdividedintocategoriesbasedonthequalityof thedata.Aslightlysmallersubsetofthenetwork'slivetimewithonlysmallamounts oftimeexcisedforoverlappingwithotheraggedprocesseswithinthedetectorare consideredtobetherstcategoryofdataqualityCategory1.Thesumofthesetime segmentsisalsothelivetimeasenumeratedinSection4.1. Extensivedetectorcharacterizationstudieshavefoundthatenvironmentaleffects onthedetectorscausesignicantcorrelationsofnon-GWsignalactivityintheGW signaldatachannel.Thesecorrelationscanariseaseventsintheanalysis,whichwould 74

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Table4-2.Livetimeforthenetworkcongurationsinthesearchaftervariousdataquality agshavebeenapplied.Theobservationtimeistheamountoflivetimeafter intersectionwithdataqualitycategories. H1H2L1H1H2L1V1 Category1livetime248966006142582 Category2livetime247156545983776 Category3livetime206428165179904 beofnon-GWsignalorigin,andshouldthereforeberejected.Thereareamyriadof originsforthesedisturbances,suchasseismicactivity,misalignmentwithinoptical systems,severewindspeeds,andglitchesinthedataacquisitionsystems.Duetothe severityoftheseproblemsinthedata,thisdatashouldbeexcludedfromtheanalysis. Therefore,whenthedataisanalyzed,anothersetofsegmentsisfedintotheanalysis pipelinetobeblockedoff.ThisdataisvetoedatCategory2. IncWB,thesesegmentsareusedonlytorejectevents.AllCategory1timeisused inthedataconditioningsteps,andCategory2timeisusedtoproducealistofevents. Typically,Category2'soverlapwithCategory1ismorethan90%,sothelossoflivetime isnotconsideredtobesevere.AnoverviewoftimeafterCategory1andCategory2is inTable4-2. Alsoincludedinthepost-productionproceduresareasetofsocalledevent vetoes.Thesevetoesaretypicallysecondorsubsecondsegmentsoftimewherealoud eventwasdetectedbyoneofthedetectorcharacterizationtoolsinboththeGWsignal channelaswellasoneormorenon-GWchannels.Therefore,theevent,ifitappearsin theanalysis,anditsdurationoverlapswithoneoftheseeventvetoes,itisrejected. Othersegmentsofdatamayhavecorrelationswithnon-GWsignalchannels,but thecorrelationisnotwellunderstoodandcouldintroduceamoresignicantlossof livetimeintoananalysis.CalledCategory3,itwasdecidedthatthesesegmentsofdata shouldnotbesafetovetoeventsforthepurposeofdetectionasCategory2wouldbe butshouldbeincludedforthecalculationofupperlimits.Therefore,eventsfallinginside 75

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oneofthesesegmentsofdatawillberejectedwhencalculatingtheloudesteventupper limitasdescribedinSection4.10. 4.6Simulation WhilenotrequiredfordetectionofGWsignals,itisnecessarytohavemeasures ofsensitivitytoasourcewhencalculatingtheratedensityupperlimits.Toobtainthe sensitivityofthenetwork,simulatedwaveformsdrawnfromthesourcepopulation areinjectedintothedatastream,andthepipelineisengagedtodetectthem.From theefciencyofdetectionasafunctionofthesourceparameters,theoverallnetwork sensitivitycanbederived. Withtheuseofsimulatedinjections,theprocedureoutlinedinSection4.2is modiedslightly.InadditiontotheGWstraindatafromtheinterferometer,separately preparedsimulatedGWsignalswithrandomlyselectedparametersareaddedintothe detectordata.Thepipelinealsoisfedinformationaboutthesourceanditsinjection timestampinthedata.Eventeventsthatarefoundincoincidencewiththedataare taggedwiththeirsourceinformation,whichisthendumpedtodiskalongwiththeother eventinformation.Notimelagsaredonewithsimulateddata. 4.6.1NumericalRelativityInspiredWaveforms PrevioussearchesforBBHsystemswereperformedusingmatchedlteringwith templatesgeneratedwithpost-Newtoniananalyticaltechniques.Thesetechniques usevariouspost-NewtonianapproximationsintheStationaryPhaseApproximation[7, 8691]tothephaseevolutionofthebinarytogeneratewaveformsfromagivenlow frequencyuptothefrequencyoftheinnermoststablecircularISCOorbitallowed bygeneralrelativity.ThisisagoodapproximationforinitialLIGOandforbinarieswith lowtotalmassessuchasNS-NSorNS-BHbinariesbecausethefrequenciesofthe mergerandring-downaremostlyindistinguishableintheshotnoise.Hence,themajority ofSNRcomesfromtherelativelylonginspiralportionofthesignalwhichispresent inthemostsensitivefrequencyregionoftheLIGOandVirgodetectors.However,the 76

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importanceofthemergerandring-downbecomesgreaterwhenhighermasssystems areconsidered,becausetheinspiralfallsfurtherbelowtheseismicnoise,andthe mergerandring-downprovideincreasingcontributionstothetotalSNRastheymove intomoresensitivepartsofthenetwork'sbandwidth. Recently,parameterizedfullIMRwaveformshavebecomeavailableforanalysisof GWdata[92].Thisisduetothesuccessesofnumericalrelativityinaccuratelyevolving thebinaryblackholesystemthroughitsmergerandsubsequentring-down[9395]. TheresultsofnumericalrelativityBBHsimulationhavebeenutilizedintheproductionofanalysistemplatesforGWsearches[96,97].Thesimulationsarestillcomputationallyexpensive.Thelong,analyticalinspiralportionsaregeneratedviathe post-Newtoniantechniquesdescribedbeforeandtheseinspiralsaresmoothlyconnectedtothemergerandring-downportions.Procedurestosystematicallygenerate waveformshavebeendevelopedandnowareabletocoveralargerangeoftheBBH parameterspace. 4.6.1.1EffectiveOneBodyNumericalRelativitywaveforms OnefamilyofnumericalrelativityinspiredwaveformsisformulatedintheEffective OneBodyEOBapproach[98,99].TheBBHsystemissimulatedbyevolvingthe EOBHamiltonian,andextractingtheGWwaveformsinthefar-eldlimit.TheEOB HamiltonianisabletoemulatethewaveformevolutionuptotheplungeofthetwoBH justbeforetheymerge.Theplunge-mergerportionofthewaveformisthenmatched smoothlytothemerger-ring-downquasi-normalmodestocompletetheevolution.The plungeeffectsareeffectedbyanadjustablepseudo4PNtermintheEOBdynamics. Thevalueofthe4PNtermhasbeencalibratedwiththeresultsofnumericalrelativity[93, 95,100]. Theinitialwaveformfamilycalled EOBNR [101104]isusedtogenerate waveformsforthedetectionMonteCarlosimulationsdescribedin4.6.2.Evenhigher ordertermshavebeenintroducedandcalibratedagainstnumericalrelativityresultsin 77

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newerversionsofthefamily.Acomparisonwiththeresultsobtainedfromnumerical relativityisbecomingincreasinglyfaithful[105].Thisnewversioniscalledthe EOBNRv2 family,andisusedtovalidatetheresultsofthe EOBNR familyinSection4.9.2. 4.6.1.2IMRPhenomfamily The IMRPhenom waveformfamilyisaparameterized,phenomenologicalbank ofBBHwaveforms[106,107].ItisconstructedintheFourierdomainbysmoothly connectingthethreeevolutionstagesasoutlinedinEquation4. h f / f = f merg )]TJ/F21 7.9701 Tf 6.586 0 Td [(7 = 6 inspiralstage h f / f = f merg )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 = 3 mergerstage h f / w 4 2 f )]TJ/F24 11.9552 Tf 11.955 0 Td [(f ring 2 + 2 = 4 ring-downstage WhereinEquation4, h f istheFourierdomainstrainamplitude, f merg f ring are themergerandring-downfrequency,respectively,and isthewidthoftheLorentzian. Thenormalization w inthering-downstageischosensuchastosmoothlytransition betweenmergerandring-downe.g. w = f ring = f merge )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 = 3 .Theinitialformulationis calledthe IMRPhenomA family. TheinclusionofspineffectshasalsobeenintroducedforsystemswithBHspins whicharealignedwiththeangularmomentumofthesystem[108].Thisupdatedfamily alsoincludesothercorrections,suchasbettermatchingtothepointmassapproximation tobeusedwithhighlyunequalmassratiosystems.Thoughnotexplicitlyincludedinthe resultsofthissearch,theeffectofspinisexploredinSection6.1.Thisfamilyisreferred toasthe IMRPhenomB family. 4.6.2EfciencyEstimation Simulatedsignaldetectionefciencyisperformedtoestimatethesensitivityof thesearchtothetargetsignalfamily.Ameasureofthesensitivityofthenetworkis 78

