Cosmological consequences of the noncommutative spectral geometry as an approach to unification
aa r X i v : . [ h e p - t h ] O c t Cosmological consequences of the noncommutativespectral geometry as an approach to unification
Mairi Sakellariadou
Department of Physics, King’s College, University of London, Strand WC2R 2LS, London,U.K.E-mail: [email protected]
Abstract.
Noncommutative spectral geometry succeeds in explaining the physics of theStandard Model of electroweak and strong interactions in all its details as determined byexperimental data. Moreover, by construction the theory lives at very high energy scales,offering a natural framework to address early universe cosmological issues. After introducing themain elements of noncommutative spectral geometry, I will summarise some of its cosmologicalconsequences and discuss constraints on the gravitational sector of the theory.
1. Introduction
Approaching Planckian energies, gravity can no longer be considered as a classical theory, thequantum nature of space-time becomes apparent and geometry can no longer be described interms of Riemannian geometry and General Relativity. At such energy scales, all forces shouldbe unified so that all interactions correspond to one underlying symmetry. The nature of space-time would change in such a way, so that its low energy limit implies the diffeomorphism andinternal gauge symmetries, which govern General Relativity and gauge groups on which theStandard Model is based, respectively. Thus, near Planckian energies one should search for aformulation of geometry within the quantum framework. Such an attempt has been realisedwithin NonCommutative Geometry (NCG) [1, 2]. Even though this approach is still in termsof an effective theory, it has already led to very encouraging results and is now at a stage to beconfronted with experimental and observational data.Since all physical data are of a spectral nature, we will use the notion of a spectral triple,which is analogous to Fourier transform we are familiar with in commutative spaces. Distancesare measured in units of wavelength of atomic spectra, while the notion of a real variable, takenas a function on a set X , will be replaced by a self-adjoint operator D in a Hilbert space. Thespace X will be described by the algebra A of coordinates, represented as operators in a fixedHilbert space H . Thus, the usual emphasis on the points of a geometric space is replaced by thespectrum of an operator and the geometry of a noncommutative space is determined in termsof the spectral triple ( A , H , D ).Noncommutative geometry spectral action, in its current version which is still at the classicallevel and it certainly considers almost commutative spaces, it nevertheless offers [3] an elegantgeometric interpretation of the most successful phenomenological model of particle physics,namely the Standard Model (SM). Certainly if the Large Hadron Collider (LHC) supportsevidence for physics beyond the SM, one should consider extensions of the effective theorye will adopt here. Closer to the Planck era one should consider the full, still unknown,theory, while at even higher energy scales the whole concept of geometry may lose its familiarmeaning. Nevertheless, the effective theory we will consider here lives, by construction, atvery high energies, offering an excellent framework to address open questions of early universecosmology [4].In what follows, I will briefly review the main elements of the NCG spectral action as anapproach to unification and then discuss some of its cosmological consequences [4, 5, 6, 7, 8, 9].
