Table Of Content

Cosmology within Noncommutative Spectral Geometry 1 Mairi Sakellariadou∗ 1 0 DepartmentofPhysics,King’sCollege,UniversityofLondon,StrandWC2R2LS,London,U.K. 2 E-mail: [email protected] n a ClosetothePlanckenergyscale, thequantumnatureofspace-timerevealsitselfandallforces, J includinggravity,shouldbeunifiedsothatallinteractionscorrespondtojustoneunderlyingsym- 1 1 metry. Intheabsenceofafullquantumgravitytheory,onemayfollowaneffectiveapproachand consider space-time as the product of a four-dimensional continuum compact Riemanian man- ] h ifold by a tiny discrete finite noncommutative space. Since all available data are of a spectral t nature,onemayarguethatitismoreappropriatetoapplythespectralactionprincipleinthisal- - p mostcommutativespace. Followingthisprocedureoneobtainsanelegantgeometricexplanation e h forthemostsuccessfulparticlephysicsmodel,namelythestandardmodel(andsupersymmetric [ extensions)ofelectroweakandstronginteractionsinallitsdetails,asdeterminedbyexperimental 1 data. Moreover, since this gravitational theory lives by construction at very high energy scales, v it offers a perfect framework to address some of the early universe cosmological questions still 4 awaitingforananswer. 7 1 Afterintroducingsomeofthemainmathematicalelementsofnoncommutativespectralgeometry, 2 Iwilldiscussvariouscosmologicalandphenomenologicalconsequencesofthistheory,focusing . 1 inparticularonconstraintsimposedonthegravitationalsectorofthetheory. 0 1 1 : v i X r a CorfuSummerInstituteonElementaryParticlesandPhysics-WorkshoponNonCommutativeField TheoryandGravity, September8-12,2010 CorfuGreece ∗Speaker. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ CosmologywithinNoncommutativeSpectralGeometry MairiSakellariadou 1. Introduction AtenergiesmuchbelowthePlanckscale,gravitycanbeconsideredasaclassicaltheory,however as energies approach the Planck scale, the quantum nature of space-time becomes apparent, and the simple prescription, dictating that physics can be described by the sum of the Einstein- Hilbert and the Standard Model (SM) action ceases to be valid. At such high energy scales, all forces, including gravity, are expected to be unified so that all interactions correspond to one un- derlyingsymmetry. Thus,nearPlanckianenergies,theappropriateformulationofgeometryshould bewithinaquantumframework,whilethenatureofspace-timewouldchangeinsuchawaysothat one can recover the low energy picture of diffeomorphism and internal gauge symmetries, which govern General Relativity (GR) and gauge groups on which the Standard Model is based, respectively. Apromisingattempttoobtainaquantumnatureofspace-timehasbeenrealisedwithinthe realmofNonCommutativeGeometry(NCG). NoncommutativeGeometry[1,2]isabeautifulandrichmathematicaltheory,accordingwhich geometry can be described through the functions defined on the geometry, while the geometric propertiesofspacescanbedescribedbythepropertiesoffunctionsdefinedonthespaces. Anim- portantnewfeatureofnoncommutativegeometryistheexistenceofinnerfluctuationsofthemet- ric, which correspond to the subgroup of inner automorphisms. Besides the mathematical beauty ofNCG,whichbyitselfexplainswhyonemaywanttostudythistheory, NCGoffersavarietyof phenomenologicalconsequences,whichturnthistheoryintoafertileframeworktoaddressfunda- mentalissuesofearlyuniversecosmologyandhighenergyphysicsphenomenology. InwhatfollowswewillfollowConnes’approach[1,2]andconsideramodelofatwo-sheeted spacemadefromtheproductofacontinuousspacebyadiscretespace. Thismodelledtoageomet- ricexplanationoftheStandardModel;inparticularthemodelshowsthatthevacuumExpectation ValueoftheHiggsfieldisrelatedtothenoncommutativedistancebetweenthetwosheets. Within Connes’ model of NCG, the Higgs field