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encodedinthevisiblevolume.Theratedensityupperlimitplacesaone-sidedstatistical condenceregionontherateofeventsoccurringinthesensitivevolume. ThevisiblevolumemustbeestimatedviaMonteCarlosimulations.Inpractice, thisisperformedbyselectingawaveforminthefamilyandinjectingitcoherentlyintoa randompointintimeinthedataforalldetectorsinthenetworkunderconsideration.The searchpipelinethenattemptstodetectit.Theresultsofthesetrialsarecompiledintoa tableofdetectionefciency. Thissearchusessimulatedsignalsoverabroadparameterspacetobestcapture theentiredomainofthetargetsignalfamilyinthesetofintrinsicparameterssuchas componentmass.Additionally,extrinsicparameters,suchasskyposition,inclination, anddistancetothesourcemustbeadequatelysampledinordertohaveareasonable estimatewhichemulateswhatmightexistinnature.Thedistributionofinjections areuniformininclinationangle,rightascension,andthecosineofthedeclination angle.However,theresponseofthenetworkcanvarygreatlyfromoneextremeofthe componentmassspacetotheother,andthereforetheradialdistributionofsourcesin spacemustalsobehandledcarefully.Thesamplingstrategymustefciencycapturethe entirerangeofparameterswithareasonablenumberoftrials. 4.7InjectionStrategy Sincethereisalargerangeofpotentialparameterstosample,manyinjectionsmust bedonetoadequatelycovertheparameterspaceandobtainareasonablestatistical experimentalerror.Inasinglerunoverthedata,thereisapracticallimitonthenumber ofinjectionsthatcanbedone.Forinstance,thetotallivetimeavailabletothequadruple networkLivingston-Hanford-Virgoisabout62days.Theminimumsafespacingof injectionsapartfromeachothershouldbenolessthanaminute,toavoidpossibly overlappinginjections.Therefore,withonlythissetofdata,only90,000injectionscan bemade.Toincreasethisnumber,multiplerunsoverthesamedatasetareperformed. 79

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Figure4-4.Injectiondistributioninthecomponentmassplane.Injectionsaremade betweenamassratioof1:1and1:100,butonlyupto1:4wasuseddueto uncertaintiesinthewaveformfamiliesseeSection4.9.2. 4.7.1MassDistribution ConcerningtheaccretingmodelSection1.3offormation,thereisreasonto believethatthedistributionofIMBHwoulddecreaseinnumberasthemassofthe IMBHincreases,sincetheaccretionprocesswouldtransformlargepopulationsof smallerBHsintoasmallernumberoflargeIMBHs.However,withoutasolidtheoretical basistoconstructaplausiblemassdistributionforthisrangeofblackholes,thechoice madewastouniformlypopulatethecomponentmassplane.Thisapproachisnearly identicaltothesearchforBBHintherangeof25to100 M [30].Thedistributionof theseinjectionsisshowninFigure4-4. Itshouldbenotedthatwhileinjectionswithmassratio q upto q =1:100were made,inabilitytovalidatewaveformswithmassratioslargerthan q =1:4arenotused inthecalculationofsensitivityseeSection4.9.2. 80

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4.7.2RadialPositionDistribution Itisknownthatthehorizondistancedetectionofanoptimallyorientedsystem withasingledetectorSNRof8foraring-downfroma200 M systemisalmost500 Mpc[109].ThisdistancewascalculatedagainsttheLIGO4kmStandardReference DesignSRD[110]whichwasachievedinS5/VSR1.WiththeadditionalSNRfromthe mergerstageandmultipledetectorsinthenetwork,itisconceivabletodetectoptimally orientedbutmarginallypowerfulwaveformswithallthreestagesofthecoalescenceout to1Gpcormore. However,somecaremustbegiventothestatisticaluncertaintyonthemeasurementofvolume.UncertaintyarisingfromthenitesamplingdoneinMonteCarlo simulationswillbedominatedbytheregionswheretheefciencyindistancetransitions fromnearunitytonearzeroseeSection4.9.Itisthereforeimportanttopopulatethis areawithasufcientnumberofinjections,toreducetheoveralluncertaintyfromthe simulations.Therefore,inordertoensurecompletecoverageindistanceandobtainthe mostaccurateefciencyestimatepossible,anominalducialsphericalvolumewitha radiusof2Gpcwaschosenastheinitialinjectionvolume.Toincreasethenumberof injectionsandfurtherreducestatisticaluncertainty,thisvolumewasthendividedinto eleventhickshellswhoseinnerandouterradiiarelogarithmicallyspaced.Elevensimulationrunspopulatedelevenshells.Theinjectiondensitydistributionandnumberisthe samebetweenallshells.Eachshellisindependentoftheother,andthereforeincreases thenumberofinjectionsbyeleven-fold. However,theinitialdistanceestimateaboveisthemostoptimalcase,andasthe totalmassofasystemincreases,themergerandring-downfrequenciesmoveoutofthe sensitiveregionoftheLIGOandVirgostrainnoisespectrum.Thus,thesesystemshave decreasingSNRasoneapproachestherapidlyincreasingpartoftheseismicnoise spectrumreferagaintoFigure2-4.Additionally,themassratioofthesystemaffects 81

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Table4-3.Overviewoftheparameterdistributionusedinthesearch. ParameterRange BinarysourceinclinationUniformin[0, ] Binarysourcedistance[0,2000 = R ]Mpc Skyposition Uniformin[ )]TJ/F27 7.9701 Tf 10.494 4.707 Td [( 2 2 ]declination Uniformin[0,2 ]rightascension PolarizationUniformin[0,2 ] Totalmass m 1 + m 2 100to450 M Massratio q 1:1to1:100injected1:4used thefrequencycontentandSNRofasignal.Forsystemswiththesametotalmassbut massratiosawayfrom1:1willdecreaseinSNR. Ana veapproachwhichinjectedanarbitrarymasspairwithalinearlyincreasing injectiondensityintoa2Gpcspherewouldplacemostinjectionsoutsideofthesensitive volume.Whilethisvolumeisappropriatetoanequalmasssystemwithtotalmass 200 M whichisoptimallyoriented,itwouldnotbeappropriateforthesamesystem withamassratioof1:5becausetheSNRdecreaseswithhighermassratio.Therefore, allinjectionshavetheirdistance r andinversely,theiramplituderescaledbyan ad-hocruleusingtheirtotalmass M T andsymmetricmassratio m 1 m 2 = M 2 T whenthey areaddedintothedata,asgiveninEquation4. R = M 3 T 400 m 1 m 2 Wheretheactualdistancetotheinjectedsourceis r = R ,and400isanormalization factorwhichcarriesunitsof M inordertokeepthescalingfactorunitless.Therefore, anequalmassbinaryof100 M willnothaveitsdistancerescaled,buta450M binary withamassratioof1:4willnotbeinjectedbeyond160Mpc.Thisrescalingservesto limitthevolumeforinjectionsthatdonothaveasmuchpowerinthesensitiveband:see Figure4-5forthecutoffinthe m 1 : m 2 plane. 82

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Figure4-5.Injectedducialradius r max = R overthecomponentmassplaneinthe S5/VSR1IMBHsearch. Figure4-6.Distributionofinjections'radiusbeforeredandaftergreenrescaling. Beforerescaling,theinjectionsaremadeintoconcentricthickshells,hence thedistinctiveregions. 83

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4.8VisibleVolume Thevisiblevolumeofasearchisanestimateofthevolumeofspacetowhichthe searchissensitivetodetection.Itisdened,ingeneral,byEquation4. V vis = Z V inj r dV = N det N inj InEquation4, isthedetectionefciencyofanetworkatapoint r in theducialvolume V inj ,ortheratiodetectedandinjectedevents.Expandingoutthe differentialelement,thevisiblevolumeformulainEquation4whichissimilarto[30] isobtained.Angulardependenceisimplicitlyaveragedoverwhendistributinginjections inskylocations,andso r becomesonlyafunctionof r 0 r V vis =4 Z R inj 0 0 r r 2 dr Nowthe V inj isreplacedwiththeducialradius R inj .Withoutanexplicitformof 0 ,or understandingoftheinjectiondistribution,thisintegralcannotbeevaluatedanyfurther. Therefore,afurthersimplifyingargumentismade.Sincethedensityofthedistribution ofinjectionscanbeconstructedinadvance,thefollowinginjectionprocedureisused. Ifoneinjectionisdoneintotheinjectedvolume,themeasurementofvisiblevolumeis either V inj ifdetectedandzeroifnot.ThisisaBernoullitrialfortheexperiment,which canberepeated.Thedistributionofinjectedversusdetectedeventsisthendrawnfrom abinomialdistributionwithaprobabilityofdetectiongivenby 0 Thevisiblevolumeisdened,inthiscase,byEquation4wherethesumisover thenumberofdetectedevents N det V vis = N det X i V inj N inj However,itisrecognizedthatthesummandinEquation4istheinjectednumber density.Inthesummationofoverdetectedevents,thedensityneednotbeuniformin 84

