InRef1wefocusonthislatterissueassumingthatthepredarkenergyproblemcaneventuallybeaddressedWepointoutthatthesameingredientsinvokedintheselftuningmechanismthatistheextradimensionsandthebrane ID: 431472
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APossibleSolutiontotheSmallnessProblemofDarkEnergyPisinChenStanfordLinearAcceleratorCenter,StanfordUniversity,Stanford,CA94309,USAJe-AnGuDepartmentofPhysics,NationalTaiwanUniversity,Taipei106,Taiwan,R.O.C.Thesmallnessofthedarkenergydensityhasbeenrecognizedasthemostcrucialdicultyinunderstandingdarkenergyandalsooneofthemostimportantquestionsinthenewcentury.Inarecentpaper[1],weproposed InRef.[1]wefocusonthislatterissue,assumingthatthepre-dark-energyproblemcaneventuallybead-dressed.Wepointoutthatthesameingredientsin-vokedintheself-tuningmechanism,thatis,theextradimensionsandthebrane-worldscenario,whenaug-mentedwithsupersymmetry(SUSY),caninprinciplesolvethepost-dark-energyproblemwithveryrelaxedconstraints.SUSYguaranteestheperfectcancellationofthevacuumenergy.However,weknowthatSUSYhastobebroken,atleastinour(3+1)-dimensionalworld,withthesymmetry-breakingscaleTeV.Conven-tionallythiswouldentailavacuumenergythatismuchtoolargefordarkenergy.Butweassumethebranetensionsoinducedshouldbeperfectlycan-celledduetotheself-tuningmechanismoritsvari-ant.IfwefurtherassumethatSUSYisbrokenonthebranethrough,forexample,agauge-mediatedSUSY-breakingmechanism(forareview,see[21])whilepre-servedinthebulk,theninthiscon“gurationwe“ndthattheleadingcontributiontothevacuumenergyontheimbedded3-branealaCasimireectcanbedra-maticallysuppressedrelativetotheSUSY-breakingWeemphasizethatthereisafundamentaldierencebetweensuchaCasimirenergyandtheconventionalvacuumenergy.TheCasimirenergyinourSUSYcon-“gurationisnontrivialonlyaroundthe3-brane,and,inparticular,itentailstheequationsofstate:and0,whereandareitspressuresalongthe3-braneandtheextradimensions,respectively.Incontrast,thebranetensionfromtheconventionalvacuumenergyobeysthefollowingequationsofstate:and=0,whichareanecessarycondi-tionforitscancellationviatheself-tuningmechanism.ThustheCasimirenergycannotberemovedbythesameself-tuningprocedureandshouldsurviveastheleadingcontributiontovacuumenergyonthebrane.AsimilarconceptofSUSY-breakinginabrane-world,calledsupersymmetriclargeextradimensions(SLED),hasbeeninvokedtoaddressthevacuumen-ergyandthecosmologicalconstantproblem[28].InthisproposalSUSYinthebulkisslightlybrokenbythepresenceofnon-supersymmetric3-brane.Byin-corporatingtheself-tuningmechanismfortwoextradimensions,thebranetensionisexactlycancelledwiththecurvature[27],andasaresulttheleadingcon-tributiontovacuumenergyisinducedbytheweakSUSY-breakinginthebulk.Thesmallnessofdarkenergyinthisproposalreliesontherequirementoftwolargeextradimensionsofthesizearound01mm,whichisontheedgeofthecurrentexperimentalcon-straint[19].3.CasimirDarkEnergyinaSupersymmetry-BreakingBrane-WorldThepossibilitythattheCasimirenergyinextradi-mensionscanplaytheroleofdarkenergyhasbeenexploredbyMilton[17].Ingeneral,apositiveCasimirenergyinordinary(3+1)dimensionscannotentailanegativepressure,whileinextradimensions,onthecontrary,itcaninprincipleinduceacosmologicalcon-stantwiththedesiredattribute.Itssigndependsonthegeometry,theboundarycondition,andthe“eldcontentsintheextraspace.Ifallextradimensionshavethesamesize,thevalueoftheCasimiren-ergydensitymeasuredonanimbedded3-dimensionalspaceisingeneralproportionalto.Imposingtheobserveddarkenergydensityoftheuniverse,whichisveryclosetothecriticaldensityasindicatedbyCMBdata[18],onecanarriveatalowerlimitontheextra-dimensionsize.FromthisrequirementMiltonobtainedastringentconstraint.Thatis,thenum-berofextradimensionscannotexceedtwoandtheextra-dimensionsizeshouldbelargerthanmicrons[17].Thisconstraintontheextra-dimensionsizeisnotfarfromthecurrentexperimentalupperlimitaround100microns[19].Stringentconstraintsand“ne-tuningsarecommonfeaturesinmanydarkenergy/darkgeometrymodels.Generallyspeaking,itisverydiculttoretainthesmallnessofthedarkenergydensityiftheenergyscaleinvokedinthedarkenergy/darkgeometrymodelisnotanaturalscalerequiredbysomesymmetryprin-ciple.Inourrecentlyproposeddarkenergymodel,whichinvokestheCasimireectinasupersymmetry-breakingbrane-world,thisdicultyisapparentlyre-solved[20].Inourmodelweconsidera(3++1)-dimensionalspace-timewithcompactextradimensions,inwhichthestandardmodel“eldsandtheirsuperpartnersarecon“nedonaimbedded3-branewhilethegrav-ity(graviton-gravitino)sectorresidesinthe(higher-dimensional)bulk.Weassumethatsupersymmetryisbroken,forexample,throughagauge-mediatedsupersymmetry-breakingmechanism(forareview,see[21]),onlyaroundthebranewhilepreservedinthebulk.Inthiscon“gurationwe“ndthattheleadingcontributiontothevacuumenergyalaCasimiref-fectcanbedramaticallysuppressedrelativetothesupersymmetry-breakingscale.Asanexamplefordemonstration,herewepresenttheresultofthecaseinwhichextradimensionsge-ometryistoroidalandonlygravitonandgravitinore-sideinthebulk.Asaresultofoursupersymmetry-breakingassumption,gravitinoacquiresamassonlyaroundthebraneandgravitonremainsmasslesseverywhere.TheresultantnetCasimirenergydensity 