Inflation in models with Conformally Coupled Scalar fields: An application to the Noncommutative Spectral Action
aa r X i v : . [ h e p - t h ] A ug Inflation in models with Conformally Coupled Scalar fields: An application to theNoncommutative Spectral Action
Michel Buck ∗ , Malcolm Fairbairn † and Mairi Sakellariadou ‡ Department of Physics, King’s College London, Strand WC2R 2LS, London, U.K.
Slow-roll inflation is studied in theories where the inflaton field is conformally coupled to the Ricciscalar. In particular, the case of Higgs field inflation in the context of the noncommutative spectralaction is analyzed. It is shown that while the Higgs potential can lead to the slow-roll conditionsbeing satisfied once the running of the self-coupling at two-loops is included, the constraints imposedfrom the CMB data make the predictions of such a scenario incompatible with the measured valueof the top quark mass. We also analyze the rˆole of an additional conformally coupled massless scalarfield, which arises naturally in the context of noncommutative geometry, for inflationary scenarios.
I. INTRODUCTION
Cosmological inflation is the most widely acceptedmechanism to resolve the shortcomings of the standardHot Big Bang model. This mechanism, leading to a phaseof exponential expansion in the very early universe, isdeeply rooted in the fundamental principles of GeneralRelativity and Field Theory, and once combined with theprinciples of Quantum Mechanics, it can account for theorigin of the observed large scale structures and the mea-sured temperature anisotropies of the Cosmic MicrowaveBackground (CMB). However, despite its success, cosmo-logical inflation remains a paradigm in search of a modelwhich should be motivated by a fundamental theory. Thestrength of the inflationary mechanism is based on theassumption that its onset is generically independent ofthe initial conditions. Nevertheless, even this issue is un-der debate [1–6] given the lack of a complete theory ofQuantum Gravity.The inflaton field (usually a scalar field) is assumedto dominate the evolution of the universe at early times,but its origin and the form of its effective potential bothremain unknown; for this reason it would be attractiveif the one scalar field that is commonly thought to exist,namely the Higgs field, also doubled as the long-searchedfor inflaton. Unfortunately, it seems that if the Higgs fieldis minimally coupled to gravity this cannot be achieved,which has led some authors to consider large nonminimalcouplings of the Higgs field to gravity where inflationmight be achieved [7].It is commonly assumed/chosen that there is no cou-pling ( i.e. , minimal coupling) between the inflaton fieldand the background geometry (the Ricci curvature).However, this assumption/choice seems to lack a solidjustification. A first (and merely aesthetic) motivationcomes from the observation that in the early universe(where masses are negligible), the equations of motion ∗ [email protected] † [email protected] ‡ [email protected] for spinors and gauge bosons have a natural conformalinvariance in four space-time dimensions, while the sameis true for scalar fields only when they couple to the Ricciscalar in a specific way. More compelling is the factthat even if classically the coupling between the scalarfield and the Ricci curvature could be set equal to zero,a nonminimal coupling will be induced once quantumcorrections in the classical field theory are considered.Moreover, a nonminimal coupling seems to be needed inorder to renormalize the scalar field theory in a curvedspace-time. The precise value of the coupling constant(denoted by ξ ) then depends on the choice of the the-ory of gravity and the scalar field [8]. It has also beenargued that in all metric theories of gravity, includingGeneral Relativity, in which the scalar field is not part ofthe gravitational sector ( e.g. , when the scalar field is theHiggs field), the coupling constant should be conformalin order for the short distance propagators of the theoryto match those found in a Minkowski space-time — arequirement of the strong equivalence principle [8, 9] (inour notation, conformal coupling means ξ = 1 / ξ tendseither to its conformal value or increases exponentiallyin modulus, depending on the specific structure of thetheory.In what follows, we will investigate whether scalarfields, and in particular the Higgs field, could play therˆole of the inflaton in the presence of a small positivenonminimal coupling between the scalar field and thebackground geometry. The coupling constant ξ is nota free parameter which could be tuned to achieve a suc-cessful inflationary scenario avoiding severe fine-tuningof inflationary parameters ( e.g. , the self-coupling of theinflaton field), ξ should instead be dictated by the un-derlying theory. For negative values of ξ , exponential ex-pansion is more easily achieved than in the minimal case,and it can in fact