is conformally coupled to the Ricci curvature, while the generalised Einstein-Hilbert action contains in addition a minimally coupled massless scalar field related to the distance between the two sheets. Connes’ approach is based upon a spectral action principle, stating that the bare bosonic Euclidean action for any noncommutative model based on a product (noncommutative) space is the trace of the heat kernel associated with the square of the noncommutativeDiracoperatoroftheproductgeometry. Withinnoncommutativespectralgeometry,welookforahiddenstructureinthefunctionalof gravitycoupledtotheSMattoday’slowenergyscales,andavoidanextrapolationbymanyorders ofmagnitudetoguesstheappropriatestructureofspace-timeatPlanckianenergyscales. Noncommutative spectral geometry offers an elegant approach to unification, based on the symplecticunitarygroupinHilbertspace,ratherthanonfinitedimensionalLiegroups. Themodel offers a unification of internal symmetries with the gravitational ones. All symmetries arise as automorphisms of the noncommutative algebra of coordinates on a product geometry. Due to the lack of a full quantum gravity theory, which a priori should define the geometry of space- time at Planckian energy scales, we will follow an effective theory approach and consider the simplestcasebeyondcommutativespaces. Thus,belowbutclosetothePlanckenergyscale,space- time will be considered as the product of a Riemanian spin manifold by a finite noncommutative space. At higher energy scales, space-time should become noncommutative in a nontrivial way, 2 CosmologywithinNoncommutativeSpectralGeometry MairiSakellariadou while at energies above the Planck scale the whole concept of geometry may altogether become meaningless. As a next, but highly nontrivial, step one should consider noncommutative spaces whoselimitisthealmostcommutativespaceconsideredhere. Unfortunately,thebirthofgeometry mayremainanunsolvedpuzzleforstillquitesometime. Let me draw the attention of the reader to the fact that the noncommutative spectral geometry approach discussed here, goes beyond the noncommutative geometry notion employed in the literaturetoimplementthefuzzinessofspace-timebymeansof[xi,xj]=iθij,whereθij isananti- symmetric,real,d×d (d isthedimensionofspace-time)matrix,andxi denotespatialcoordinates. 2. ElementsofNoncommutativeSpectralGeometry To extend the Riemanian paradigm of geometry to the notion of metric on a noncommutative space, the latter should contain the Riemanian manifold with the metric tensor (as a special case), allow for departures from commutativity of coordinates as well as for quantum corrections ofgeometry,containspacesofcomplexdimension,andofferthemeansofexpressingtheStandard Model coupled to Einstein gravity as pure gravity on a suitable geometry. Noncommutative spec- tralgeometryconsiderstheSMasaphenomenologicalmodel,whichshouldhoweverdeterminethe geometry of space-time, so that the Maxwell-Dirac action functional leads to the SM action. The geometricalspaceturnsouttobetheproductofacontinuummanifoldforthespace-timegeometry andanoncommutativespacefortheinternalgeometryoftheSM. Adopting the simplest generalisation beyond commutative spaces, we consider the geometry of space-time to be described by the tensor product M ×F of a continuum compact Riemanian manifoldM andatinydiscretefinitenoncommutativespaceF composedofjusttwopoints. Itis worthnotingthatwhilethemetricdimensionofF iszero,itsK-theoreticdimensionisequalto6 modulo8. Followingthisprescription,theLagrangianoftheSM,includingmixingandMajorana masstermsforneutrinos,minimallycoupledtogravityisrecoveredfromspectralinvariantsofthe innerfluctuationsoftheproductmetriconM ×F. ThenoncommutativenatureofthediscretespaceF isgivenbyaspectraltriple(A,H ,D), where A is an involution of operators on the finite-dimensional Hilbert space H of Euclidean fermions, and D is a self-adjoint unbounded operator in H . For commutative geometries, the classical notion of a real variable is described as a real-valued function on a space, described by thecorrespondingalgebraA ofcoordinates,whichfornoncommutativegeometriesisrepresented as operators in a fixed Hilbert space H . Since real coordinates are represented by self-adjoint operators, all information about a space is encoded in the algebra of coordinates A, which is re- latedtothegaugegroupoflocalgaugetransformations. Bydroppingthecommutativityproperty, the infinitesimal line element ds, employed to define geometry through the measurement of dis- (cid:82) tances d(x,y) through d(x,y)=inf ds where the infimum is considered over all possible paths γ from point x to y, does not need to be localised. The absence of commutation of the line element with the coordinates renders possible the measurement of distances through the formula d(x,y)=sup{|f(x)−f(y)|: f ∈A,||D,f||≤1},whereD denotestheinverseofthelineelement. AssumingthealgebraA tobesymplectic-unitary,impliesA =M (H)⊕M (C),withk=2a a k and H denoting the algebra of quaternions, which plays an important rôle here and its choice re- mains to be explained; we assume quaternion linearity to obtain the SM. The choice k=4 is the 3 CosmologywithinNoncommutativeSpectralGeometry MairiSakellariadou first value that produces the correct number (k2 = 16) of fermions in each of the three generations [3], with the number of generations being a physical input. If at the Large Hadron Collider newparticlesarediscovered,onemaybeabletoaccommodatethembyconsideringahighervalue fork. WhilethechoiceofalgebraA constitutesthemaininputofthetheory,thechoiceofHilbert spaceH isirrelevant,sinceallseparableinfinite-dimensionalHilbertspacesareisomorphic. The operatorD correspondstotheinverseoftheEuclideanpropagatoroffermions,andisgivenbythe YukawacouplingmatrixwhichencodesthemassesoftheelementaryfermionsandtheKobayashi– Maskawa mixing parameters. We thus describe geometry by focusing on the Dirac operator D, insteadofthemetrictensorg ,usedforspaceswhosecoordinatescommute. Thefermionsofthe µν SMprovidetheHilbertspaceH ofaspectraltripleforthealgebraA,whilethebosonsoftheSM areobtainedthroughinnerfluctuationsoftheDiracoperatoroftheproductgeometry. Allexperimentaldataareofaspectralnature,thusouraimistoextractinformation,fromour noncommutativegeometryconstruction,whichisofaspectralnature. Luckilytheappropriatetool innoncommutativegeometryhasbeendeveloped;theSpectralActionFunctionalinnoncommuta- tivespacesistheanalogoustotheFourierTransformusedinspacescharacterisedbycommutation ofthecoordinates. ToobtainthefullLagrangianoftheSM,minimallycoupledtogravity,wewill apply the Spectral Action Principle, stating that the bosonic part of the spectral action functional depends only on the spectrum of the Dirac operator and its asymptotic expression, and for large energy Λ is of the form Tr(f(D/Λ)), with f being a cut-off function, whose choice plays only a small rôle; both D and Λ have physical dimensions of a mass and there is no absolute scale on which they can be measured. The rôle of the cut-off scale Λ is equivalent to keeping only fre- quencies smaller than the mass scale Λ. According to the spectral action principle, Tr(f(D/Λ)) is the fundamental action functional that can be used not only at the classical level but also at the quantumlevel,afterWickrotationtoEuclideansignature. Thecut-off-dependentEuclideanaction isviewed(intheWilsonianapproach)asthebareactionatmassscaleΛ. ThephysicalLagrangian has also a fermionic part, which has the simple linear form (1/2)(cid:104)Jψ,Dψ(cid:105), where J is the real structureonthespectraltripleandψ arespinorsdefinedontheHilbertspace. Tostudycosmolog- icalimplicationsofthisgravitationalmodelonemayconsideronlythebosonicpartoftheaction; thefermionicpartisimportanttodeduceparticlephysicsphenomenology. The formalism of spectral triples favours Euclidean rather than Lorentzian signature. The discussionofphenomenologicalaspectsofthetheoryreliesonaWickrotationtoimaginarytime, into the Lorentzian signature. While sensible from the phenomenological point of view, there exists as yet no justification on the level of the underlying theory. It is worth noting however that the issue of Euclidean versus Lorentzian signature is not a kind of pathology only for the case of noncommutative spectral geometry, it is for instance encountered in the nonperturbative path- integralapproachtoquantumgravity. Applying the spectral action principle to the inner fluctuations of the product M ×F of an ordinarycompactspin4-manifoldwiththefinitenoncommutativegeometry,onerecoverstheStan- dardModelactioncoupledtoEinsteinandWeylgravityplushigherordernonrenormalisableinter- actionssuppressedbypowersoftheinverseofthemassscaleofthetheory[4]. Thismodelprovides specific values of some of the SM parameters at unification scale (denoted by Λ). Following the Wilsonian approach, one can then obtain physical predictions for the SM parameters by running themdowntolow(present)energyscalesthroughtheRenormalisationGroupEquations(RGE). 4 CosmologywithinNoncommutativeSpectralGeometry MairiSakellariadou Using heat kernel methods, the trace Tr(f(D/Λ)) can be written in terms of the geometrical Seeley-deWittcoefficientsa ,whichareknownforanysecondorderellipticdifferentialoperator, n as∑∞n=0F4−nΛ4−nan ,wherethefunctionF isdefinedsuchthatF(D2)= f(D). Thus,thebosonic partofthespectralactioncanbeexpandedinpowersofΛintheform[5,6] (cid:18) (cid:18)D(cid:19)(cid:19) (cid:90) Tr f ∼ ∑ fkΛk −|D|−k+ f(0)ζD(0)+O(1). (2.1) Λ k∈DimSp Themomenta f aredefinedas f ≡(cid:82)∞ f(u)uk−1dufork>0and f ≡ f(0),thenoncommutative k k 0 0 integrationisdefinedintermsofresiduesofzetafunctionsζD(s)=Tr(|D|−s)atpolesofthezeta function,andthesumisoverpointsinthedimensionspectrumofthespectraltriple. ConsideringtheRiemaniangeometrytobefour-dimensional,theasymptoticexpansionofthe tracereads (cid:18) (cid:18) (cid:19)(cid:19) D Tr f ∼2Λ4f a +2Λ2f a + f a +···+Λ−2kf a +··· . (2.2) 4 0 2 2 0 4 −2k 4+2k Λ Thesmoothevenfunction f,whichdecaysfastatinfinity,onlyentersinthemultiplicativefactors: (cid:90) ∞ (cid:90) ∞ f = f(u)u3du , f = f(u)udu, 4 2 0 0 k! f = f(0) , f =(−1)k f(2k)(0). (2.3) 0 −2k (2k)! Since f is taken as a cut-off function, its Taylor expansion at zero vanishes, thus its asymptotic expansion,Eq.(2.2),reducestojust (cid:18) (cid:18) (cid:19)(cid:19) D Tr f ∼2Λ4f a +2Λ2f a + f a . (2.4) 4 0 2 2 0 4 Λ Thecut-offfunctionentersthroughitsmomenta f ,f ,f ;thesethreeadditionalrealparame- 0 2 4 tersarephysicallyrelatedtothecouplingconstantsatunification,thegravitationalconstantandthe cosmologicalconstant. Inthefour-dimensionalcase,theterminΛ4inthespectralaction,Eq.(2.1), givesacosmologicalterm,theterminΛ2givestheEinstein-Hilbertactionfunctionalwiththephys- icalsignfortheEuclideanfunctionalintegral(provided f >0),andtheΛ-independenttermyields 2 the Yang-Mills action for the gauge fields corresponding to the internal degrees of freedom of the metric. The scale-independent terms in the spectral action have conformal invariance. Note that thearbitrarymassscaleΛcanbemadedynamicalbyintroducingascalingdilatonfield. SincethephysicalLagrangianisentirelydeterminedbythegeometricinput,thephysicalim- plicationsofthisapproacharecloselydependentontheunderlyingchosengeometry. Theobtained physical