1110 2 TeV 6 9 12 15 Mpl log[M/GeV] 6 9 12 15 Mpl log[M/GeV] TeV 6 9 12 15 Mpl TeV 6 9 12 15 18 n=1n=2n=22MS=MG MSUSY~MG Figure2:Constraintonandundertheassumption(i.e.1).Thesolidcurvescorrespondtosolutionsunderthefurtherassumptionofandthedashedlineindicatesthecondition4.PerspectivesInourapproachthesmallnessofthedarkenergyisattainedthroughthesharpcontrastbetweenthevolumeofthesupersymmetry-breakingregionaroundthebraneandthatofthesupersymmetry-preservingregioninthebulk,wheretheirrationaturallyariseswhenoneintegratesouttheimbeddingextradimen-sionstoobtaintheCasimirenergyonthe3Dbrane.Todemonstratethepowerfulnessofthisnewwayofhandlingthepost-dark-energyproblem,wehaveex-aminedsuchCasimireectunderawiderangeofextradimensions,withoutlimitingourselvestothespeci“c1and2extradimensionspreviouslyinvokedintheself-tuningmodels.Thereexistvarioushierarchyproblemsinphysics,suchastheweaknessofgravityandthesmallnessofthecosmologicalconstant.Torevealthefundamentallawsofnature,itisoftendesirabletorelatetheori-ginsofthehierarchiestomoreprofoundphysics.Forexample,theweaknessofgravitycanbeexplainedasaresultofthelargenessofextradimensions(asintheArkani-Hamed-Dimopoulos-Dvali(ADD)model[23])orthewarpageofhigher-dimensionalgeometry(asintheRandall-Sundrum(RS)model[24]).Inthisre-gardourmodelfollowsthesamespiritasthatoftheADDandtheRSmodels.Notethatbyfurtherin-voking,ourmodelmanagestosolvebothhierarchyproblemsatonce.Inaddition,sinceourmodelisbasedonthesimilaringredientsasthoseintheself-tuningmechanism,itprovidesmuchhopethatasynergybetweenthesetwoconceptsmaybefoundforacompletesolutiontobothpre-andpost-dark-energyproblemsinonestroke.Inthecurrentconstructionofourmodelthecon“g-urationoftheextradimensionsisADD-like[23].ItisinterestingtofurtherexploreourscenarioundertheRScon“guration[24].Thepossibilityoftimeevolutionoftheextra-dimensionsizeisanotherimportantissueyettobestudied.Inmanymodelsthatinvokeextradimen-sions,theirspeci“csizesarerequiredtobestableinordertosatisfytheexperimentalconstraintsfromthesolar-systemtestsofgravity.Thisisgenerallyas-sumedbyintroducingacertainstabilizationmecha-nism(e.g.,see[22]).Byinvokingsimilararguments,theCasimirenergyonthebraneinourmodelisex-actlyacosmologicalconstantinour3-dimensionalworld.Ontheotherhand,itispossiblethatatsomecosmicepochsthestabilizationmechanismwasnotef-fectivesothattheextradimensionscouldevolve.InthiscasetheCasimirenergyinourmodelmaychangeintime.Thispossibilityofatime-varyingdarkenergyisworthyoffurtherinvestigation.ThenotionofCasimirenergyinvolvestheconceptofareferenceenergy.Inourmodelthisreferenceen-ergyisassociatedwithaMinkowskian,non-compact,higher-dimensionalspace-timewithoutboundarycon-dition,withnoparticles,andnootherenergyforms.Whethersuchaenergyreferencepointisconceptuallymeaningfulrequiresfurtherinvestigation.Adierentenergyreferencepointmayberequired,especiallyun-derthesituationwherecertainextradimensionssta-bilizationmechanismisintroducedtothesystem.Theultimatechallengeofthedarkenergyissueisevidently“ndingacomplete,self-consistentsolutiontotheproblem.Inourmodelwebypassedthepre-dark-energyproblemandassumedthatitwouldeven-tuallybeaddressedbytheself-tuningmechanismoritsvariant.Itisknownthattheself-tuningideahasitsownchallenges[29,32],anditclearlyawaitsfurtherdevelopmentsbeforethepre-dark-energyproblemcanbefullyaddressed.Butevenifthatcanbedone,onestillfacesthechallengeofreinstatingtheverysmallamountofvacuumenergyself-consistentlytocomplywiththeobservation.Withthisinmind,thefurtherintegrationofourmodelwiththeself-tuningmecha-nismwouldbeextremelyinteresting.AcknowledgmentsThisworkissupportedbytheUSDepartmentofEnergyunderContractNo.DE-AC03-76SF00515,andinpartbytheMinistryofEducation(MoE89-N-FA01-1-4)andtheNationalScienceCouncil(NSC93-2811-M-002-033;NSC93-2112-M-002-047)ofTaiwan,R.O.C.References[1]P.ChenandJe-AnGu,arXiv:astro-ph/0409238.[2]A.G.Riessetal.[SupernovaSearchTeamCol-laboration],Astron.J.,1009(1998). 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