lead to inflation consistent with obser-vational data in the strong coupling limit [7, 12]. In fact,the slow-roll parameters for large | ξ | are independent of ξ and only depend on the number of e-folds. However,exponential expansion is less favoured for positive values(in our conventions) such as conformal coupling [13]. Inlight of the motivations for a small positive ξ outlinedabove, we will investigate whether quantum correctionsto the Higgs potential can lead to a slow-roll inflationaryera and if so, whether the constraints imposed from theCMB temperature anisotropies are satisfied.We will apply this analysis to the Spectral Action ofNonCommutative Geometry (NCG). This theory leadsnaturally to a Lagrangian with a conformal couplingbetween the Higgs field and the background geometry,in the form of a boundary condition at high energies E ≥ Λ, where Λ is a characteristic scale of the model.NCG provides an elegant way of accounting for the Stan-dard Model (SM) of Particle Physics and its phenomenol-ogy [14]. Our motivation is to investigate cosmologicalconsequences of the NCG Spectral Action and, in partic-ular, to test whether slow-roll inflation driven by one ofthe scalar fields arising naturally within NCG could berealized in agreement with experimental data and astro-physical measurements.In a previous study, we (one of us and a collaborator)have studied [15] the conditions on the couplings so thatthe Higgs field could play the rˆole of the inflaton in thecontext of the NCG. Since however the running of cou-plings with the cut-off scale had been only analyzed [14]neglecting the nonminimal coupling between the Higgsfield and the curvature, we were not able to reach a def-inite conclusion. In this respect, the study below is afollow-up of Ref. [15]. Moreover, it has been argued [16]that inflation with a conformally coupled Higgs bosoncould be realized in the context of NCG due to the run-ning of the effective gravitational constant. In what fol-lows, we will also analyze the validity of this statement.Finally, the NCG Spectral Action provides, in additionto the Higgs field, another conformally coupled (mass-less) scalar field, which exhibits no coupling to the mat-ter sector [17]. One may a priori wish/expect that thisfield could be another candidate for the inflaton; we willexamine this scenario as well.Concluding, we analyze slow-roll inflation within mod-els that exhibit a conformal coupling between the Higgsfield and the Ricci curvature. Our motivation is to inves-tigate whether any of the two scalar fields arising natu-rally within the NCG Spectral Action could be identifiedas the inflaton. As we will explicitly show, our analy-sis leads us to the conclusion that unfortunately such aslow-roll inflationary scenario fails to remain in agree-ment with current data from high energy physics exper-iments and astrophysical measurements.This paper is organized as follows: In Section II, westudy the issue of the realization of slow-roll inflationwithin theories with a nonminimal coupling between thescalar field and the Ricci curvature, classically. The anal-ysis is first performed in the Jordan frame and then inthe Einstein frame. In Section III, we consider correc-tions to the Higgs potential through two-loop renormal-ization group analysis of the minimally coupled StandardModel; we then enlarge this study in the case of a confor-mal coupling. We focus on the gravitational and Higgs field sector of the Lagrangian density, obtained withinthe noncommutative spectral action, which has a confor-mal coupling, in Section IV. We find that even though wecan accommodate an era of slow-roll inflation, it seemsdifficult to reach an agreement with the CMB data. Thisconclusion holds not only for the Higgs field but also forthe other scalar field which appears generically in the the-ory. We then examine in Section V, whether running ofthe gravitational constant could modify our conclusionswith regards to the realization of a successful inflation-ary scenario driven through one of the scalar fields in theNCG theory. We round up our conclusions in Section VI.Our signature convention is ( − + + +); the Riemannand Ricci tensors are defined as R σµνρ = Γ σµρ,ν − Γ σνρ,µ + Γ τµρ Γ στν − Γ τνρ Γ στµ ,R µν = R ρµρν , respectively. Note that within our definition of ξ , confor-mal coupling means ξ = 1 / II. SLOW-ROLL INFLATION WITHNONMINIMALLY COUPLED SCALAR FIELDS
In this section, we will study whether slow-roll infla-tionary scenarios can be realized within models with animplicit nonminimal coupling between the inflaton fieldand the scalar curvature. We will first work in the Jordanframe and then we will perform the analysis in the Ein-stein frame. The Jordan frame is natural (physical) andoffers some useful insights on the effect of conformal cou-pling, while the Einstein frame is mathematically moreconvenient, especially when including more complicatedcorrections to the potential.