Lagrangian contains, in addition to the full Standard Model Lagrangian, the Einstein- Hilbert action with a cosmological term, a topological term related to the Euler characteristic of the space-time manifold, a conformal Weyl term and a conformal coupling of the Higgs field to gravity. ThebosonicactioninEuclideansignaturereads[4] (cid:90) (cid:18) 1 1 1 SE= R+α C Cµνρσ+γ +τ R(cid:63)R(cid:63) + Gi Gµνi+ Fα Fµνα 2κ2 0 µνρσ 0 0 4 µν 4 µν 0 +1BµνB +1|D H|2−µ2|H|2−ξ R|H|2+λ |H|4(cid:1)√gd4x, (2.5) 4 µν 2 µ 0 0 0 5 CosmologywithinNoncommutativeSpectralGeometry MairiSakellariadou where 12π2 3f κ2= , α =− 0 , 0 96f Λ2− f c 0 10π2 2 0 (cid:18) (cid:19) 1 f 11f γ = 48f Λ4− f Λ2c+ 0d , τ = 0 , 0 π2 4 2 4 0 60π2 f e 1 π2b µ2=2Λ2 2 − , ξ = , λ = ; (2.6) 0 f a 0 12 0 2f a2 0 0 √ H is a rescaling H = ( af /π)φ of the Higgs field φ to normalize the kinetic energy, and the 0 momentum f is physically related to the coupling constants at unification. Notice the absence 0 of quadratic terms in the curvature; there is only the term quadratic in the Weyl curvature and thetopologicaltermR(cid:63)R(cid:63). Inacosmologicalsetting,namelyforFriedmann-Lemaître-Robertson- Walker(FLRW)geometries,theWeyltermvanishes. Noticealsothetermthatcouplesgravitywith theSM,atermwhichshouldalwaysbepresentwhenoneconsidersgravitycoupledtoscalarfields. Writing the asymptotic expansion of the spectral action, a number of geometric parameters appeared, which describe the possible choices of Dirac operators on the finite noncommutative space. These parameters correspond to the Yukawa parameters of the particle physics model and theMajoranatermsfortheright-handedneutrinos. Theyaregivenby[4] (cid:16) (cid:16) (cid:17)(cid:17) a = Tr Y(cid:63) Y +Y(cid:63) Y +3 Y(cid:63) Y +Y(cid:63) Y , (↑1) (↑1) (↓1) (↓1) (↑3) (↑3) (↓3) (↓3) (cid:18) (cid:19) (cid:16) (cid:17)2 (cid:16) (cid:17)2 (cid:16) (cid:17)2 (cid:16) (cid:17)2 b = Tr Y(cid:63) Y + Y(cid:63) Y +3 Y(cid:63) Y +3 Y(cid:63) Y , (↑1) (↑1) (↓1) (↓1) (↑3) (↑3) (↓3) (↓3) c = Tr(Y(cid:63)Y ) , R R (cid:16) (cid:17) d = Tr (Y(cid:63)Y )2 , R R (cid:16) (cid:17) e = Tr Y(cid:63)Y Y(cid:63) Y , (2.7) R R (↑1) (↑1) withY ,Y ,Y ,Y andY being (3×3) matrices, withY symmetric. TheY matrices are (↓1) (↑1) (↓3) (↑3) R R used to classify the action of the Dirac operator and give the fermion and lepton masses, as well as lepton mixing, in the asymptotic version of the spectral action. The Yukawa parameters run with the RGE of the particle physics model. Since running towards lower energies implies that nonperturbativeeffectsinthespectralactioncannotbeanylongerneglected, anyresultsbasedon the asymptotic expansion and on renormalisation group analysis can only hold for early universe cosmology. Hence, the spectral action at the unification scale Λ offers a framework to investigate early universe cosmological models [7, 8, 9, 10, 11, 12]. For later times, one should consider the fullspectralaction,adirectionwhichrequiresthedevelopmentofnontrivialmathematicaltools. It is important to emphasise that the relations given in Eq. (2.6) above are tied to the scale at which the expansion is performed. There is a priori no reason for the constraints to hold at scales belowtheunificationscaleΛ,sincetheyrepresentmereboundaryconditions. Oneshouldtherefore beverycarefulandkeepinmindthatitisincorrecttoconsidertherelationsinEq.(2.6)asfunctions oftheenergyscale;theserelationsareonlyvalidatunificationscaleΛ. 