2. NCG spectral action
In the context of noncommutative geometry, all information about a physical system is containedwithin the algebra of functions, represented as operators in a Hilbert space, while the actionand metric properties are encoded in a generalised Dirac operator. The geometry is specified bya spectral triple ( A , H , D ), defined by an algebra A , a Hilbert space H and a generalised Diracoperator D .Adopting the simplest generalisation beyond commutative spaces, we take the extension ofour smooth four-dimensional manifold, M , by taking its product with a discrete noncommutingmanifold F . Thus, M describes the geometry for the four-dimensional space-time, whilethe noncommutative space F specifies the internal geometry for the SM and is composed ofjust two points. Considering the Standard Model of electroweak and strong interactions as aphenomenological model, we will look for a geometry, such that the associated action functionalleads to the SM with all specifications as determined by experimental data.Within NCG, a space described by the algebra of real coordinates is represented by self-adjoint operators on a Hilbert space. Since real coordinates are represented by self-adjointoperators, all information about a space is encoded in the algebra of coordinates A , which isthe main input of the theory. Under the assumption that the algebra constructed in M × F issymplectic-unitary, it turns out that A must be of the form A = M a ( H ) ⊕ M k ( C ) , (1)with k = 2 a and H denoting the algebra of quaternions. The choice k = 4 is the first value thatproduces the correct number of fermions, namely k = 16, in each of the three generations [10].The number of generations is a physical input. Certainly if at LHC new particles are discovered,one may be able to accommodate them by including a higher value for the even number k .The operator D corresponds to the inverse of the Euclidean propagator of fermions, and isgiven by the Yukawa coupling matrix which encodes the masses of the elementary fermions andthe Kobayashi–Maskawa mixing parameters. The fermions of the SM provide the Hilbert space H of a spectral triple for the algebra A , while the bosons of the SM are obtained through innerfluctuations of the Dirac operator of the product M × F geometry.The spectral action principle states that the bosonic part of the spectral functional S dependsonly on the spectrum of the Dirac operator and its asymptotic expression, and for large energyΛ is of the form Tr( f ( D/ Λ)), with f being a cut-off function, whose choice plays only a smallrˆole. The physical Lagrangian has also a fermionic part, which has the simple linear form(1 / h J ψ, Dψ i , where J is the real structure on the spectral triple and ψ are spinors defined onthe Hilbert space [11]. Applying the spectral action principle to the inner fluctuations of theproduct geometry M× F , one recovers the Standard Model action coupled to Einstein and Weylgravity plus higher order nonrenormalisable interactions suppressed by powers of the inverse ofthe mass scale of the theory [3].To study the implications of this noncommutative approach coupled to gravity for thecosmological models of the early universe, one can concentrate just on the bosonic part ofthe action; the fermionic part is however crucial for the particle physics phenomenology of themodel.sing the heat kernel method, the bosonic part of the spectral action can be expanded inpowers of the scale Λ in the form [3, 12, 13]Tr (cid:18) f (cid:18) D Λ (cid:19)(cid:19) ∼ X k ∈ DimSp f k Λ k Z −| D | − k + f (0) ζ D (0) + O (1) , (2)with the momenta f k of the cut-off function f given by f k ≡ Z ∞ f ( u ) u k − d u for k > , and f ≡ f (0) ; (3)the noncommutative integration is defined in terms of residues of zeta functions, ζ D ( s ) =Tr( | D | − s ) at poles of the zeta function, and the sum is over points in the dimension spectrum of the spectral triple.In this way, one obtains a Lagrangian which contains in addition to the full SM Lagrangian,the Einstein-Hilbert action with a cosmological term, a topological term related to the Eulercharacteristic of the space-time manifold, a conformal Weyl term and a conformal coupling ofthe Higgs field to gravity. Writing the asymptotic expansion of the spectral action, a number ofgeometric parameters appear, which describe the possible choices of Dirac operators on the finitenoncommutative space. These