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volume,andthereforewilldependonthepositionoftheinjectioninthevolume.Hence, thesummationisredenedtobeoneoverthedensityatthelocationoftheinjection,as showinEquation4. V vis = N det X i 1 i Therescalingoftheducialvolumemeansonefurtheramendmentmustbemade tothedenitionof toaccountfortherescalingby R ,suchasinEquation4. 1 i = 4 r 2 i d N inj = dr Where d N dr representstherescaledradialinjectionnumberdensity.Forexample,in thecaseofuniforminathree-dimensionalvolume,thenormalizationinEquation4 providestheexactformoftheintegralovertheinjectiondensity,whichisreplacedin Equation4. N inj = Z R = R 0 r dr = 1 3 R inj R 3 V vis = V inj N inj N det X i 400 m 1 m 2 M 3 T 3 i InthecaseoftheinjectiondistributionfortheIMBHsearch,aninjectiondensity whichwaslinearlyincreasingindistancewasused,andthusthevisiblevolumecalculationisdifferent.ThisformulaisexpressedinEquation4. V vis = 2 R 2 inj N inj N det X i 400 m 1 m 2 M 3 T 2 i r i TheeffectiverangeisdenedinEquation4fromthevisiblevolumeascalculatedinSection4.8.Itistheradiusofthevisiblevolume. 85

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R eff = 3 4 V vis 1 = 3 4.9UncertaintiesandEstimationofError BecauseeachoftheBernoullitrialsisindependent,thentheuncertaintyonvisible volumeissimplythequadraturesumoftheuncertaintiesoneachtrial.Beginningwith thedenitioninEquation4,thentheuncertaintyonthetotalvisiblevolumeisgiven byEquation4 2 V = i 2 AstheestimationofefciencyisessentiallyasumofBernoullitrials,theestimated varianceisthatofthebinomialdistribution.Specifying asinEquation4 2 V = N inj 0 i )]TJ/F26 11.9552 Tf 11.955 0 Td [( 0 i 2 i Thetermproportionalto 0 2 canonlydetractfromtheerrorestimation,andtherefore droppingitmakesthecalculationconservative.Onceagain,areplacementismadewith 0 N inj = N det .ThisisagainrecognizedasasumoverdetectedeventsasinEquation 4. 2 V = X i 1 2 i Dividingbythevisiblevolumegivesusthefractionaluncertainty. V V vis = q P i 1 2 i P i 1 i 86

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Table4-4.FractionalerrorontheamplitudeofthecalibratedGWstraininthe40Hz 2kHzbandwidth. Detector1 errorontheamplitude H110.4% H210.1% L114.4% V16.1% Asitiswritten,Equation4isgeneralandmakesnoreferencetotheactual distributionoflocaldensity.Inthecaseofadistributionwhichisuniforminathreedimensionalvolume, i = ,aconstant,andcanberemovedfromthesuminthe numerator.Inthiscase,theuncertaintyisgivenby: V V vis = s 1 V vis = 1 p N det 4.9.1CalibrationUncertainties InS5/VSR1,thegravitationalwavestrainwascalculatedfromachannelwhichmeasuresthedifferentiallengthbetweenthearms.Thetransferfunctionamathematical modelofreconstructedstrainisusedtocalculatetheGWstrainfromthischannel. Whilethisintroducesaconvenienceforanalysts,italsocanintroduceahigherdegree ofuncertaintyinameasurementbecausethetransferfunctionhasitsownassociated errors[111].TheseerrorsarepropagatedtotheGWstraindatainanestimatederroron boththeamplitudeandtimingofthesamplesfromtheGWstraindata.InS5/VSR1,the calibrationuncertaintiesfortheLIGO[112]andVirgo[113]instrumentsarelistedinTable 4-4. TheamplitudeoftheGWscalesinverselywiththedistanceseeEquation1 so,thefractionalerrorontheamplitudecanbetranslatedintoafractionalerroron theeffectiverangeonadetector-by-detectorbasis.However,becauseeachdetected injectionisreconstructedfromacoherentcombinationofdetectors,itisnotstraightforwardtoassignadetector-by-detectorstatisticalerroronthevisiblevolumedueto 87

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thecalibrationerror.IntheS5/VSR1all-skysearch,theeffectofcalibrationuncertainty wasincludedbycalculatingtheeffectofthoseuncertaintiesontheoverallefciency. Theuncertaintieswerecombinedbycalculatingtheroot-square-sumamplitudeSNR deviationandassumingtheerrorsareindependent.Thisassumptionhastheeffectof makingtheestimateconservative,astheamplitudeerrorsarejustaslikelytoincrease theamplitudeastheyaretodecreaseit.Thisdeviationwasfoundtoshifttheefciency oftheS5/VSR1allskysearchlowerby11.1%[81]in h rss theroot-square-sumofthe strainonagiveninterferometer.Giventhatthecoherentnetworkamplitudeisdirectly proportionaltothetotalstrainonadetectorbyaGW,againstwhichtheefciencyof thesearchwascalculated,theeffectofcalibrationuncertaintyisintroducedbyshifting thecoherentnetworkamplitudethresholdofthesearchupwardby11.1%andallowing thiseffecttopropagatethroughtheremainderoftheanalysise.g.loweringefciencyat largedistancesandhencethevisiblevolume.Theeffectoftimingorphaseuncertainty isnotconsideredinthissearch. 4.9.2WaveformFamilyBiases Thesearchpresentedhere,aswellastheS5highmasssearchusedolderversion ofthe EOBNR familytodoMonteCarlosimulations.Later,improvedfamiliesbecame available.Thesefamiliesarebettertunedtotheresultsofnumericalrelativity.The maindifferencefoundinexaminationofthenewfamilywasasignicantreductionin theenergyreleasedinthemergerresultinginthelowerSNRofdetectedeventsand thereforeasmallervisiblevolume.Figure4-7showsthedifferenceinSNRbetween theold EOBNR andnew EOBNRv2 familyrelativetowaveformsdrawnfromnumerical relativitysimulations.Ascanbeseen,thenewerfamilyismuchmorefaithfultoitsNR counterparts,whiletheoldfamilyisbiasedtowardshigherSNRforthemergersignal. Formostsystemsconsideredinthissearch,onlythemergersignalisvisible. Hence,thebiaswillhaveasignicanteffectontheestimationofsensitivevolumesand consequentlybiastheupperlimit.Inordertoaccountforthisbias,thedistancesof 88

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Figure4-7.ThedeviationinSNRversustotalmassforvariousmassratiosisshownfor boththefamilyusedinthesearch EOBNR andalsofortheimproved versionofthefamily EOBNRv2 .SpecialthankstoYiPanforprovidingthe datapointsusedinthisgure. detectedwaveformshavebeenscaleddownbytheratioofSNRsbetween EOBNR and NR.Figure4-8showsthedifferenceineffectiverangetheradiusofthesphereofvisible volumeofthebiascorrectedEOBNRresultversusthenon-correctedresult. Dataforthesebiasesisavailableonlyforaselectnumberofwaveformswith massrationogreaterthan q =1:6.Giventheinabilitytoreliablycheckthevalidityof waveformsbeyondthisregime,volumesandratesareonlycalculatedformassratios notexceeding q =1:4.Whilesimulationsweredonewithupto q =1:100,thiscutoffisnot expectedtohavealargeeffectonthetotalvisiblevolumesincethevisiblevolumeinthis regionofthemassparameterspacedecreasesrapidly. 4.10CombinedSearchUpperLimits ItisdesirabletocombinetheresultsoftheH1H2L1V1andH1H2L1searchesinto asinglemeasurement.Thismeasurementcouldthenbeusedtodetermineanupper 89

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Figure4-8.PercentdifferenceintheuncorrectedEOBNRresultrelativetothebias correctedEOBNRresultoverthecomponentmassplane. limitoncoalescencesderivedfrombothnetworks.Previoussearchesbothun-modeled andtemplatebasedhaveusedeitheraxeddetectionthreshold[81,114]ortheloudest eventstatistic[115].Bothofthesemethodsinvolvemeasuringthesignicanceofa candidatebycomparingitwiththesearchbackgrounddistribution. Nostraightforwardprocesshasbeenusedtocombinetheresultsofmultiple searchesforthesamesource,eveninthecaseofthesamepipeline.Sincetheranking statisticsofsearchesarecalculatedfordifferentnetworksonanindividualbasis,itis impossibletocomparethemdirectlyusingthecoherentnetworkamplitude.Itispossible tocomparethemusingthefalsealarmrate,butthistypeofcomparisonwouldnottake intoaccountthedifferentsensitivitiesofdifferentnetworks.Hence,itisdesirableto formulateawaytodeterminethesignicanceofcandidateeventsusingthecombined search.Thisapproachshoulduseboththebackgrounddistributionaswellasthe sensitivitytothesourcemodeloftheindividualsearchnetworks. 4.10.1FalseAlarmRateStatistics ThefalsealarmrateFARcanbeusedtodeterminethesignicanceofaevent relativetothebackgrounddistributionofasearch.TheFARiscalculatedbysumming 90

