Relic gravitational waves and their detection

arXiv:astro-ph/0604458v422Jul2006RelicgravitationalwavesandtheirdetectionWenZhao,YangZhangAstrophysicsCenterUniversityofScienceandTechnologyofChinaHefei,Anhui,ChinaAbstractAsthestrongevidenceforinflation,therelicgravitationalwaves(RGW)havebeenextensivelystudied.Although,ithasnotbeendetected,yetsomeconstraintshavebeenachievedbytheobservations.FutureexperimentsfortheRGWdetectionaremainlytwokinds:theCMBexperimentsandthelaserinterferometers.Inthispaper,westudythesecurrentconstraintsandthedetectiveabilitiesoffutureexperiments.WecalculatethestrengthofRGWΩg(k)intwomethods:theanalyticmethodandthenumericalmethodbysolvingtheinflationaryflowequations.BythefirstmethodweobtainaboundΩg3.89×10−16atν=0.1Hz,wherewehaveusedthecurrentconstraintsonthescalarspectralindex,thetensor-scalarratio,furthermore,wehavetakenintoaccountoftheredshift-suppressioneffect,theacceleratingexpansioneffect,theneutrinodampingeffectontheRGW.ButtheanalyticexpressionofΩg(k)dependsonthespecificinflationarymodelsandnotwellappliesforthewaveswithveryhighfrequencies.Thenumericalmethodismorepreciseforthewaveswithhighfrequenciesandappliestoanysingle-fieldinflationarymodel.ItgivesaboundΩg8.62×10−14,whichisindependentoftheinflationaryparameters,andappliestoanysingle-fieldslow-rollinflationarymodel.Afterconsideringthecurrentconstraintsontheinflationaryparameters,thisboundbecomesΩg2×10−17.Thesetwomethodsgivetheconsistentconclusions:thecurrentconstraintsontheRGWfromLIGO,bigbangnucleosynthesis,andpulsartimingaretooloosetogiveanyconstraintforthesingle-fieldinflationarymodels,andtheconstraintsfromWMAParerelativelytighter.ThefuturelaserinterferometersaremoreeffectiveondetectingtheRGWwiththesmallertensor-scalarratio,buttheCMBexperimentsaremoreeffectivefordetectingthewaveswiththelargerratio.ThesedetectionmethodsarecomplementarytoeachotherforthedetectionofRGW.PACSnumbers:98.80.-k,98.80.Es,04.30.-w,04.62.+ve-mail:[e-mail],anumberofobservationsonthecosmicmicrowavebackgroundradiation(CMB)powerspectra[1,2,3]andonthelargescalestructure(LSS)[4]havesupportedinflationasthegoodphenomeno-logicalmodeltodescribetheevolutionoftheuniverseatveryearlystage,whichnaturallygivesrisetotheoriginoftheprimordialfluctuationswithanearlyscale-invariantandgaussianspectrum.Inadditiontotheprimordialdensityperturbations,inflationarymodelsalsopredictastochasticbackgroundofrelicgravitationalwaves(RGW),whichisthetensorperturbations.Thedetectionofsuchabackgroundwouldprovideincontrovertibleevidencethatinflationactuallyoccurredandwouldalsosetstrongconstraintsonthedynamicofinflation[5].TherearemainlytwokindsofexperimentstodetectRGWatdifferentfrequencies.ForRGWofverylowfrequenciesν≃(10−17∼10−15)Hz,onecanobservethembydetectingthepowerspectrumofCMBB-polarizations[6].Now,thefirst-three-yearresultsofWMAP[2]havenotyetfoundtheevidenceofRGW.TheexperimentofthePlancksatellite[7]withhighersensitivitytopolarizationisscheduledforlaunchin2007,andtheClover[8]andCMBPol[9]projectswithmuchhighersensitivitiesthanPlanckarealsounderdevelopment.ForRGWofhighfrequenciesν≃(10−4∼104)Hz,anotherkindofexperimentsapply,i.e.thelaserinterferometersdetectors,includingthecurrentTAMA[10],VIRGO[11],LIGO[12,13],andthefutureLISA[14],ASTROD[15],BBO[16],andDECIGO[17].Besidesthesetwokinds,othermethodshavealsobeenusedtoconstrainthestrengthofRGW.Forexamplethetimingstudiesonthemillisecondpulsars,whichcanconstraintheamplitudeofthegravitationalwavesbystudyingthesignalresidualsofthemillisecondpulses[18].Thismethodissensitivetothewaveswithfrequenciesat10−9∼10−7Hz.Theobservationalresultsofbigbangnucleosynthesis(BBN)alsocanconstrainthestrengthofRGW[19,20,21]atallfrequencies.AlthoughtheRGWhavenotbefoundyet,someconstraintsonithavealreadybeenobtainedbytheseexperimentsorobservations.ThispaperistostudythevariousmodificationsonthepowerspectrumofRGWandexaminetheconstraintsonitfromtheexperiments.Onthespectrum,wewillconsiderthemodificationsduetosuchimportanteffectsastheredshift-suppressioneffect,theacceleratingexpansioneffectandtheneutrinodampingeffect.Twomethodswillbeappliedinthisstudy:theanalyticmethodandthenumericalmethodbysolvingtheinflationaryflowequations.Afterconsideringallthesedampingeffects,wewillgetananalyticformulaofthestrengthofRGW,which,asafunctionofthewavenumberk,dependsonthescalarspectralindexnsandthetensor-scalarratior.Bytakingintoaccountofthecurrentobservationalconstraintsonnsandr,wewillobtainanupperlimitofthestrengthofRGWΩg3.89×10−16atν=0.1Hz.Fromther−Ωgplane,wefindtheBBOexperimentscandetecttheRGWifr8.3×10−3,whichismoresensitivethanPlancksatellite,butlessthanCloverandCMBPol.Butthewould-beultimateDECIGOcandetectifr6.8×10−6,muchmoresensitivethanthoseCMBexperiments.Inthisanalyticmethod,Ωgdependsontheratior,anundeterminedparameter,whosevaluevariesforthevariousspecificinflationarymodels.Furthermore,theapproximatepowerlawofprimordialpowerspectrumalsomayyieldfairlylargeerror.Toovercometheseshortcomings,wemoveontothenumericalmethod.BecausetheRGWdependssensitivelyoninflationarystage,duringwhichitisgenerated,andthereareanumberofinflationarymodels,sot