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Relic Gravity Waves Investigation by Advanced Space-Based Gravitational Waves Detector

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Relic Gravity Waves Investigation by Advanced Space-Based Gravitational Waves Detector

    Relic Gravity Waves Investigation by

    Advanced Space-Based Gravitational

    Waves Detector

    Commun.Theor.Phys,(Beijing,China)53(2010)PP.496498

    @ChinesePhysicalSocietyandIOPPublishingLtdVo1.53,No.3,March15,2010 RelicGravityWavesInvestigationbyAdvancedSpaceBasedGravitationalWaves

    Detector

    LIJin(李瑾)71,ZHONGYuanHong(仲元红),andZHONGTap(钟涛)

    DepartmentofTheoreticalPhysics,ChongqingUniversity,Chongqing400030,China 2DepartmentofCommunication,

    ChongqingUniversity,Chongqing400030,China

    (ReceivedMarch27,2009;revisedmanuscriptreceivedOctober19,2009) AbstractTh

    paperfoCHSeSontherelicgravitywavesproducedduringthetransitionfromatad1ation

    dominated

    inflationaryphasetoadustdominatedFHedrnan-Robertson

    Walker-typeexpansion.Wediscusshowtoinvestigatethe

    spectralenergydensibthelatestspace

    basedGWsdetectorsatf=01Hze.DECIGO).Inthecaseofpower-law andexponentialinflation,weapptheCROSS

    correlationmethodtothelatestdetectorandgetthetimedependenceof theveryearlyHubbleparameter.

    PACSnumbers:04.30.Tv.04.80.Nn.04.20.Cv

    Keywords:relicgravitywaves,spacebasedGWsdetectors,spectralenergydensity,crosscorrelationmethod

    1Introduction

RelicgravitywavesfRGW1asafundamentalconse

    quenceofinflationstoretheinformationofcosmological overallevolution.Inparticular,itsenergydensityspec

    trumrepeatsexactlythetimedependenceofthevery earlyHubbleparameter.Frpintheself-consistentsolu

    tionsoftheEinsteinequation,thepowerlawinflation

    andexponentialinflationpredictdifierentexpressionof energyspectrum.Thisinvestigationcanprovideapracti

    ca1methodtofindwhethertheinflationmodelsindifief

    entphaseofuniversearereasonable.Althoughuntilnow peoplehavenotobservedRGW,alotofconstraintshave beenachievedonthem.Theseconstraintscanbeclassified intotwokinds:oneconstrainsRGWatverylowfrequen

    ciesf?f10?7,l0一均)HzmainlyfromtheCMBobserva

    tions.especiallytherecentWMAPresults.Theotheris theconstraintsfromLIGO.LISABBN.andpulsartim

    ing,andissensitivetothewaveswithhighfrequencies ,?(10_..,10)Hz.However,theseconstraintsactually aretooloosetogiveanyconstraintonthesinglefieldin

    flationmodels.Hereweintroducethecurrentsensitivity ofdifferentGWdetectors:theadvancedLIGOcandetect thewaveswith2h>10atf100Hz(Qisthe

    strengthofaGw);theLIsAprojectisexpectedtodetect waveswithQ.h>10一?atf0.005HztheSATR0D,

    aspaceprojectwhichissensitivetothewaveswithfe- quenciesatf?(10.,10)Hz,isexpectedtodetectthe

    waveswithQ.h>10?l0atf5x10?Hz.TheBBO

    candetectabackgroundRGWwith2>2.2x1017at

    fO.11Hz.theDECIGOprojectwithamuchhigher sensitivitybydesignisexpectedtodetectRGWwith

I2h2>1020atf0.1Hz.1]Fortunatelythefuture

    laserinterferometerDECIG0canachievetherequired sensitivityhms10tomeetthenewestupperlimit

    energyofRGWenergy2Gw10..atf0.1Hz.[.JThe

    purposeofthispaperistofindtheresponseofrelicGWs energydensityspectrumthroughthecrosscorrelation

    measurementfbetweentwodetectorsfordecadesobser. vation1.Thatcanprovideameanstounderstandthe evolutionoftheveryearlyHubbleparameterandmatter density.