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thenumberof orderedbackgroundeventsanddividingbythebackgroundlivetime ofthesearch.ThisprocedurebuildstheFARversusrankingstatistic,asshownby Equation4. FAR = N T InEquation4 N isthenumberofeventswithrankingstatistic i greaterthan orequalto ,and T isthecombinedobservationtimeofalltimelaganalysesafterdata qualityhasbeenapplied.Bydenition,theFARisdeneduptotheloudesteventinthe setwhichwillhaveaFARof1 = T Forasinglenetworksearch,theeacheventwillbeassignedtheFARcorresponding totheir statistic.ToestimatethesignicancefromFAR,itisrstnotedthattheprocess whichcreatesthebackgroundeventsisaPoissonianprocess,andtherefore,the estimatedmeanofthisprocessisgivenbytheinverseoftheFAR.Thenumberof expectedeventsaboveathreshold fromthisprocessis = T obs FAR .Thefalse alarmprobabilityFAPofthebackgroundproducing N sucheventswithstrengthnot lessthanrankingstatistic intheforegroundsamplecharacterizedby T obs isgivenby Equation4. FAP n =1 )]TJ/F24 7.9701 Tf 11.955 14.944 Td [(N )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 X n =0 n n exp )]TJ/F26 11.9552 Tf 9.298 0 Td [( Ifnoeventsaresignicantenoughtoclaimdetection,thentheloudestevent statisticcanbeusedtoderivea90%condenceupperlimitontherateofthetargeted source[116,117]asinEquation4. R 90% = 2.303 T obs Where,inthisequation,theefciency isevaluatedfromnumberofrecovered injectionswith atleastasgreatastheloudestevent intheforegroundassumed 91

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tonotbeadetectionand T obs istheindividualsearchobservationtimeafteralldata qualityisconsidered. 4.10.2FalseAlarmDensityandCombinedSearchStatistics WhilethemethodinSection4.10.1canbeappliedtomultiplesearcheswithvery differentrankingstatistics,itoverlookstherelativesensitivitiesofdifferentsearches. Thisproblemhasbeenaddressedbefore[118].However,thissearchwillemployamore straightforwardapproach. Inordertorankaneventinthismethod,twopiecesofinformationarerequired:a senseofhownoisyadetectornetworkisi.e.howoftenasearchnetworkproduces backgroundeventsofacertainstrengthandthesensitivityofasearchtothesources asafunctionofinjectionstrength.Therefore,thefalsealarmdensityFADcombines theFARameasureofthefrequencyofloudeventsofaeventwithitsvisiblevolume ameasureofthesensitivityofthesearchnetworktothesourceasafunctionofthe rankingstatistic.CandidateswithlargerankingstatisticwillhaveasmallFAR,buta relativelylowvalueofvisiblevolumeaseventswhicharefartherawayareexpectedto haverelativelyweakdetectionstatistics.ItisexpectedthatarealGWsignalismuch morelikelytobeproducedfarawaybecausethevisiblevolumeincreaseslikeacubic functionindistanceandhavealowervaluetherankingstatistic.Hence,astrong candidatecouldberuledlessprobablerelativetomultipleweakerevents. TheFARofaeventiscalculatedviathesingledetectormethodinEquation4 23.Thisiscombinedwiththevisiblevolumeofthesearchatthethresholdsetbythe loudness ofthecandidateeventasshowninEquation4. FAD= FAR V vis 92

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Figure4-9.TheplotaboveshowstheFADascalculatedfromEquation4.Forlarge valuesoftherankingstatistic,theFADisnolongermonotonicwiththe rankingstatistic. Unfortunately,thisstatisticdoesnotprovideamonotoniccorrespondencebetween theFADandtheinputrankingstatistic ,ascanbeseeninFigure4-9.Thiscorrespondenceisrequiredtocorrectlysetthethresholdsonvisiblevolumeviatheloudestevent statistic. Giventhatthereisnostraightforwardwaytodealwiththereversemapping, analternativebutmoreconservativestatisticisproposedinEquation46.This statisticproducesadistributionwhichisguaranteedtobemonotonicwith becausethe summedvolumeisuniquein FAD = 1 T bkg X i > 1 V i 93

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InEquation4,thesumisoveralleventswithvolume V i whichhaveranking statistic i greaterthanthethreshold T bkg isthetotalcombinedlivetimeinthetimeshiftanalysiswithnon-zerolagsseeSection4.5.TheFADisconstructedforeach searchnetworkwiththerankingsdonerelativetotheindividualsearches. Theoverallsearchproductivitytheestimateofthedensityofbackgroundevents inthevisiblevolumeiscalculatedbyEquation4. = X k T obs, k V vis FAD Where,again,inEquation4,thevolume V vis FADiscalculatedfromthe injectionvolumedistributionand T obs isthetotalnon-laggedobservationtimefornetwork k .TheFADstatisticisexpectedtoobeyPoissoncountingstatisticswiththeexpected meanbeingdeterminedbythecombinationwithproductivity.Hence,thefalsealarm probabilityofanyeventascalculatedfromtheoverallcombinedsearchisgivenin Equation4,withthePoissonmean = FAD. Thethresholds k usedtocalculatethenalvisiblevolumeswillbedeterminedby theloudesteventinFADdenoted d FADfromtheforeground.Hence,theratedensity upperlimit,asderivedfromtheloudesteventstatisticisthencalculatedinEquation 4. R 90% = 2.303 d FAD 94

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CHAPTER5 RESULTSOFTHES5/VSR1IMBHBINARYSEARCH Uponcompletionofthesearchtuningstudies,thepost-productionthresholds werefrozen.Thebackgroundsetshadpost-productioncriteriaappliedaswellasdata qualityvetoestoformthenalbackgrounddistribution.Finally,allcutsanddataquality wereappliedtotheforegroundset.Theforegroundofthetwosearchnetworkswere examinedforloudoutliers.Thesignicanceofallforegroundeventswasexamined throughboththeirindividualnetworkfalsealarmprobabilities,aswellasthecombined searchfalsealarmprobabilities.Noeventwasconsideredsignicantenoughtobe consideredaneventofgravitationalwaveorigin.Hence,theeventwiththesmallest FADwasusedtoxthethresholdsontherankingstatisticforthedifferentnetworks, andthesevalue,alongwiththeotherselectedpost-productioncriteriawereappliedto theMonteCarloinjectionrunstodetermineanalvisiblevolumeforthenetworks.In theabsenceofadetection,theresultsofthesesimulationsareusedtoderiveanupper limitontherateofIMBHbinarycoalescences. Disclaimer : Theresultspresentedhere donotrepresentthescienticopinionoftheLIGOorVirgoScienticCollaboration,nor havetheresultsbeenreviewedbyeitherentity. 5.1ApplicationofCategory2EventVetoes Afterthedataisprocessedwiththepipeline,eventvetoesareappliedtothe resultingevents.Theeventvetoesservetoeliminateloudeventswhichcancondently becorrelatedtoterrestrialorigins.AsdescribedinSection4.5.3,eventvetoesdescribe shortperiodsoftimewhereanenvironmentaleffectcausedaloudeventinadetector. Theseloudsingledetectoreventsthenshowupinthebackgroundsamplebyhaving multiple,nearlyidentical,loudnetworkeventsinasinglesegmentoftime,causinga towerofeventsincoherentnetworkamplitudearoundthesametime.Iftheevent overlapswithanenvironmentaldisturbance,thentheeventisrejected. 95

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Figure5-1.Thecoherentnetworkamplitudeoftheeventsintimebeforeandaftercuts andCategory2vetoesareappliedfortheH1H2L1V1network.Theeffectof thevetoesistoremoveloudeventswhichareknowntobecausedby environmentaleffects,thiscanbeseenbytheremovalofmostofthe towersofeventsintime.EventsinredhavenocutsorCategory2data qualityvetoesapplied.Eventsingreenhavethecutsapplied,butno Category2vetoes.EventsinbluesurviveallcutsandCategory2data qualityvetoes. Theeventvetosegmentsaredenedonadetectorbydetectorbasis.Therefore, ifthedurationofaeventinanydetectoroverlapswithasegmentinitsrespectiveveto listing,thentheentirenetworkeventisrejected.Inthecaseofbackgroundevents, wheretheanalysisisperformedontime-shifteddatastreams,itisnecessarytorejecta eventifthevetolistforthatdetectorshiftedappropriatelywouldoverlapwiththeevent. Thesevetoesareappliedinboththebackgroundandforeground.Theeffectofthese vetoesonthebackgroundeventsetwithrespecttothestrengthoftheeventcanbe seeninFigures5-1and5-2. 96