parameters correspond to the Yukawa parameters of the particlephysics model and the Majorana terms for the right-handed neutrinos. The Yukawa parametersrun with the Renormalisation Group Equations (RGE) of the particle physics model. Sincerunning towards lower energies, implies that nonperturbative effects in the spectral action cannotbe any longer neglected, any results based on the asymptotic expansion and on renormalisationgroup analysis can only hold for early universe cosmology. For later times, one should insteadconsider the full spectral action.More precisely, the bosonic action in Euclidean signature reads [3] S E = Z (cid:18) κ R + α C µνρσ C µνρσ + γ + τ R ⋆ R ⋆ + 14 G iµν G µνi + 14 F αµν F µνα + 14 B µν B µν + 12 | D µ H | − µ | H | − ξ R | H | + λ | H | (cid:1) √ g d x , (4)where κ = 12 π f Λ − f c ,α = − f π ,γ = 1 π (cid:18) f Λ − f Λ c + f d (cid:19) ,τ = 11 f π ,µ = 2Λ f f − ea ,ξ = 112 ,λ = π b f a , (5)ith a , b , c , d , e given by [3] a = Tr (cid:16) Y ⋆ ( ↑ Y ( ↑ + Y ⋆ ( ↓ Y ( ↓ + 3 (cid:16) Y ⋆ ( ↑ Y ( ↑ + Y ⋆ ( ↓ Y ( ↓ (cid:17)(cid:17) , b = Tr (cid:18)(cid:16) Y ⋆ ( ↑ Y ( ↑ (cid:17) + (cid:16) Y ⋆ ( ↓ Y ( ↓ (cid:17) + 3 (cid:16) Y ⋆ ( ↑ Y ( ↑ (cid:17) + 3 (cid:16) Y ⋆ ( ↓ Y ( ↓ (cid:17) (cid:19) , c = Tr ( Y ⋆R Y R ) , d = Tr (cid:16) ( Y ⋆R Y R ) (cid:17) , e = Tr (cid:16) Y ⋆R Y R Y ⋆ ( ↑ Y ( ↑ (cid:17) , (6)with Y ( ↓ , Y ( ↑ , Y ( ↓ , Y ( ↑ and Y R being (3 ×
3) matrices, with Y R symmetric; the Y matricesare 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. Note that H is arescaling of the Higgs field, so that the kinetic terms are normalised. One should be cautiousthat the relations in Eq. (5) are only valid at unification scale Λ; it is incorrect to consider themas functions of the energy scale.The noncommutative spectral geometry approach leads to various phenomenologicalconsequences. Normalisation of the kinetic terms implies g f π = 14 and g = g = 53 g , while sin θ W = 38 ; (7)a relation which holds also for SU(5) and SO(10), while assuming the big desert hypothesis,one can find the running of the three couplings α i = g i / (4 π ). One-loop RGE for the runningof the gauge couplings and the Newton constant, shows that they do not meet exactly at onepoint, the error is though within just few percent. Therefore, the model in its simplified form,does not specify a unique unification energy, it however leads to the correct representations ofthe fermions with respect to the gauge group of the SM, the Higgs doublet appears as part ofthe inner fluctuations of the metric, and Spontaneous Symmetry Breaking mechanism arisesnaturally with the negative mass term without any tuning. Moreover, the see-saw mechanism isobtained, the 16 fundamental fermions are recovered, and a top quark mass of M top ∼
179 Gev ispredicted. The mass of Higgs in zeroth order approximation of the spectral action is ∼ a priori .
3. Cosmological consequences
To use the formalism of spectral triples in NCG, it is convenient to work with Euclidean ratherthan Lorentzian signature. Thus, the analysis of the cosmological consequences of the theoryrelies on a Wick rotation to imaginary time, into the Lorentzian signature.The Lorentzian version of the gravitational part of the asymptotic formula for the bosonicsector of the NCG spectral action, including the coupling between the Higgs field and the Riccicurvature scalar, reads [3] S Lgrav = Z (cid:18) κ R + α C µνρσ C µνρσ + τ R ⋆ R ⋆ ξ R | H | (cid:19) √− g d x , (8)eading to the equations of motion [5] R µν − g µν R + 1 β δ cc h C µλνκ ; λ ; κ + C µλνκ R λκ i = κ δ cc T µν matter , with β ≡ − κ α and δ cc ≡ [1 − κ ξ H ] − . (9)In the low energy weak curvature regime, the nonminimal coupling between the backgroundgeometry and the Higgs field can be neglected, implying δ cc = 1. For a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) space-time, the Weyl tensor vanishes, hence the NCG correctionsto the Einstein equation vanish [5], rending difficult to restrict β (or equivalently α , or f )via cosmology or solar-system tests. Imposing however a lower limit on β is very important,since it implies an upper limit to the moment f , corresponding to a restriction on the particlephysics at unification. This has been achieved in Refs. [8, 9], by considering the energy lost togravitational radiation by orbiting binaries.Deriving the weak field limit of noncommutative spectral geometry, we have shown [8] thatthe