    DECIGOwillbeaspaceantennawithanarmlength 500000kinas1/10arnllengthofLISA.Theyhavethe sameassumptionofthetechnologylevel,suchasalaser powerof1W.theopticsof30am.etc.ButtheDE- CIGO'ssensitivityhasafactorof10betterthanLISA around0.1Hz.Beinglimitedonlybyradiationpressure noisebelow01Hzandshotnoiseabove0.1Hz.itssen

    sitivityisoptimizedatsuchfrequency,andittargetsthe frequencywindowaroundf0.1-1Hzwherecontami

    nationofastrophysicalforegroundsisseparable.Theul

    timatesensitivityofaspaceantennainthefuturecould behrms3×102around0.1Hzintermsofstrain.

    Therefore,adetector1ikeDECIGOisnecessarytomea- suresomeimportantparametersincosmology. 2EnergyDensityofRelicGravityWaves

    Thestudyofthequantumgravitonsereationin Friedman?RebertsonWalkerfFRW1cosmologywasfirst demonstratedbyGrishchuk.4JAsaresultoflinearizedap

    proximation,thebehaviourofgravitywavesinanFRW universebecomesidenticaltothatofamasslessminimally

SupportedbytheNationalBasicResearchProgramofChinaunderGrantNo.2003CB71630

    0,theNationalNaturalScienceFoun-

    dationofChinaunderGrantNo.10575140,CAEPFoundationunderGrantNo.2008T0401a

    nd2008T0402,andChongqingUniversity

    PostgraduatesScienceandInnovationFund,ProjectNo.200811BIA0100299,andChinese

    StateScholarshipFund

    fCorrespondingauthor

    ,Email:cqstarv~hotmail.com

    No.3RelicGravityWavesInvestigationbyAdvancedSpace

    BasedGravitationalWavesDetector497

    coupledscalarfield,witheachofthetwopolarization statesofthemetrictensor.[]Herewedirectlyresortto thesolutionoftheKlein——GordonequationinanFRW

    universe(seeRef.6]formoredetails).

    Theenergydensityofthecreatedgravitonsisgiven by:[?8]

    =

    ()27r2a2

    (j+k2i),(1)

    whereisconformaltime,aisthecosmologicalscalar factor,adrl=cdt,kisthecomovingwavenumber

    =

    2rca/A,rloistheinitialconformaltimefintheground state).[.]Thespectralenergydensityforgravitywavesfin unitsoferg/cm.):

    5(f)

    whereb=

    d~g

    =

6s2?(27r2(

    4re(+j'(27r/3)/.r()I,(1+l1)/[(z/

    1/2)(II+1/2)],(aretheparametersdeterminedby theinflationstage)ands"zf(27r/)(Hifisthe

    Hubbleparameterattheboundarybetweentheadia- baticandsuperadiabaticregimes[9j).m.isthePlank mass.bisthedimensionlessparameterfirstintroduced bystarobinsky,[]?3([/xf+1/2).(8Ga..7r)isthe

    backgroundmatterdensity.Oflargerpracticalinterestis theproblemofreconstructionthematterenergydensity PandHubbleparameterHintheinflationfromthemea?

    suredspectrum?(1,).Fortunately,PandHasfunctions offrequencyfweregivenbyGrishchuk:[11] }{2l}=8rrP

    E

    ~_vSf,

    wherePpisPlankdensity(1),ristheenergydensity ofthe3-Kmicrowavebackgroundatpresent.Therefore. weshouldgetthe(f)knownfromtheactualmeasure?

    mentbyDECIGOtocalculateP(f)and(1,).

    IntermsofHa=n(n:wavenumber),weget:

    dft

    =

    d(Ha)(1+H/H..:(1,7)nn,(5)一—一十万/?..In,.J

    =

    d/_(1H.

    (6)

    d

    d(1nf)

    Then,wecanobtainthefollowingequation:

dH(t)=

    (1-7.