6 CosmologywithinNoncommutativeSpectralGeometry MairiSakellariadou 3. HighEnergyPhenomenologyoftheNoncommutativeSpectralGeometry In what follows we assume that the function f is well approximated by the cut-off function andignorehigherorderterms. Normalisationofthekinetictermsimplies g2f 1 5 3 0 = and g2=g2= g2 , 2π2 4 3 2 3 1 while 3 sin2θ = ; (3.1) W 8 a relation which was also found for SU(5) and SO(10). Assuming the big desert hypothesis, the runningofthecouplingsα =g2/(4π)withi=1,2,3canthenbeobtainedviatheRGE. i i Thephenomenologicalconsequencesofthenoncommutativespectralgeometryasanapproach tounificationhavebeendiscussedinRef.[4],wheretheauthorsconsideredonlyone-loopcorrec- tions,forwhichtheβ-functionsareβ =(4π)−2bg3withi=1,2,3andb=(41/6,−19/6,−7). It gi i i isworthnoticingthatonlyatone-loopordertheRenormalisationGroupEquationsforthecoupling constantsg areuncoupledfromtheotherStandardModelparameters. i Performingone-loopRGEfortherunningofthegaugecouplingsandtheNewtonconstant,it was shown [4] that these do not meet at a point, the error being within just few percent. The fact that the predicted unification of the coupling constants does not hold exactly, implies that the big desert hypothesis is only approximately valid and new physics are expected between unification andpresentenergyscales. Intermsofourassumptionforthecut-offfunction,thelackofaunique unification energy, implies that even though the function f can be approximated by the cut-off function,thereexistsmalldeviations. Besidesthisresultitishoweverworthnotingthatthemodel leadstothecorrectrepresentationsofthefermionswithrespecttothegaugegroupoftheSM,the Higgs doublet appears as part of the inner fluctuations of the metric, and Spontaneous Symmetry Breakingmechanismarisesnaturallywiththenegativemasstermwithoutanytuning. Inaddition, the see-saw mechanism is obtained, the 16 fundamental fermions are recovered, and a top quark massofM ∼179GeVispredicted. top The model predicts a heavy Higgs mass; in zeroth order approximation, it predicts a mass of the Higgs boson approximately equal to 170 GeV, which strictly speaking is ruled out by cur- rentexperimentaldata. DuetothisdiscrepancybetweentheNCGpredictionandtheexperimental data,noncommutativespectralgeometryhasbeen(ratherunfairly)criticised,eventhoughtheresult quoted above depends on the value of the gauge couplings at unification scale, which is uncertain andwasobtainedneglectingthenonminimalcouplingbetweentheHiggsfieldandtheRiccicurva- ture. Ibelievethatoneshouldinsteaddrawtheconclusionthatnoncommutativespectralgeometry as an approach to unification, even in its present (and certainly simple) version, it still led to the correctorderofmagnitudefortheHiggsmass,aresultwhichwasbynomeansobvious. ConsideringanenergyscaleΛ∼1.1×1017GeV,thestandardformofthegravitationalaction andtheexperimentalvalueofNewton’sconstantatordinaryscalesimplyκ−1∼2.43×1018 GeV. 0 Finally,thisapproachtounificationdoesnotprovideanyexplanationofthenumberofgener- ations,norleadstoconstraintsonthevaluesoftheYukawacouplings. 7 CosmologywithinNoncommutativeSpectralGeometry MairiSakellariadou 4. CosmologicalConsequencesoftheNoncommutativeSpectralGeometry The Lorentzian version of the gravitational part of the asymptotic formula for the bosonic sectoroftheactionobtainedwithinnoncommutativespectralgeometryreads[4] (cid:90) (cid:18) 1 (cid:19)√ SL = R+α C Cµνρσ+τ R(cid:63)R(cid:63)ξ R|H|2 −gd4x. (4.1) grav 2κ2 0 µνρσ 0 0 0 Thisactionleadstothefollowingequationsofmotion[7] 1 1 (cid:104) (cid:105) Rµν− gµνR+ δ 2Cµλνκ+CµλνκR =κ2δ Tµν , 2 B2 cc ;λ;κ λκ 0 cc matter wherewehaveintroducedB2≡−(4κ2α )−1,relatedtothe f momentofthecut-offfunctionand 0 0 0 wehavecapturedthenonminimalcouplingbetweentheHiggsfieldandtheRiccicurvaturescalar intheparameterδ ,definedasδ ≡[1−2κ2ξ H2]−1. cc cc 0 0 In the low energy weak curvature regime, the nonminimal coupling between the background geometry and the Higgs field can be neglected, thus δ =1. For a FLRW space-time, the Weyl cc tensorvanishes,hencetheNCGcorrectionstotheEinsteinequationvanish[7],rendingdifficultto restrictBviacosmologyorsolar-systemtests. ImposingalowerlimitonBwouldimplyanupper limittothemoment f ,andthusrestrictparticlephysicsatunification. 