A. Analysis in the Jordan frame
Let us consider the action of a Higgs boson (or anyother scalar field φ ) nonminimally coupled to gravity: S = Z d x √− g (cid:26) κ f ( φ ) R −
12 ( ∇ φ ) − V ( φ ) (cid:27) , (1)where f ( φ ) = 1 − κ ξφ , with κ ≡ √ πG = m − and g being the determinant ofthe metric tensor. The scalar potential of φ is: V ( φ ) = λφ − µ φ . (2)The term − ξφ R in the action encodes the explicit non-minimal coupling of the scalar field φ to the Ricci curva-ture R .The background geometry during inflation is of theFridemann-Lemaˆıtre-Robertson-Walker (FLRW) form:d s = d t − a ( t )dΣ , (3)where t stands for cosmological time, a ( t ) is the scalefactor and dΣ describes spatial sections of constant cur-vature.Einstein’s equations read R µν − g µν R = κ f ( φ ) − T µν ( φ ) , (4)where the energy-momentum tensor, obtained by varyingthe action with respect to the metric, is [18, 19] T µν ( φ ) = (1 − ξ ) ∇ µ φ ∇ ν φ + 4 ξφ ( g µν (cid:3) − ∇ µ ∇ ν ) φ + g µν (cid:20) − (cid:18) − ξ (cid:19) ∇ ρ φ ∇ ρ φ − V ( φ ) (cid:21) . (5)Here (cid:3) ≡ g µν ∇ µ ∇ ν is the Laplace-Beltrami operator andGreek and Latin indices take values 0,1,2,3 and 1,2,3,respectively . The equation of motion (Klein-Gordonequation) of the Higgs field reads (cid:3) φ − ξRφ − dVdφ = 0 . (6)For vanishing and quartic potentials, Eq. (6) is invariantunder conformal transformations g µν → Ω( x ) g µν and φ → Ω( x ) − φ at conformal coupling ξ = 1 / φ , Eqs. (4),(6) combine to H = κ f ( φ ) (cid:20)
12 ˙ φ + V ( φ ) + 12 ξHφ ˙ φ (cid:21) , (7)0 = ¨ φ + 3 H ˙ φ − ξ (1 − ξ ) κ φ ˙ φ − ξ (1 − ξ ) κ φ + 8 ξκ φV ( φ ) + f ( φ ) V ′ ( φ )1 − ξ (1 − ξ ) κ φ , (8)where overdots denote time derivatives and primes standfor derivatives with respect to the argument ( e.g. , V ′ ( φ ) ≡ d V / d φ ). Note that 2 ξ (1 − ξ ) is zero at both,minimal ( i.e. , ξ = 0) and conformal ( i.e. , ξ = 1 /
12) cou-plings.Inflationary models are usually built upon the slow-rollapproximation, consisting of neglecting the most slowlyvarying terms in the equation of motion for the inflatonfield. However, in the case of nonminimal coupling ( i.e. , ξ = 0), it is more difficult to achieve the slow-rolling ofthe inflaton field. More precisely, the nonminimal cou-pling term in the action, − ξφ R , plays the rˆole of aneffective mass term for the scalar field, distorting theflatness of the scalar potential. Thus, in the case of anonminimal coupling, inflationary requirements such as − ˙ H < H (where H denotes the Hubble parameter) do Note that it is really the tensor T µν ( φ ) = f ( φ ) − T µν ( φ ) whichis covariantly conserved rather than T µν ( φ ) [13], but this ambi-guity in the choice of the energy-momentum tensor will not berelevant in our analysis. not translate in an equally straight-forward manner to re-lations on the inflaton fields and their scalar potentials.Indeed, there is no common choice of conditions ( see e.g., Refs. [18, 20, 21]), and no analog of slow-roll parametersin terms of which quantities such as the number of e -foldsof expansion or perturbation amplitudes are evaluated.With a tentative choice of conditions [18] | ¨ φ ˙ φ | ≪ H, | ˙ φφ | ≪ H and 12 ˙ φ ≪ V ( φ ) , (9)and a negligible mass term in the potential at high en-ergies, the energy constraint, Eq. (7), and field equation,Eq. (8), reduce to H ≈ λκ φ f ( φ ) (cid:20) − ξ − ξ (1 − ξ ) κ φ (cid:21) , (10)3 H ˙ φ ≈ − λφ − ξ (1 − ξ ) κ φ , (11)respectively. These equations determine the backgroundsolution, given by a ( φ ) = (cid:0) − ξκ φ (cid:1) exp (cid:20) − − ξ κ φ (cid:21) . (12)It is the second factor, in Eq. (12) above, which has thepotential to generate sufficient number of e -folds, as thefirst one will only lead to logarithmic corrections. For ξ = 1 /