    FOraradiationdominateduniverse(/2=3/2:

    

    1/2),=H/ZZ?2,werecoverthetheoretical

    trunlE(,)firstobtainedbyStarobinsk:

    ~gw(,)28zc

    m

    "

    spec

    H(f)

    dH(t)

    dt

    8

    

    "lrp

    

    p

    

    2S2C

    rr~

    (2,))

    ,

    3.\\/

    0

    Inthiscase,H(t)=constant.

    Foradustdominatedepochofexpansion

    =

    3/2),

~gw(.)

    H(f)

    H(t)

    382?m

    32rrf0

    (10)

    (/2=3/2

    ().,

    s

    Pp

    

    ~m1(t

    o=constant)

    (12)

    (13)

    ForthePower..1awinflationinradiation..dominated

    epochofexpansion(>3/2)leadstothesimilarsolu

    tiontoEq.(14),whichis:

    H()821

    ttn

    (15)

    Dueto<0,thespectraldensity?(,)=,decreases

    towardtothehigherfrequencies. 3Cross-CorrelationMethodonRelicGWs InVestigation

    Asapartofstochasticgravitywavebackground,RGW

    atfrequenciesaround0.1Hzisthesensitivesignalsofthe

    futurespacebasedGWinterferometersfDECIGO1.Two interferometers(D1andD2)crosscorrelationwillmea-

sureanRGWbackgroundf0.1Hz.whichweassumeto

    beisotropic,unpolarized,stationary,andGaussian.Such backgroundwillgenerateacrosscorrelationbetweenthe strainsnleasuredbyD1andD2.theapproachissimilar withcorrelationanalysisofLISA.12J

    Themostgeneralexpressionforthecrosscorrelation

    oftwodetectorsisgivenby

    C:J——T/2J——T/2S1(t)S2(t)(?一t,).(16)

    whereQ(tt)isafilterfunction,Tistheobservation timeandtheSarethestrainamplitudesreadoutfrom thei-thdetector.

    IntheFourierdomain:

    ()

    (ST(fl,)(,)(,)(,).(17)

    llJ

    o.J..

    5T(fI,)((,)(.,))(,,).(18)

    Thestrainamplitudeinthei-thdetector. (,)(,)+sift)+ni(_,)(19)

    hascontributionsfromresolvableastrophysicalsources h(1),thestochasticRGWbackground8i,andintrinsic detectornoise/zi.

    /H

    

    01

    叩一

    不叩

    2

    ,

    \,,,

/

    

    

    ??

    ,,

    

    ,

    H

    o)

    ?

    qle(

    

    t

    S

    d

    e

    y

    6

    q

    E

    498LIJin,ZHONGYuan?Hong,andZHONGTaoVo1.53 Afterremovingresolvableastrophysicalsources,the

    crosscorrelationisgivenby (C)=(81,82)+(81,n2)+(nl,82)+(nl,n2)

    =

    (81,82).(20)

    Thereforethe((,)(,,))inEq.(18)canbetakeninto

    (S~(f)S2(f,)),andperformingtheensembleaveragewe

    get

    1

    (Sx-*(f)s2(f,))=(_,,)(,)(.,)sin.0,(21) 0istheanglebetweenthetwoarmsofdetectors.The

    overlapreductionfunction7(f)is 7(f):5

    2sin2.

    F

    +(,,)(,,))exp

    Fl(h,,),F2(h,,)are

    Thelabels+,xdenote

    tationalwave,yielding

    27r'Q(1x2)

    C

    (22)

    thedetectorresponsefunctions. thetwopolarizationsofthegravi

    totheexpectationvalueofC

    (.d,仇?)(23)

    ThenoiseofCisgivenbyN=C(),thesignaltonoise

    ratio(SNR)is:

    =

    =(24)InordertomaximizetheSNR,wefindthefilterfunction

    Q(f)shouldsatisfy:

    where

    (=IQ(f)l.M(f)df,(25)

    M(f)=Snl(f)Sn2(f)+Snl(f)Sq(f)R2(f) +Sn2(f)S9(f1R1tf1+s(f1

    ×

    (sin4l,y(,)l+R(_,)R.(,))(26) 1(,),sn2(f)arethenoisepowerspectrumofD1,D2.

    Rl(f),R2(f)arethetransferfunctionsofD1,D2.n(f):

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