Massive Gravity in Curved Cosmological Backgrounds

arXiv:0808.1394v2[hep-th]11Aug2008DPUR/TH/11August,2008MassiveGravityinCurvedCosmologicalBackgroundsMasahiroMaeno1andIchiroOda2DepartmentofPhysics,FacultyofScience,UniversityoftheRyukyus,Nishihara,Okinawa903-0213,JAPANAbstractWestudythephysicalpropagatingmodesinamassivegravitymodelincurvedcos-mologicalbackgrounds,whichwehavefoundasclassicalsolutionsinourpreviouspaper.Weshowthat,generically,thereexistsuchthecosmologicalbackgroundsolutionscon-sistentwiththeequationsofmotionwhereweassumetheghostcondensationansatzes.Usingthe(1+3)-parametrizationofthemetricfluctuationswith’unitary’gauge,wefindthatthereisneitherascalarghostnoratachyoninthespectrumofthepropagat-ingmodes,thetensormodesbecomemassiveowingtogravitationalHiggsmechanism,andthemodelisfreeoftheBoulware-Deserinstability.ThepricewehavetopayisthatthescalarsectorbreakstheLorentz-invariance,buttherearenopathologiesinthespectrumandleadtointerestingphenomenology.Moreover,wepresentaproofoftheabsenceofnon-unitarymodesforaspecificghostcondensationmodelinacosmologicalbackground.1E-mailaddress:[e-mail]2E-mailaddress:[e-mail]1IntroductionInrecentyears,wehavewatchedrevivalofintereststoconstructmassivegravitytheoriesfromdifferentphysicalmotivations[1]-[16].Itconcernsaverysimplequestion:Canthegravitonhavea(small)massconsistentwithexperiments?Ifso,how?Onemotivationbehindthisquestioncomesfromtheastonishingobservationalfactthatouruniverseisnotjustexpandingbutisatpresentinanepochofundergoinganaccelerat-ingexpansion[17,18].Althoughthestandardmodelofcosmologyisremarkablysuccessfulinaccountingformanyofobservationalfactsoftheuniverse,itisfairtosaythatwearestilllackingafundamentalunderstandingofthelate-timecosmicaccelerationinadditiontoproblemsassociatedwithdarkmatteranddarkenergy.Severalideashavebeenthusfarputforwardforexplainingthisperplexingfact[19].ItisherethatmassivegravitytheoriesmightplayanimportantrolesincemassivegravitytheoriescouldmodifyEinstein’stheoryofgeneralrelativityatlargecosmologicalscalesandmightleadtothepresentacceleratedexpansionoftheuniversewithoutintroducingstillmysteriousdarkmatteranddarkenergyatall.Sincegeneralrelativityisalmosttheuniquetheoryofmasslessspin2gravitationalfieldwhoseuniversalityclassisdeterminedbylocalsymmetriesundergeneralcoordinatetransformations,anyinfraredmodificationofgeneralrelativitycannotaviodintroductionofsomekindofmassforthegraviton.Theothermotivationforattemptingtoconstructmassivegravitytheoriesliesinstringtheoryapproachtoquantumchromodynamics(QCD)[8].Forinstance,asinspiredinthelarge-Nexpansionofthegaugetheory,whichdefinestheplanar(orgenuszero)diagramandisanalogoustothetreediagramofstringtheory,ifwewishtoapplyabosonicstringtheorytothegluonicsectorinQCD,masslessfieldssuchasspin2gravitoninstringtheory,mustbecomemassiveorberemovedsomehowbyaningeniousdynamicalmechanismsincesuchthemasslessfieldsdonotappearinQCD.Notethatthismotivationisrelevanttothemodificationofgeneralrelativityintheshortdistanceregionwhilethepreviouscosmologicaloneistotheinfraredmodification.ItisworthwhiletopointoutthatthemodificationofbothshortandlongdistanceregionsremindsusofT-dualityR↔l2s/Rinstringtheorywherelsisthefundamentalstringlengthscale.Ifthereistheinterestingsymmetrybehindthebothdistanceregionsingeneralrelativity,suchamodificationcouldperhapsopenafruitfuldialogbetweencosmologyandQCD.Anapproachfortheconstructionofmassivegravitytheoriesistotakeaccountofthespontaneoussymmetrybreakdown(SSB)ofgeneralcoordinatereparametrizationinvariance[1]-[8].NotethattheSSBofgeneralcoordinatereparametrizationinvariancemightprovidearesolutionforcosmologicalconstantproblem[6,8].Thoughthecosmologicalconstantproblemneedsatheoryofquantumgravity,solvingtheproblemrequiresalowenergymechanism,forinstance,apartialcancellationofvacuumenergydensitystemmingfromquantumloops.WeexpectthatfromtheanalogywiththeHiggsmechanisminconventionalgaugetheories,theNambu-Goldstonemodemixeswiththemasslessgraviton,therebychangingthevacuumstructureofgravitationalsectorinanon-trivialway.Recently,asadeviceforperformingaconsistentinfraredmodificationofgeneralrelativity1anideaofghostcondensationhasbeenproposed[9].Inthisscenario,the’unitary’propagat-ingscalarfieldappearsastheNambu-Goldstonebosonforaspontaneouslybrokentime-likediffeomorphism,andyieldsapossibilityforresolvingcosmologicalproblemssuchasinflation,darkmatteranddarkenergy.Thekeyobservationhereisthatascalarghostisconvertedtoanormalscalarwithpositive-energyexcitations.Morerecently,’tHoofthasproposedanewHiggsmechanismforgravitywherethemass-lessgraviton’eats’fourrealscalarfieldsandconsequentlybecomesmassive[8].Inhismodel,vacuumexpectationvalues(VEV’s)ofthescalarfieldsaretakentobethefourspace-timecoordinatesbygauge-fixingdiffeomorphisms,sothewholediffeomorphismsarebrokenspon-tanously.Ofcourse,thenumberofdynamicaldegreesoffreedomisleftunchangedbeforeandaftertheSSB.Actually,beforetheSSBofdiffeomorphismstherearemasslessgravitonsoftwodynamicaldegreesoffreedomandfourrealscalarfieldswhereasaftertheSSBwehavemassivegravitonsoffivedynamicaldegreesoffreedomandone