12 ( i.e. , nonconformal coupling), a large enoughchange in p | ξ | κφ can lead to sufficient inflation to resolvethe horizon problem. This leaves some room to play withthe coupling and the field values, and it has indeed beenshown [12, 22] in recent literature that inflation can beachieved in a manner consistent with CMB data for largenegative | ξ | ∼ .At conformal coupling ( ξ = 1 /
12) however, the argu-ment in the exponential vanishes identically. For thisparticular value the smallness of p | ξ | can thus not becompensated by a larger value of φ during inflation togenerate the required expansion.What about quantum corrections to ξ ? For values closeto conformal coupling, δξ = ξ − /
12, the number of e -folds is approximately N ( φ ) = 32 δξκ (cid:0) φ − φ (cid:1) , (13)( φ e denotes the value of φ at the end of the inflationaryera) which requires a minimum initial Higgs field of theorder of φ ≈ p N/ | δξ | . Renormalization group analysisshows that δξ (as a function of the energy scale) is smallin the inflationary region, namely less than O ( ξ ) [10, 11].The initial Higgs amplitude required for sufficient num-ber of e -folds with such values of δξ generally lies abovethe Planck scale. Whether this implies energies above thePlanck mass relies in turn on the value of the parameter λ . Note however that the same renormalization groupanalysis of the nonminimally coupled Standard Modelsuggests that there are no quantum corrections to ξ , if itis exactly conformal at some energy scale [10, 11]. This isbased on the observation that there are no nonconformalvalues for the coupling ξ for which there is a renormaliza-tion group flow towards the conformal value as one runsthe Standard Model parameters up in the energy scale.It thus indicates that if one expects an exactly conformalcoupling for the Higgs field at some specific scale, it willbe exactly conformal at all scales, hence δξ = 0.The fact that conformal coupling destroys the accel-erated expansion has been noted previously [13]. Howcan conformal invariance be connected to the conditionsfor inflation? The implications of conformal invarianceon the stress-energy tensor are well-known: if the mat-ter sector of the theory is invariant under the conformaltransformation g µν → Ω g µν , φ → Ω − φ , (14)then the trace of T µν vanishes covariantly, and hence thescalar curvature R is zero. However, for a FLRW universethe scalar curvature reads R = 6( ˙ H + 2 H ) , (15)and therefore R = 0 implies − ˙ HH = 2 , (16)which is, for example, satisfied during the radiation-dominated period of the evolution of a universe in thecontext of General Relativity. However, it rules out in-flationary solutions which require − ˙ H/H <
1. Indeed,taking T µν ( φ ) from Eq. (5), its trace evaluates to T µµ ( φ ) = − [1 − ξ ] ∇ ρ φ ∇ ρ φ + [12 ξφV ′ ( φ ) − V ( φ )]+ 24 ξ Rφ , (17)having used the equation of motion for the scalar,Eq. (6). However, from Eq. (4), the trace of the energy-momentum thensor of φ reads T µµ ( φ ) = − κ − f ( φ ) R = − κ − (1 − ξκ φ ) R . (18)Thus, Eqs. (17), (18) imply − [1 − ξ ] ∇ ρ φ ∇ ρ φ + [12 ξφV ′ ( φ ) − V ( φ )] + 24 ξ Rφ = − κ − (1 − ξκ φ ) R . (19)Let us analyze Eq. (19): For vanishing ( V = 0) or quartic( V = λφ ) potential, conformal invariance ( ξ = 