production and dynamics of gravitational waves are significantly altered and, in particular,the graviton contains a massive mode that alters the energy lost to gravitational radiation, insystems with evolving quadrupole moment. Considering the rate of energy loss from a binarypair of masses m , m , in the far field limit, we have shown [8, 9] that the orbital frequency ω = | ρ | − / p G ( m + m ) , (10)where | ρ | stands for the magnitude of their separation vector, has a critical value2 ω c = βc , (11)around which strong deviations from the familiar results of General Relativity are expected.This maximum frequency results from the natural length scale, given by β − , at whichnoncommutative geometry effects become dominant.The form of the gravitational radiation from binary systems can be used to constrain β . Thereare several binary pulsars for which the rate of change of the orbital frequency has been wellcharacterised, and the predictions of General Relativity agree with the data to high accuracy.Thus, on can restrict the parameter β by requiring that the magnitude of deviations from GeneralRelativity be less than this uncertainty. Requiring that β > ω/c , we have found [9] β > . × − m − . (12)Since the strongest constraint comes from systems with high orbital frequencies, one expectsthat future observations of rapidly orbiting binaries, relatively close to the Earth, could improveit by many orders of magnitude.Considering the background equations, the corrections to Einstein’s equations can only beapparent at leading order for anisotropic models. Calculating the modified Friedmann equationfor the Bianchi type-V model, we have shown [5] that the correction terms come in two types.The first one contains terms which are fourth order in time derivatives, hence for the slowlyvarying functions usually used in cosmology they can be neglected. The second one occurs at thesame order as the standard Einstein-Hilbert terms, however, it vanishes for homogeneous versionsof Bianchi type-V. Thus, although anisotropic cosmologies do contain corrections due to theadditional NCG terms in the action, they are typically of higher order. Inhomogeneous models doontain correction terms that appear on the same footing as the ordinary (commutative) terms.In conclusion, the corrections to Einstein’s equations can only be important for inhomogeneousand anisotropic space-times.Certainly one cannot always neglect the coupling of the Higgs field to the curvature. Namely,as energies approach the Higgs scale, this nonminimal coupling can no longer be neglected,leading to corrections even for background cosmologies. To understand the effects of thesecorrections let us neglect the conformal term in Eq. (9), so that the equations of motion read [5] R µν − g µν R = κ (cid:20) − κ | H | / (cid:21) T µν matter . (13)Thus, | H | plays the rˆole of an effective gravitational constant [5].The nonminimal coupling between the Higgs field and the Ricci curvature may turn out tobe particularly useful in early universe cosmology [6, 7]. Such a coupling has been introduced ad hoc in the literature, in an attempt to drive inflation through the Higgs field. However, thevalue of the coupling constant between the scalar field and the background geometry should bedictated by the underlying theory.In a FLRW metric, the Gravity-Higgs sector of the asymptotic expansion of the spectralaction, in Lorentzian signature, reads S LGH = Z h − κ ξ H κ R −
12 ( ∇ H ) − V ( H ) i √− g d x , (14)where V ( H ) = λ H − µ H , (15)with µ and λ subject to radiative corrections as functions of energy. For large enough valuesof the Higgs field, the renormalised value of these parameters must be calculated, while therunning of the top Yukawa coupling and the gauge couplings must be evolved simultaneously.At high energies the mass term is sub-dominant, and can be neglected. For each value ofthe top quark mass, there is a value of the Higgs mass where the effective potential is about todevelop a metastable minimum at large values of the Higgs field and the Higgs potential is locallyflattened [7]. Since the region where the potential is flat is narrow, the slow-roll must be veryslow in order to provide a sufficiently long period of quasi-exponential expansion. Besides theslow-roll parameters, denoted by ǫ and η , which may be slow enough to allow sufficient numberof e-folds, the amplitude of density perturbations ∆ R in the Cosmic Microwave Backgroundmust be within the allowed experimental window. Inflation predicts that at horizon crossing(denoted by stars), the amplitude of density perturbations is related to the inflaton potentialthrough (cid:18) V ∗ ǫ ∗ (cid:19) = 2 √ π m Pl ∆ R , (16)where ǫ ∗ ≤