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Figure5-2.Thecoherentnetworkamplitudeoftheeventsintimebeforeandaftercuts andCategory2vetoesareappliedfortheH1H2L1network.Theeffectofthe vetoesisverypronouncedhere,withmanytowersbeingremoved.Events inredhavenocutsorCategory2dataqualityvetoesapplied.Eventsin greenhavethecutsapplied,butnoCategory2vetoes.Eventsinblue surviveallcutsandCategory2dataqualityvetoes. 5.2Post-productionTuning IntheS5all-skysearch,thepost-productionthresholdsweretunedsuchthatthe lossofinjectionswasminimized.Thisconservativetuningoptimizedthedetectionof awidevarietyofwaveforms.Asaconsequence,theeffectivenessofthesecutsin rejectingeventsinthebackgroundpopulationisreduced.Amoreaggressivesetof cutsshouldfurthersuppressthebackground,andhenceincreasethesignicance ofapotentialdetectioncandidateorleadtoabetterupperlimit.ManysetsofpostproductionvaluesweretestedseeTable5-1todetermineboththeireffectivenessin eliminatingthebackground,aswellasensuringthatinjectionrecoveryefciencywasnot systematicallyaffected. 97

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Table5-1.Listingofthecutsetstested.Theall-skysetsarerepresentativeofthetypes ofcutsusedintheS5all-skyburstsearch.Weaklymodeledcutsetsmake useofthedualstreamenergydisbalance. CriteriaCutsetvalue Standardall-skyTighterall-sky Networkcorrelationcoefcientcc0.60.7 Energydisbalance 0.350.3 Penaltyfactor P f 0.30.3 StandardweaklymodeledTighterweaklymodeled Networkcorrelationcoefcientcc0.60.6 Dualstreamenergydisbalance 0.20.15 Stringentweaklymodeled Networkcorrelationcoefcientcc0.7 Dualstreamenergydisbalance 0.15 Figure5-3.Inthisgure,thetwoselectioncriteria,ccx-axisand colorforinjections areplottedagainstthecoherentnetworkamplitude onthey-axis.The statisticissuppressedforthebackground,sothatitisvisibleincontrastto theinjections. Todemonstratetheeffectivenessofthesecuts,acomparisonofthepost-production cutsbetweenrecoveredinjectionsandthebackgroundisshowninFigures5-3and 5-4.Inthemajorityofcases,thecutsremoveweakerinjectedsignalswhileconsistently removingstrongbackgroundeventswithlowvaluesofccandlargevaluesof 98

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Figure5-4.Inthisgure,thetwoselectioncriteria, x-axisandcccolorforinjections areplottedagainstthecoherentnetworkamplitude onthey-axis.Thecc statisticissuppressedforthebackground,sothatitisvisibleincontrastto theinjections. Incontrasttothepreviousall-skysearch,thissearchtargetsaspecicsource modelandanincreasedlossofinjectionsdoesnotaffecttheupperlimittowithinthe uncertainties.Therefore,thethresholdscanbeappliedinamoreaggressivewayas longastheyarenotsettosystematicallyrejectlargepartsoftheparameterspace.The effectiverangeversusdistanceformultiplesetsofcutsisshowninFigure5-5.Cutsets withmorestringentvaluesofthresholds,suchasccand decreaseoverallefciency, butonlybylessthan15%. Inthethreedetectorcase,itwasapparentthattailsinthebackgroundstrong eventswhichwerenotrejectedbyCategory2eventvetoeswouldneedtobecut awayinordertoincreasethesignicanceofpotentialdetectioncandidates.Therefore, forthethreedetectornetwork,themoststringentsetofcutswasused.Inthefour detectorcase,allcutsproducednearlydegeneratebackgroundsets,withveryfew outliers.Hence,thelessstringentcutswereused,soastomaximizethevisiblevolume. 99

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Figure5-5.Thecumulativeeffectiverangeradiusofasphereofvisiblevolumeasa functionofdistancefortheH1H2L1V1network.Errorbarsarepresentbut toosmalltobeseen. Table5-2.Listingofthepost-productioncutthresholds. NetworkH1H2L1V1H1H2L1 Dualstreamenergydisbalance < 0.20.15 Networkcorrelationcoefcientcc > 0.60.7 Oncethethresholdsonthe andcccutshadbeenselected,thesecutswere appliedtothebackgroundtogeneratethenalbackgroundsetforbothnetworks.The eventwithlowestFADsetsthethresholdson .Fromthesetofselectioncutsavailable fromSection4.4,thoselistedinTable5-2alongwiththeirvalueswereusedinthenal analysiswiththesimulatedinjections. 100

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Figure5-6.Twodimensionalplotofeventrankingstatistic incolorversustimeand centralfrequency F c fortheH1H2L1network.Thetimeisrelativetothe startoftheS5run. 5.3BackgroundSets Twonetworkswereanalyzed,providingtwobackgrounddistributions.Toavoid correlationsofthebackgroundsets,thefourdetectortimeobservationtimewasexcised fromthethreedetectortime.Therefore,anyH1H2L1eventappearingintheH1H2L1V1 timewasnotincludedintheH1H2L1distribution.Allresultsshownareafteralldata qualityuptoCategory2forforegroundcandidatesandCategory3forsimulated injectionsandpost-productionthresholdshavebeenapplied. 5.3.1H1H2L1Network Figure5-6presentsthedistributionofallbackgroundeventswhichsurviveall vetoesandpost-productioncutsversusfrequencyandtime.Theseeventsarethen orderedbytheirrankingstatisticandaccordinglyassignedaFARasdescribedin Equation4.TheFARversusrankingstatisticispresentedinFigure5-7. 101

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Figure5-7.Cumulativerateofbackgroundeventsversustheirrankingstatisticforthe H1H2L1network.Theloudesteventsrepresentthetailofthedistribution withalowvalueofFAR.Theoverlayedgreeneventsaretheforeground events.Errorbarsonthebackgroundpointsrepresent1 errorsexpected fromaPoissoniandistribution. 5.3.2H1H2L1V1Network Figure5-8presentsthedistributionofallbackgroundeventswhichsurviveall vetoesandpost-productioncutsversusfrequencyandtime.Theseeventsarethen orderedbytheirrankingstatisticandaccordinglyassignedaFARasdescribedin Equation4.TheFARversusrankingstatisticispresentedinFigure5-9. 5.4LoudestEvents 5.4.1LoudestEventsintheH1H2L1Search Thethreeloudesteventsinthethreedetectorsearcharelisted,withtheirproperties,inTable5-3.Noneareconsideredsignicantenoughforpossibledetectionfollow up. 102

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Figure5-8.Twodimensionalplotofeventrankingstatistic incolorversustimeand centralfrequency F c fortheH1H2L1V1network.Thetimeisrelativetothe startoftheS5/VSR1run. Table5-3.ThreeloudesteventsintheH1H2L1search.Noneoftheseeventswouldbe signicantenoughcandidatestobeinterestinginasinglenetworksearch. GPStimes cc F c Hz 8576928703.740.740.1375 8467357543.690.760.1372 8200910223.550.830.0545 5.4.2LoudestEventsintheH1H2L1V1Search Theloudestthreeeventsinthefourdetectornetworksearcharelisted,along withtheirreconstructedproperties,inTable5-4.Thestrongesteventinthefourdetectorsearch,andalsothestrongesteventoverallasmeasuredby wasatGPStime 874465554.7.Thiseventisknowntobeofnon-GWorigin.Periodically,thecollaborationoperatingtheinstrumentsinjectssimulatedGWsignalsintothedetectorsinorder tochecktheefcacyofsearches.Someoftheseinjectionsarekeptsecretuntilall searcheshaveannouncedtheirresults.Thestrongesteventisknowntobeoneofthese blindinjections.Itisasine-Gaussiantypeinjectionwithacentralfrequencyof120Hz. 103

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Figure5-9.Cumulativerateofbackgroundeventsversustheirrankingstatisticforthe H1H2L1V1network.Theloudesteventsrepresentthetailofthe distributionwithalowvalueofFAR.Theoverlayedgreeneventsarethe foregroundevents.Errorbarsonthebackgroundpointsrepresent1 errors expectedfromaPoissoniandistribution.Theblindinjectioneventisplotted inblue.ItsFARistheFARthatwouldbeassigneditifitwerearealevent, whiletheFARoftheotherforegroundeventsignorethiseventintheir calculation. Table5-4.ThreeloudesteventsintheH1H2L1V1search.Thetopmosteventisablind injection,andthereforeofnon-GWorigin.Theothertwoeventsarenot signicantenoughtoclaimdetection. GPStimes cc F c Hz 8744655555.290.780.06116 8714743933.160.900.17582 8702611702.960.720.06409 ThisinjectionwasalsorecoveredbythecWBS5/VSR1all-skysearchwithverysimilar parameters.Asecondinspiral-typeblindinjectionwasalsomadeduringthistime,the totalmassofwhichwaswellbelowthetargetofthissearch.Itwasnotdetectedbythe all-skysearchnorthissearch. 104