1 / Note that conformal invariance is considered here solely in thematter sector. The Einstein-Hilbert term is not conformally in-variant. implies that the terms in square brackets vanish and thelast term on the left-hand side cancels with the last termon the right-hand side, leading to zero scalar curvature R and thus zero trace of the energy-momentum tensor.However, when conformal invariance is broken, due forexample to a nonzero mass term for the inflaton field ( i.e. , µ = 0), the induced corrections to the scalar curvatureare δR = 2 µ κ φ . (20)In this case, the inflationary condition − ˙ H/H < µκφ > √ | H | , which is not satisfied by a lightscalar inflaton.For a V ( φ ) = λφ potential, classical analysis there-fore seems to exclude an inflationary regime. However,it is worth investigating whether quantum corrections tothe quartic self-coupling λ can induce potential termsthat break conformal invariance, and whether this canhave a sufficiently strong effect as to enable inflation-ary solutions. This can happen if these corrections aredrastic enough to generate terms in the effective poten-tial which alter the local profile of the potential, i.e. , V ( φ ) → V eff = V ( φ ) + αδφ with O (( δφ ) ′ ) ∼ O ( V ′ ).Then the slow-roll parameters will have a different formand may allow inflation.For slow-roll analysis with more complex potentials,it is convenient to perform a transformation to the Ein-stein frame, where the action is formulated in terms ofa rescaled metric and a new scalar field with a minimalcoupling to the curvature scalar of the new metric. Anymeaningful conclusions should of course be independentof the choice of conformal frame used during the calcu-lation. B. Analysis in the Einstein Frame
Performing a suitable Weyl transformation, the action,Eq. (48), can be recast in terms of a new metricˆ g µν = f ( φ ) g µν = (cid:0) − ξκ φ (cid:1) g µν , (21)and a canonical scalar field χ ( φ ) that is minimally cou-pled and related to the Higgs field byd χ d φ = p − ξ (1 − ξ ) κ φ f ( φ ) . (22)It should be noted that the transformation is singular for φ s = 1 / ( κ √ ξ ). In fact, solving for the canonical field χ , one can show that it covers only the range | φ | ≤ φ s ,implying that the analysis in the Einstein frame is validonly for this restricted domain of the original scalar. Thevalue φ s also has special status in the Jordan frame itself.At ξ = 1 /
12 in particular, it was shown [23] that althoughthe scalar field evolves smoothly through φ s in isotropicbackground cosmologies, its anisotropic shear diverges.We will safely stay below this point in the Einstein frameanalysis, still having access to Higgs field values all theway up to the Planck scale, as long as ξ ≤ x µ = x µ .Now a (ˆ t ) = a ( t ) is not the FRWL scale factor of the uni-verse described by the Einstein frame variables. However,by defining a new time coordinatedˆ τ = a (ˆ t )dˆ t = a ( t )d t , (23)the metric takes the FRWL form in the Einstein framewith a scale factorˆ a ( τ ) = p f ( φ ) a ( t ) , (24)and the Hubble parameter can be defined asˆ H = 1ˆ a dˆ a dˆ τ . (25)This leaves us with an Einstein frame action S E = Z d x p − ˆ g (cid:26) κ ˆ R −