1. Its value, as measured by WMAP7 [14], requires (cid:18) V ∗ ǫ ∗ (cid:19) = (2 . ± . × − m Pl , (17)where m Pl stands for the Planck mass.Calculating [7] the renormalisation of the Higgs self-coupling up to two-loops, we haveconstructed an effective potential which fits the renormalisation group improved potential aroundhe flat region. We have found [7] a very good analytic fit to the Higgs potential around theminimum of the potential: V eff = λ eff0 ( H ) H = [ a ln ( bκH ) + c ] H , (18)where the parameters a, b are related to the low energy values of top quark mass m t as [7] a ( m t ) = 4 . × − − . × − (cid:16) m t GeV (cid:17) + 1 . × − (cid:16) m t GeV (cid:17) ,b ( m t ) = exp h − . (cid:16) m t GeV − . (cid:17)i . (19)The third parameter, c , encodes the appearance of an extremum and depends on the values fortop quark mass and Higgs mass. An extremum occurs if and only if c/a ≤ /
16, the saturationof the bound corresponding to a perfectly flat region. It is convenient to write c = [(1 + δ ) / a ,where δ = 0 saturates the bound below which a local minimum is formed.This analysis was performed in the case of minimal coupling, so let us investigate themodifications introduced in the case of a small nonminimal coupling; within NCG the couplingis ξ = 1 /
12. We have found [7] that the induced corrections to the potential imply that flatnessdoes not occur at δ = 0, but for fixed values of δ depending on the value of the top quarkmass. Thus, for inflation to occur via the Higgs field, the top quark mass fixes the Higgs massextremely accurately. Scanning carefully through the parameter space, we concluded [7] thatsufficient e -folds are indeed generated provided a suitably tuned relationship between the topquark mass and the Higgs mass holds. However, while the Higgs potential can lead to the slow-roll conditions being satisfied once the running of the self-coupling at two-loops is included, theconstraints imposed from the CMB data make the predictions of such a scenario incompatiblewith the measured value of the top quark mass [7]. Running of the gravitational constant andcorrections by considering the more appropriate de Sitter, instead of a Minkowski, backgroundwe found [7] that do not improve substantially the realisation of a successful inflationary era.
4. Conclusions
Noncommutative spectral geometry is a beautiful mathematical construction with a richphenomenological arena. In its present simple form, which still remains classical and refersonly to almost commutative spaces, it offers an elegant explanation for the Standard Model ofelectroweak and strong interactions. The prediction for the top quark mass is in agreement withcurrent experimental data, while the Higss mass is within the correct order of magnitude; itsprecise value is excluded from the most recent experimental data but it is still remarkable howclose it remains to the experimental allowed value besides the simplifications under which it wascalculated.Noncommutative spectral geometry lives by construction at very high energy scales, offering anatural set-up to study early universe cosmology. Late time astrophysics is a more difficult taskdue to technical issues at the current stage of the NCG spectral action [4]. Expecting furtherprogress in computing exactly the spectral action in its nonperturbative form and performing theappropriate renormalisation group analysis, we expect that we will be able to tackle astrophysicalissues in the near future.Here, I have reviewed possible cosmological fingerprints of noncommutative spectral geometry,and proposed a mechanism to constrain physics at unification through the implications ofproduction and dynamics of gravitational waves.
5. Acknowledgments
It is a pleasure to thank the organisers of the 14th Conference on Recent Developments inGravity (NEB14) “
N ǫωτ ǫρǫ s E ξǫλιξǫι s στ η B αρυτ ητ α
14” in Ioannina (Greece) for invitinge to present these results. This work is partially supported by the European Union throughthe Marie Curie Research and Training Network
UniverseNet (MRTN-CT-2006-035863).
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