0 Onecanimposeanupperlimittothemoment f , bystudyingtheenergylosttogravitational 0 radiationbyorbitingbinaries[11,12]. ConsideringlinearperturbationsaroundaMinkowskiback- groundmetric,theequationsofmotionread[12] (cid:0)(cid:3)−B2(cid:1)(cid:3)hµν =B216πGTµν , (4.2) c4 matter where Tµν is taken to lowest order in hµν. Since B plays the rôle of a mass, it must be real and matter positive,thusα mustbenegativeforMinkowskispacetobeastablevacuumofthetheory. 0 Consider the energy lost to gravitational radiation by orbiting binaries. In the far field limit, |r|≈|r−r(cid:48)|(randr(cid:48)denotethelocationsofobserverandemitter,respectively),thespatialcompo- nentsofthegeneralfirstordersolutionforaperturbationagainstaMinkowskibackgroundread[12] hik(r,t)≈ 2GB(cid:90) t−1c|r| dt(cid:48) J (cid:18)B(cid:113)c2(t−t(cid:48))2−|r|2(cid:19)Dïk(cid:0)t(cid:48)(cid:1) ; (4.3) 3c4 (cid:113) 1 −∞ c2(t−t(cid:48))2−|r|2 Dik isthequadrupolemoment,definedasDik(t)≡ 3 (cid:82) xixkT00(r,t)dr,andJ aBesselfunction c2 1 of the first kind. While for B→∞ the theory reduces to GR, for finite B gravitational radiation containsmassiveandmasslessmodes,bothsourcedfromthequadrupolemomentofthesystem. Forabinarypairofmassesm ,m incircularorbitinthe(xy)-plane,therateofenergylossis 1 2 dE c2 − ≈ |r|2h˙ h˙ij , (4.4) ij dt 20G with[12] 128µ2|ρ|4ω6G2B2 (cid:20) (cid:18) 2ω(cid:19) (cid:18) 2ω(cid:19)(cid:21) h˙ijh˙ = × f2 B|r|, + f2 B|r|, , (4.5) ij c8 c Bc s Bc 8 CosmologywithinNoncommutativeSpectralGeometry MairiSakellariadou (cid:90) ∞ ds (cid:16) (cid:112) (cid:17) f (x,z) ≡ √ J (s)sin z s2+x2 , (4.6) s 1 0 s2+x2 (cid:90) ∞ ds (cid:16) (cid:112) (cid:17) f (x,z) ≡ √ J (s)cos z s2+x2 . (4.7) c 1 0 s2+x2 The orbital frequency ω, defined in terms of the magnitude |ρ| of the separation vector between (cid:112) thetwobodies,isconstantandequaltoω =|ρ|−3/2 G(m +m ). 1 2 The integrals in Eqs. (4.6), (4.7), exhibit a strong resonance behavior at z=1, which corre- spondstothecriticalfrequency[12] 2ω =Bc, (4.8) c aroundwhichstrongdeviationsfromtheGRresultsareexpected. Thismaximumfrequencyresults fromthenaturallengthscale,givenbyB−1,atwhichNCGeffectsbecomedominant. Thereareseveralbinarypulsarsforwhichtherateofchangeoftheorbitalfrequencyhasbeen wellcharacterised,andthepredictionsofGeneralRelativityagreewiththedatatoahighaccuracy. RequiringthemagnitudeofdeviationsfromGRobtainedinthecontextofnoncommutativespectral geometry,tobelessthanthealloweduncertaintyinthedata,oneconstrainsB,namely[11] B>7.55×10−13 m−1 . (4.9) Thisobservationalconstraintmayseemweak,howeveritiscomparabletoexistingconstraints on similar, ad hoc, additions to GR. In particular, constraints on additions to the Einstein-Hilbert action, of the form R2 and R Rµν, are of the order of B ≥3.2×10−9m−1, where B is the µν R2 R2 B parameter associated with the couplings of these terms [13]. Note also that since the strongest constraintcomesfromsystemswithhighorbitalfrequencies,futureobservationsofrapidlyorbiting binaries,relativelyclosetotheEarth,couldimproveitbymanyordersofmagnitude. Remaininginthelow-energylimit,inotherwordsneglectingthecouplingbetweentheHiggs field and the background geometry, on may consider the corrections to the background Einstein’s equations. It turns out that noncommutative corrections do not occur at the level of a FLRW background, since then the modified Friedmann equation reduces to its standard form [7]. One may have naively claimed that this was expected, arguing that in a spatially homogeneous space- timethespatialpointsareequivalentandanynoncommutativeeffectsarethenexpectedtovanish. However, this