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Figure5-10.FalseAlarmDensityforbothnetworks,includingthebackgroundevents andforegroundevents,asafunctionoftherankingstatistic. 5.4.3LoudestEventsintheCombinedSearchbyIFAD TheFADofallforegroundeventsrelativetothebackgroundareshownin5-10and theproductivityattheFADofeacheventisshownin5-19.TheFADofeacheventand searchproductivityisthenusedtodeterminethesignicanceofforegroundeventsinthe combinedsearchbytheprocedureoutlinedinSection4.10.2.Falsealarmprobabilities forforegroundeventswiththeexceptionoftheblindinjectionwiththesmallestFAD arelistedinTable5-5.TheeventwiththesmallestFADistheloudesteventinthe H1H2L1V1search,andbecausethisnetworkisconsiderablylessnoisythanthethe H1H2L1network,itisrankedhigherinFADstatistic.EventcandidateswithaFAP oflessthan1%wouldbefollowedup.Withafalsealarmprobabilityof44%itisnot consideredsignicantenoughfordetection.Noothereventsaresignicantenoughto claimthedetectionofaGWevent. 105

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Table5-5.TriggersrankedbythecombinedsearchstatisticalongwiththeirFAPand networkinformation.Theblindinjectionisexcludedfromthistablebecause itsinclusionwouldbiasthesignicanceresultfortheactualloudesteventin thecombinedsearch.Thetoprankedeventisnotsignicantenoughfor detection. GPStimeNetworkCoherentnetworkamplitude FADFAP 871474393H1H2L1V13.166.79 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(8 43.75% 857692870H1H2L13.742.03 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(7 63.89% 846735754H1H2L13.692.28 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(7 45.61% 820091022H1H2L13.553.24 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(7 51.89% 5.5ApplicationofCategory3Vetoes Withnosignicantcandidateevents,thesearchhasnotmadeadetection,and thereforeupperlimitsarecalculated.Thedetectionstudiesrequiredonlytheapplication ofCategory2vetodataquality.Category3dataqualityisconsideredtobeunsafe tousefordetectionstudies,sincethecorrelationswithpossibleenvironmentaleffects islesscertain.Afterdetectionsignicanceisestablished,Category3dataqualityis appliedtothebackgrounddistributionbeforedeterminingthenalthresholdsonthe coherentnetworkamplitudefromtheloudesteventstatistic.Noreevaluationofthe foregroundeventswasdoneaftertheapplicationofCategory3dataquality.Theeffect ofCategory3dataqualityonthebackgroundrateversusrankingstatisticisshownfor theH1H2L1V1networkinFigure5-11andH1H2L1inFigure5-12. Table4-2showsthesearchlivetimeafterCategory3dataqualityisapplied.The Category3vetoesareeffectiveineliminatingfurtherbackgroundandimprovingtherate versusrankingstatisticofthebackgrounddistributions. ThenalcheckunderCategory3dataqualityistoseeiftheloudesteventinthe combinedsearchisremoved.Iftheloudesteventisvetoed,thenthenextloudestunvetoedeventinthecombinedsearchwouldbeselectedtosetthethresholds.However, theloudesteventbyFalseAlarmDensityisnotrejectedbyCategory3dataquality vetoes.Thenalthresholdssetbytheloudesteventafteralldataqualityvetoeshave beenappliedtothebackgroundandinjectiondistributionsarederivedfromthenewFAD 106

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Figure5-11.Cumulativerateofbackgroundeventsversustheirrankingstatisticforthe H1H2L1V1networkafterCategory2andCategory3dataquality. Figure5-12.Cumulativerateofbackgroundeventsversustheirrankingstatisticforthe H1H2L1networkafterCategory2andCategory3dataquality. 107

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Figure5-13.FalseAlarmDensityforbothnetworksafterCategory3dataquality, includingthebackgroundeventsandforegroundevents,asafunctionof therankingstatistic. distributionsasshowninFigure5-13.Theexplicitthresholdsforeachnetworkarelisted intheTable5-2. 5.6EffectiveRange TheeffectiverangeisshownfortheH1H2L1V1networksearchin5-14andthe H1H2L1networkisshownin5-15.Asexpected,thevolumedecreasessharplyasthe massratiodecreasesfromequalmassvaluesbecause,whilethefrequencybandwidth isincreased,theenergyreleasedinthemergerbecomessmaller.Additionally,asthe totalmassincreases,theportionofthesignalwhichremainsinbandbecomessmaller, andhencetheeffectiverangealsodecreases. 5.7StatisticalandSystematicUncertainties AsdiscussedinSection4.9,thesearchvisiblevolumeisaffectedbyuncertainties frommultiplesources.Inthatsection,thestatisticaluncertaintyfromthenitesampling availabletothesimulatedinjectionstudieswasdescribed.Thenumberofinjectionsin 108

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Figure5-14.TheeffectiverangemeasuredfortheH1H2L1V1searchoverthe componentmassplane. Figure5-15.TheeffectiverangemeasuredfortheH1H2L1searchoverthecomponent massplane. 109

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Figure5-16.Fractionaluncertaintyabsolutestatisticaluncertaintynormalizedbythe totalvisiblevolumeonthevisiblevolumefortheH1H2L1V1search. eachnetworkwaschosensoastogivestatisticaluncertaintiesnearthesamevaluesof thecalibrationuncertainties.FortheH1H2L1V1network,about700,000injectionswere performed,andfortheH1H2L1networkabout2,000,000injectionswereperformed. GiventhesubsequentcutonmassratioseeSection4.9.2onlyabouttwo-thirdsof eachinjectionsetwasused.Figures5-16and5-17displaythefractionalstatistical uncertaintiesonthevisiblevolumedividedoverthecomponentmassplane. Thetotaluncertaintyonthevisiblevolumeasafunctionofcomponentmassis displayedinFigures5-16and5-17.Thestatisticaluncertaintyperbinisabout5%.For theentiresearch,intheH1H2L1V1casethestatisticaluncertaintyis0.5%,and,inthe H1H2L1caseitis0.4%.Visiblevolumestakeintoaccountthe11.1%upwardshiftin duetocalibrationuncertainties. Inadditiontothebiascorrectionappliedtothevolumecalculation,thereisan additionalsystematicuncertaintyduetotheuncertaintyfromtheNRwaveformswhich areusedtounbiasthe EOBNR waveformsandthesubsequentvolumecalculations. TheNRuncertaintiesarestillunderstudy,andtheyareexpectedtobebetween10-20% permassbin.Acomparisonoftheeffectiverangefromthe IMRPhenomB familywith 110

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Figure5-17.Fractionaluncertaintyabsolutestatisticaluncertaintynormalizedbythe totalvisiblevolumeonthevisiblevolumefortheH1H2L1search. thebiascorrected EOBNR willprovideameasurementofthesystematicuncertainties betweenthetwofamiliesofNRwaveformsusedtounbiastheeffectiverangeresults. 5.8CombinedUpperLimit Theupperlimitoverthecomponentmassplaneforthecombinedsearchiscalculatedintheloudesteventstatisticformulation[115].Thisrateiscalculatedinthe absenceofdetectionsat90%condencebyEquation4.Whenthetwosearch networksarecombinedbytheFADstatistics,theratedensityupperlimitoverthecomponentmassplaneisshowninFigure5-18.Themostsignicanteventinthissearch listedinTable5-5setsacombinedsearchproductivityof5.51 10 6 Mpc 3 yr.This canbeseeninFigure5-19.SubstitutingthisproductivityintoEquation4setsan upperlimitonratedensityof4.19 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(7 Mpc )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 yr )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 at90%condence.Theforthe bestcomponentmasstile+75 M theupperlimitontheratedensityis7.8 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(8 Mpc )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 yr )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 111

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Figure5-18.TheratedensityinMpc )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 yr )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 upperlimitonIMBHBcoalescenceoverthe componentmassplaneforthecombinedsearch. Figure5-19.CombinedsearchproductivitygreenasafunctionofFAD.Theevent settingtheproductivityisthelowestblackpointonthecurve. 112

