Calculation of the Cosmological Constant by Unifyi

arXiv:gr-qc/0609004v11Sep2006c2006InternationalPressAdv.Theor.Math.Phys.00(2006)000–000CalculationoftheCosmologicalConstantbyUnifyingMatterandDarkEnergyTorstenAsselmeyer-MalugaandHelgeRos´eFhGFIRST,Kekulestr.7,12489Berlin,Germanytorsten@first.fhg.de,rose@first.fhg.deAbstractWeshowthatthedifferential-geometricdescriptionofmatterbydiffer-entialstructuresofspacetimeleadstoaunifyingmodelofthethreetypesofenergyinthecosmos:matter,darkmatteranddarkenergy.UsingthismodelweareabletocalculatethevalueofthecosmologicalconstantwithΛ=p14/278πG/c4ρobs≈(1.4±0.2)·10−52m−2.1IntroductionForcenturiesithasbeenourfirmconvictionthatmatterandenergyofthesamekindasissurroundingusalsoconstitutetherestoftheworld.Thor-oughexaminationsofsupernovae[46,49,43]andofcosmicbackgroundradiation[9,56,30],however,havereplacedthisconvictionbytheinsightthattheglobalstructureofthecosmosisdominatedat95%byanen-ergyformthathashithertobeenentirelyunknown.Abouttwothirdsofthisenergyformconsistin“darkenergy”,andonethirdin“darkmatter”.ThisisthemostradicalrevolutioninourunderstandingofthecosmosafterKopernikus.Inthelastyears,greatefforthasbeeninvestedtounderstandtheseunknownformsofenergy[45,44,51].e-printarchive:[65,58,15],orintroduceadditionalglobalscalarfields[59,48].Thesead-hocentitiescanonlybeobservedbytheirgravitationalinteraction,whichprecludesanindependentfalsificationbyotherobservableinteractions.Aslongasaconsistentintegrationofsuchadditionalentitiesintotheprovenmodelsofspacetimegeometryandbaryonicmatterhasyettobestated,itseemstobequestionablewhethersuchapproachesareworthanythingmorethananexplanationdeusexmachina.Therefore,ourbasicapproachistoseethecauseofdarkenergyinoneofthetwoprovenconcepts,viz.spacetimeormatter.Sincesofarthedarkenergyhasbeenobservedonlygravitationally,itseemstobeobvioustoseekitscauseinthespacetimeitself.Weareawareofthefactthatthis“geometrizing”approachdiffersdiametricallyfrompreviousapproachesandmayhaveanunconventionalappearancetothecommunity.However,inconsiderationoftheenormousdifficultiesofconservativeapproaches,anunorthodoxapproachseemstobenecessary.Likeeveryotheraxiom,ourgeometricconceptdoesnotpossessanintra-physicaljustification;however,itdoesavoidtheepistemicproblemsofotherapproaches:•Noadditionalentitiesareintroduced.Acertainwidelyignoredprop-ertyofspacetime,presenteversinceingeneralrelativity,isassumedtobethecauseoftheexistenceofdarkenergy.Thispropertyisitsdifferentialstructure.•Accordingtoobservations,darkenergyinteractsonlygravitationally.Allknownquantumfields,however,coupletoothergaugefieldsaswellandcanthusbedetectedindependentlyfromgravitation.Itisnotplausibletointroduceadeviatingassumption.Thus,thereisonlyoneentityleftthatinteractsonlygravitationally:Thegeometryofspacetimeitself.Spacetimeisa4-dimensionalmanifold,whosestructureisdeterminedbyitstopology,differentialstructure,andmetric.Thefirstdiscussionofdif-ferentialstructuresappearedinaseriesofpapers[13,12,11]writtenbyBrans.Afurtherrelationtoparticlephysicswasdiscussedin[54,52,53].Furthermore,in[12]Bransconjecturedaboutsourcesofgravitygivenbydifferentialstructures(Bransconjecture)whichwasprovenforsomecom-pactmanifoldsbyoneofauthors[7]andforsomenon-compactspacesin[55].In[8],thelocalpropertiesofdifferentialstructurehavebeenexam-ined.Itwasshownthat,referringto3-dimensionalsubsetsΣi,thealgebraofchangesofthedifferentialstructureisaTemperley-LiebalgebraandinTORSTENASSELMEYER-MALUGAANDHELGEROS´E3particularaCliffordalgebra.Thus,theΣicanbeidentifiedwithfermions.–Inthispaper,weshallexaminetheglobalpropertiesofthedifferentialstructureofspacetime.Weshalldevelopamodelofdarkenergyanddarkmatterthatexplainsthe95%unknownenergydensitybasingonthespace-timegeometryalone,withoutresortingtoadditionalentities.WemakejustoneBasicassumption:Spacetimeisa4-dimensionalcompactclosedman-ifoldMwhichisdifferentiableandsimplyconnected.Thecosmosisthe3-dimensionalboundaryΣofadistinguishedsubmanifoldA⊂M–theAk-bulutcorkA–,whichdeterminesthedifferentialstructureofMuniquely.TheenergydensityofanykindofmatterisdescribedbythecurvatureoftheassociatedsubmanifoldofΣ.Asweshallsee,thechoiceofaparticulardifferentialstructureandhenceoftheAkbulutcorkAisaveryrestrictivecommitment.ItalsolargelydeterminestheglobalstructureoftheboundaryΣofA.Itisawell-knownresult[28,29]thattheboundaryofasimplyconnected4-manifold,likeA,isaso-calledhomology3-sphere(seeAppendixC).ThusweassumethecosmosΣisahomology3-sphere.Fromthestructuretheoryof3-manifoldsweknowthatthereareonlythreekindsof3-manifoldsthatcanformΣ.ThisremarkablefactmotivatesthefollowingConjecture:Thethreetypesof3-manifoldsthatconstitutethecosmosasahomology3-sphere,correspondtothethreekindsofmatter:baryonicmatter,darkmatter,anddarkenergy.Thusweobtainaunifiedapproachforallobservedkindsofenergyden-sities.Theglobalstructureofthecosmoscan