argument does not apply here, since the noncommutativity is incorporated in the internalmanifoldF andthespace-timeisacommutativefour-dimensionalmanifold. Neglecting the nonminimal coupling between the Higgs field and the Ricci curvature, any modifications to the background equation will be apparent at leading order for anisotropic and inhomogeneousmodels. Letusconsider therepresentativeexampleof Bianchitype-Vmodel, for whichthespace-timemetric,inCartesiancoordinates,reads g =diag(cid:2)−1,{a (t)}2e−2nz,{a (t)}2e−2nz,{a (t)}2(cid:3) ; (4.10) µν 1 2 3 a(t)withi=1,2,3arbitraryfunctionsandnisaninteger. i 9 CosmologywithinNoncommutativeSpectralGeometry MairiSakellariadou DefiningA (t)=lna (t)withi=1,2,3,themodifiedFriedmannequationreads[7]: i i κ2T = 0 00 −A˙ (cid:0)A˙ +A˙ (cid:1)−n2e−2A3(cid:0)A˙ A˙ −3(cid:1) 3 1 2 1 2 +8α0κ02n2e−2A3(cid:104)5(cid:0)A˙ (cid:1)2+5(cid:0)A˙ (cid:1)2−(cid:0)A˙ (cid:1)2 −A˙ A˙ −A˙ A˙ −A˙ A˙ −A¨ −A¨ −A¨ +3(cid:3) 1 2 3 1 2 2 3 3 1 1 2 3 3 (cid:40) −4α0κ02∑ A˙ A˙ A˙ A˙ +A˙ A˙ (cid:16)(cid:0)A˙ −A˙ (cid:1)2−A˙ A˙ (cid:17) 1 2 3 i i i+1 i i+1 i i+1 3 i (cid:20) (cid:21) +(cid:16)A¨ +(cid:0)A˙ (cid:1)2(cid:17) −A¨ −(cid:0)A˙ (cid:1)2+1(cid:0)A¨ +A¨ (cid:1) +1(cid:16)(cid:0)A˙ (cid:1)2+(cid:0)A˙ (cid:1)2(cid:17) i i i i i+1 i+2 i+1 i+2 2 2 (cid:41) (cid:104)... (cid:16) (cid:17) (cid:105) + A +3A˙ A¨ − A¨ +(cid:0)A˙ (cid:1)2 (cid:0)A˙ −A˙ −A˙ (cid:1) (cid:2)2A˙ −A˙ −A˙ (cid:3) (4.11) i i i i i i i+1 i+2 i i+1 i+2 withi=1,2,3,whilethet-dependenceofthetermshasbeenomittedforsimplicity. Any term containing α in Eq. (4.11), encodes a modification from the standard result. Let 0 us study Eq. (4.11) above. The correction terms can be divided into two types. The first one containsthetermsinbracesinEq.(4.11),whicharefourthorderintimederivatives. Thus,forthe slowlyvaryingfunctions,usuallyconsideredincosmology,thesecorrectionscanbeneglected. The secondtype,whichappearsinthethirdlineinEq.(4.11),occursatthesameorderasthestandard Einstein-Hilbertterms. However,sincethistermisproportionalton2,itvanishesforhomogeneous versions of Bianchi type-V. Thus, although anisotropic cosmologies do contain corrections due to the additional NCG terms in the action, they are typically of higher order [7]. Inhomogeneous modelsdocontaincorrectiontermsthatappearonthesamefootingastheoriginalterms. ItisworthnotingthatbystudyingthecaseoftheBianchitypeVmodel,onecanidentifythe noncommutative geometry effects in other cases of cosmological models, for instance Bianchi I and Kasner models. In conclusion, the corrections to Einstein’s equations can only be important forinhomogeneousandanisotropicspace-times[7]. Certainly, the coupling between the Higgs field and the background geometry cannot be ne- glectedoncetheenergiesreachtheHiggsscale. ThenonminimalcouplingofHiggsfieldtocurva- tureleadstocorrectionstoEinstein’sequationsevenforhomogeneousandisotropiccosmological models. ToillustratetheeffectsofthesecorrectionsletusneglecttheconformalterminEq.(4.2), sothattheequationsofmotionread[7] (cid:20) (cid:21) 1 1 Rµν− gµνR=κ2 Tµν , (4.12) 2 0 1−κ2|H|2/6 matter 0 implyingthat|H|playstherôleofaneffectivegravitationalconstant[7]. ThenonminimalcouplingbetweentheHiggsfieldandtheRiccicurvaturemayturnouttobe crucial in early universe cosmology [8, 10]. Such a coupling has been introduced ad hoc in the literature, in an attempt to drive inflation through the Higgs field, and thus cure one of the main pathologiesoftheinflationaryparadigm,namelytheoriginoftheinflatonfield. However,thevalue ofthecouplingconstantbetweenthescalarfieldandthebackgroundgeometryshouldbedictated bytheunderlyingtheory. Actually,evenifclassicallythecouplingbetweentheHiggsfieldandthe 10