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5.9ResultsSummary Afterexaminingallcandidatesinthecombinedsearch,nocandidatewarranted furtherfollowupasapossibleGWevent.Then,theupperlimitontheeventratedensity forIMBHBsourcesinthemassrangeof100 M )]TJ/F21 11.9552 Tf 11.955 0 Td [(450 M wasdetermined. TheMonteCarlosimulateddetectionstudiesshowthattheH1H2L1searchnetwork,whenaveragedoverallmasses,hasaneffectiverangeof103Mpcwithatotal statisticaluncertaintyof0.4%.TheH1H2L1V1networkismoresensitivehavingarange of121Mpcwithatotalstatisticaluncertaintyof0.5%.Atthemostsensitivepoint,near atotalmassof75+75 M ,thefourdetectornetworkhasarangeofabout237Mpc. ThisrangeisthefurthestrangethathasbeenachievedbyanyGWsearchtodate. Atthemostsensitivepoint M +75 M ,theratedensityupperlimitis7.8 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(8 Mpc )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 yr )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 .Withoutadetectioncandidate,theratedensityupperdensityupperlimit issetat4.19 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(7 Mpc )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 yr )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 averagedovertheentiremassrange.Convertingthis numbertoGc )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 yr )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 theratedensityobtainedbytheIMBHsearchis1.4 10 3 Gc )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 yr )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 Evenforthemostsensitivetile,theratedensityupperlimit.6 10 2 Gc )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 yr )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 isstill wellabovetheratesof0.007 )]TJ/F21 11.9552 Tf 11.955 0 Td [(20Gc )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 yr )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 predictedbythemodelof[41]. 113

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CHAPTER6 FUTUREDIRECTIONS OnlyasmallpartoftheparameterspaceofIMBHbinarieshasbeensurveyed inthissearch.Consider,forexample,theomissionofBHspineffects.WhiletheunderstandingofGWfromcoalescencesofspinningbinariesisdevelopingrapidly,the integrationofthoseresultswiththeeffortsofdataanalysisremainsinitsearlystages. Oneofthemainissuesinvolvedistherapidincreaseinthedimensionalityintheproblemwhencomponentspinsareaddedintomatched-lterdetection.Therequirementof muchlargertemplatebankstoachievethesameresultsasnon-spinningsearchesis unavoidable.However,un-modeledsearchesareunaffectedbythisincreasedcomplexityintermsofdetection,andearlystudieswithaligned-spinwaveformshaveproduced promisingresults. Untilthenextgenerationofinterferometerdetectorsareconstructedandcommissioned,IMBHswithmassesgreaterthan450 M remainoutofthereachofthe GWsearches.However,improvementstothedetectornetworkshavebeenmadein S6/VSR2,asciencerunsubsequenttoS5/VSR1.Furthermore,secondgenerationinterferometerprojectswillextendthesensitivitytointermediatemassblackholebinaries IMBHBinbothmassandrange. 6.1SpinEffects TheastrophysicalinterpretationoftheresultsofthissearchforGWfromIMBHB hasneglectedtherealitythatbinarycoalescenceswillprobablycontainatleastone componentmasswhichhasspinangularmomentum[119,120].Thesespincomponents interactwiththeoverallangularmomentumofthebinarysystemandcanhaveadrastic effectonitsevolutionandcongurationofthenalmergedblackhole[121].Current matched-ltersearchesforring-downsignals[32]doincludetheeffectofthenalspin oftheremainingblackhole,butotherwise,fewsearcheshaveincludedwaveforms ortemplateswhichexplicitlyusenon-zerospinparameters[122],thoughsomehave 114

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Figure6-1.AboveisadiagramofaBHbinarysystemwithspinningcomponentmasses. The IMRPhenomB family,usedinthesimulationspresentedhere,havespin angularmomentum S 1 S 2 alignedwiththeoverallangularmomentum vector L .The S 1 S 2 alignedcaseisshownhere,butcaseswith S 1 and/or S 2 anti-alignedwerealsoexamined. beendevelopedforuseintemplatesearches[123].Thismayhavealargeeffectonthe detectionabilityoftemplatesasthetotalmassofthebinaryincreases[124],however weaklymodeledsearchesarenotexpectedtobesignicantlyaffectedbyspineffects. 6.1.1Spin-AlignedBinaries Theunderstandingofspin-orbitinteractionsinbinariesisdevelopingrapidly [125132],however,thefulldescriptionhowtheyaffecttheGWemittedisstillinits infancy[133135].SomeofthemajorprogressavailabletoGWdataanalysishascome fromthedevelopmentoffullIMRwaveformsincludingtheeffectsofblackholespinsthat arealignedwiththeorbitalangularmomentumoftherotatingsystem. Physically,thealignmentofthespinswiththeorbitalangularmomentumcanhave aneffectonthedurationofthewaveform.Whenaligned,thewaveformbecomeslonger becausemoreenergymustberadiatedbeforethesystemcanmergetheso-called orbitalhang-upeffect[136].Theoppositeeffectoccursforanti-alignedspins.Alsoof interestarespecicanti-alignedspincongurationswhichcancauseaspin-ip[137] wherethebinarytransitionsfromtherelativelylowangularmomentumanti-aligned congurationintoanalblackholewithaspinwhichisoppositeinsign. 6.1.2PhenomenologicalWaveformswithAlignedSpins Forthisstudy,aPhenomenologicalfamily IMRPhenomB enhancedwithadescriptionofaligned-spineffects[108]andothercorrectionsforhighmassratioeffects 115

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wasused.TheconstructionofthistypeoffamilywasdescribedinSection4.6.1.2.In additiontothesimulationparametersoutlinedinTable4-3,anadditionaltwoparameters pertainingtothemagnitudeofthenormalizedcomponentspinvectors s z ,1 s z ,2 was included.Forthediscussionofalignedspins,thebinary'sorbitalplaneistakentobe inthe xy planeandthedirectionofthebinary'sorbitalangularmomentumistakento beinthe z direction.Hencethespinangularmomentumisalsointhisdirection.Inthe IMRPhenomB family,thesetwoaredegeneratetoasingleparameter, ,denedin Equation6where m 1 m 2 arethecomponentmassesofthebinaryand S 1 S 2 are thespinangularmomentaofeachcomponent. i = S i m 2 i = 1+ m 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(m 2 m 1 + m 2 1 + 1 )]TJ/F24 11.9552 Tf 13.15 8.088 Td [(m 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(m 2 m 1 + m 2 2 NotethatEquation6,isconstructedsuchthat )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 i 1.Thisreectsthe physicalconstraintthatthemagnitudeofthecomponentBHangularmomentumcannot exceed m 2 i .Kerrmetricsviolatingthisprinciplewouldcontainnakedsingularities. Thesetypesofsingularitiesarenotgenerallyconsideredtobepossibleforblackholes formedfromstellarcollapse[138,139]. 6.1.3Results Thisstudywasdoneforthefourdetectornetwork,withnoothersourceparameters changedfromTable4-3.Figure6-2showsthedistributionofspinparametersforeither componentinthebinary. Overall,theefciencyinFigure6-3showsdetectionofspinsisbetterforcases wherebothcomponentspinsarealignedwiththeorbitalangularmomentumasopposed topartiallyalignedcases.Anti-alignedcasesshowtheworstefciencyoutofthe alignmentcongurations.ThereisadecreaseinGWenergyemissionwhenthespins areanti-aligned,henceasmallerSNRandreduceddetectionefciency.However, 116

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Figure6-2.Distributionofthemagnitudeofthecomponentspinvectors. averagingoverallspinalignments,thereisanoverallincreaseinGWenergyemitted whencomparedtonon-spinningcases 1 Theeffectiverangeoverthecomponentmassplaneforthe IMRPhenomB familyis showninFigure6-4.Acomparisonbetweenthespinningandnon-spinningcasesafew masstilesonthecomponentmassplaneisinTable6-1.TheoverallgainofSNRwhen averagingoverspinsyieldsbetweena10%and20%increaseintheeffectiverange whencomparingspinningandnon-spinningfamiliesforreasonablysmallmassratios. Figure6-5conrmspredictionsabouttheincreasedeffectiverangeofIMBHBwith spinforrstgenerationdetectors[140].Thisindicatesthateffectiverangeswithoutspin effectsincludedareactuallyconservativeunlessnaturestronglyprefersanti-aligned spinsoverothercongurations.InregardtodetectionofIMBHBwithalignedspin, reference[140]citesthatanoptimallydetectablesetofbinaryparameterswillhave amaximallyalignedangularmomentumparametere.g. 1andatotalmassof 200 M forLIGOand395 M forVirgo. 1 ThisincreaseontheaveragecanbenotedinFigure11of[140]byaveragingover theenergyemittedbetweentheextremesofthespinalignmentparameter. 117

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Figure6-3.Detectionefciencyovercomponentspinsforthe IMRPhenomB waveform family.Theupperrightcornerrepresentshighlyspinningalignedbinaries andthelowerleftareanti-alignedcases.Betweenthetwocornersare partiallyalignedcasesonealigned,oneanti-aligned. Table6-1.Comparisonofeffectiverangesbetweenthe IMRPhenomB nospinand IMRPhenomB withspinfamilies. Family Masstile M 100+100200+200100+325 IMRPhenonBnospin1623714 IMRPhenonBaligned-spin175465.7 6.1.4ArbitrarySpinDirections Ideally,waveformswitharbitraryspindirectionsshouldbeusedforthisstudy, however,nofamilieswiththesepropertiesareyetavailableforuse.Oneofthephysical effectofnon-alignedcongurationsisthatthesystemwillhaveprecessionaboutthe totalorbitalangularmomentumvector[141,142].Givenoftheprobableprevalence ofthesesystems,itisdesirabletounderstandtheoveralleffectonthesensitivityof un-modeledGWsearchestothesesourcesmightbe.Specically,fromthevantage 118

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Figure6-4.Effectiverangeoverthecomponentmassplaneforthe IMRPhenomB family withaligned-spinsfortheH1H2L1V1network. Figure6-5.Ratiooftheeffectiverangesforspinningovernon-spinningfamiliesoverthe componentmassplane. 119

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pointoftheEarth,thesimpleprecessionofthesysteminducesasecondaryoscillation ontheenvelope.Technically,thiswouldrequirethepolarizationratio fromEquation 3tobecometimedependent,whichcouldhaveaneffectontheellipticalpolarization constraint.However,incasesofweakspinmagnitude,itislikelythatthisprecessionis slowenoughrelativetotheorbitalfrequencythatthedifferencein a overthelifeofthe mergerissmallenoughtobeneglectedandtheellipticalpolarizationconstraintwillhold. However,ifthecouplingislarge,orthespinvectorisatalargeangletotheoverall angularmomentum,thentheeffectsonthebinaryaremuchmoredifculttopredicte.g. spinipsandlargerecoilkicks[127,143148].Thus,theGWwaveformwillnolonger beellipticallypolarizedfromthevantagepointofanEarthbaseddetector;breaking theindependenceontimeofthe parameter.Whiletheellipticalconstraintmaystill functionintheearlyinspiral,themergerislikelytobedifferentenoughtobreakthe dependence. 6.2EnhancedandSecondGenerationDetectors ThefullrangeofIMBHBareunavailabletotheS5/VSR1dataset,owingtothe poorsensitivityofgroundbasedGWinterferometrytobinarieswithmassgreater than450 M .Thepoweremittedinthemergerandring-downbecomeminisculein comparisontotherapidlyrisingnoiseoorbelow32Hz.A250+250 M mergerisout ofreachtorstgenerationdetectorssuchasLIGOandVirgobecausethefrequency bandwidthliesbelowtheregionwhereitispracticaltocalibratethedatatimeseries, givenitspoorsensitivity. SeveralavenuesexistforthefutureofIMBHBdetection.LIGOandVirgodetectors inanenhancedcongurationcompletedasecondjointdatatakingrunasofOctober of2009ofciallycalledS6/VSR2/VSR3.Thedesigntargetofthisrunwastoincrease theoverallsensitivityoftheinstrumentsbyafactoroftwo.ManyGWdetectorsarecurrentlyinconstructionwhichcouldextendthemassrangeanddeepentheastrophysical 120

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reachofIMBHBsearches.Asofthiswriting,theinstrumentsarenowbeingdisassembledtoenabletheconstructionoftheiradvancedcounterparts:AdvancedLIGO andAdvancedVirgo.Thesesecondgenerationinterferometersareexpectedtofar outperformtheinitialdetectorswithafactoroftenincreaseinsensitivityandpotentially GWdetectionsonaweeklytomonthlybasis.BothlowmassIMBHbinariesaswellas extrememassratioinspiralse.g.aneutronstarsaccretingontoanIMBHshouldbe detectablebytheseinstruments.Optimisticdetectionrateshavebeenquotedafewto afewhundredperyearforsolarmassblackholesaccumulatingontoIMBHs,aswellas oneperyeartoonepertenyearsforIMBHB[36,40,41,149].However,thesenumbers arehighlydependentonthedynamicsoftheglobularclustersthattheIMBHresidein, aswellasthenalcongurationandoperatingperformanceoftheadvanceddetector networks. Thespace-borneLISApathndermissionisslatedtolaunchin2011.Ifsuccessful, thismissionissettobeatestbedforthetechnologiesandtechniqueswhichwillmake uptheLISAmission.LISAwillhavecomparablestrainsensitivitytoinitialLIGO,butits sensitivebandwidthwillbeinthemicrohertztodecihertzrange.Thiswillprovideunique opportunitiestoseethequasi-periodicinspiralstageoflargeIMBHBbinaries[150] sof M aswellastheaccumulationofsmallerstellarmassBHsontotheIMBH withratesofafewperyearafteraccumulationofSNRthroughthelonginspiral[35,36]. Beyondthistimeline,thirdgenerationdetectors[151]havebeenproposedwhichwill furtherenhancesensitivityinthe1Hzrange.AproposalfortheEinsteinTelescope wouldprojecttohaveastrainsensitivityof10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(21 p Hz )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 at1Hz,withsignicantSNRs foranumberofbinariesintheintermediatemassrangeatredshiftsapproaching one[152].Detectionratesareestimatedinthehundredsperyear[36].Theoverlapof coveragebetweenthesetwoinstrumentscaneffectivelycovertheentirerangeofIMBH inspiralandcoalescence. 121

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Figure6-6.LIGOstrainsensitivityin p Hz )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 fortheHanfordredandLivingston greendetectorsduringS6uptoMayof2010. 6.2.1S6/VSR2/VSR3ScienceRun TheS6/VSR2/VSR3ScienceRunbeganAugust2009andcompletedinOctober 2010.Enhancementstoinputlaserpowerandlaserpowerstabilizationallowedfor improvedsensitivityinthekHzregionofthestrainsensitivity.Theonlymajordifference betweentheS5andS6runsistheremovalofthesecond2kminterferometerat Hanford,leavingtheonlythreedetectorcongurationavailabletobeH1L1V1. WhilethesensitivityimprovementisnotanimmediatebenetforIMBHBsearches, itdoesallowforadditionaldatatobefoldedintothecombinedsearchupperlimit.Since LIGO'saveragesensitivityinS6isequalorbettertoS5,seeFigure6-6[153]the combinedupperlimitcouldbeimprovedsignicantly.Additionally,theS6/VSR2/VSR3 datasetisverycloseincharactertoS5/VSR1andthereforewouldrequirelittleorno modicationtothesoftwareinfrastructuretocomplete. 122

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6.2.2AdvancedLIGOandAdvancedVirgo Projectedfora2015startup,AdvancedLIGO[154,155]aLIGOandAdvanced Virgo[156]advVirgorepresentthesecondgenerationofgroundbasedGWinterferometry.Technicalimprovementsinlaserpower,signalrecyclingcavities,andbetter seismicattenuationwillshapeashallowerlowfrequencynoiseoor.Itisexpectedthat observationalqualitydatawillnowextendto10HzseeFigure6-7[157]forAdvanced LIGOandFigure6-8[158]forAdvancedVirgo,whichshouldexpandthemassrangeof IMBHBmergersbeyond500 M .Thedesigngoalforaveragesensitivityofthesecond generationinterferometersisafactorof10overtherstgeneration.Thistranslatestoa factorof10increaseintheeffectiverangefortheIMBHBpresentedhere.Thisincrease pushestheboundaryofdetectionintocosmologicallysignicantareas,with z 1.Potentially,thisreachintotheearlyuniversecouldprovideanindependentmeasurement oftheHubbleparameter.Thescalefactor a t fromtheFriedmann-Robertson-Walker metric[159]iscoupledintothepropagationofabinaryatcosmologicaldistances.However,thiscouplingredshiftsthefrequencycontentofaninspiral,andthereforepushes morepowerofthesensitiveband[55].Additionally,thisposesnewchallengestocoherentmethods,asitisknownthatthepolarizationstateswilldecoupleandtherefore, currentcoherentmethodsmaynotbeabletodealaccuratelywiththewaveform. 123

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Figure6-7.ProjectedAdvancedLIGOstrainsensitivity.Itshouldbenotedthatthisis onlyamodelbasedonthecombinationofknownfundamentalnoise sources,anddoesnotincludeanyothertechnicalsourcesofnoisesuchas violinmodes,electricalcoupling,etc.Themostlikelytuning,earlyintherun, willbethe'Zerodetuning,highpower'. Figure6-8.ProjectedAdvancedVirgobaselinestrainsensitivity. 124

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BIOGRAPHICALSKETCH ChrisPankowwasborninPittsburgh,PAinJuneof1983.Hegraduatedfromthe UniversityofPittsburghin2005withaBachelorsofArtsinphysics&astronomy.InMay of2011,hereceivedhisPh.D.fromtheUniversityofFlorida. 133