12 ( ˆ ∇ χ ) − ˆ V ( χ ) (cid:27) , (26)and a scalar potentialˆ V ( χ ) = V ( φ ( χ ))[ f ( φ ( χ ))] = λ [ φ ( χ )] − µ [ φ ( χ )] [ f ( φ ( χ ))] . (27)The expression for φ ( χ ) is obtained from Eq. (22) andcan be solved analytically for any ξ [24]. In this studyhowever we shall express any functions ( e.g. , slow-rollparameters) of the Einstein frame in terms of φ , thephysical degree of freedom, so we leave the new potentialin terms of φ . Of course, our interpretation of the Ein-stein frame as unphysical but mathematically convenient presupposes that the “observables” computed thereinhave no immediate physical meaning. We will comeback to this point, and particularly the translation fromEinstein frame observables to physical Jordan frameobservables, later.It is now possible to look for an inflationary regimewithin the Einstein frame cosmology. We shall neglectthe mass term in the following analysis since we considerenergy scales E ≫ µ . In terms of the Higgs field φ , thecanonical first and second slow-roll parameters are given It is worth noting that the potential takes a particularly simpleform at ξ = 1 /
12 when the (conformal invariance breaking) massterm is neglected: V ( χ ) = 36 λκ − sinh ( κχ/ √ by the formulae:ˆ ǫ ( φ ) = 12 κ V d ˆ V d φ ! (cid:18) d χ d φ (cid:19) − (28)ˆ η ( φ ) = 1 κ V "(cid:18) d χ d φ (cid:19) − d ˆ V d φ − d χ d φ (cid:18) d χ d φ (cid:19) − d ˆ V d φ . (29)The number of Einstein frame e -folds isˆ N = Z t end t ˆ Hd ˆ τ = κ Z χχ end p ǫ ( χ ) d χ = κ Z φφ end p ǫ ( φ ) d χ d φ d φ , (30)and is related to the true number of e -folds in the Jordanframe by N = ˆ N + 12 ln (cid:20) f ( φ ) f ( φ end ) (cid:21) . (31)Classically, we have ˆ V ( φ ) = λφ / [ f ( φ )] , which givesˆ ǫ ( φ ) = 8 κ φ [1 − ξ (1 − ξ ) κ φ ] CC −−→ κ φ , (32)ˆ η ( φ ) = 4 h ξ κ φ − ξ (1 − ξ ) κ (cid:0) ξκφ + 1 (cid:1) φ κ φ [1 − ξ (1 − ξ ) κ φ ] i CC −−→
43 + 12 κ φ , (33)where CC denotes the conformal coupling limit. It thusemerges that the slow-roll parameters admit no slow-rollregion at all at conformal coupling. This can also be seenfrom the total number of e -folds:ˆ N ( φ ) = (1 − ξ ) κ (cid:0) φ − φ (cid:1) −
34 ln (cid:20) f ( φ ) f ( φ end ) (cid:21) , (34)which lacks the first, exponential expansion generating,term when ξ = 1 / e -folds in the Jordan frame,obtained in the Einstein frame analysis, namely fromEq. (31), with the scalar factor a ( t ) given from Eq. (12),one can confirm that it indeed agrees with the previousresult obtained within the Jordan frame. This shows thatthe canonical slow-roll conditions in the Einstein frameand the ones chosen in the Jordan frame produce agree-ing results, at least at the level of the observed expansion.Of course this does not imply the equivalence of otherquantities such as the perturbation amplitudes in the twoframes. As it has been explicitly shown in Ref. [25], thescalar two-point correlation functions evaluated in theJordan frame are different than those calculated afterthe field redefinitions in the Einstein frame. Therefore,one should keep in mind that there is a number of ambi-guities when quantum fluctuations of the scalar fields arestudied in different frames in the context of generalizedEinstein theories. Primordial spectral indices are calcu-lated to second order in slow-roll parameters in Ref. [20]for different inflationary models, in the context of the-ories with a nonminimal coupling between the inflatonfield and the Ricci curvature scalar. It has been shownthat there are inflationary models ( e.g. , new inflation) forwhich there are discrepancies between the values of thespectral index n s calculated in the Einstein and the Jor-dan frame, while for others ( e.g. , chaotic inflation) thereare not. Finally, the reader should keep in mind thatwhile realization of slow-roll inflation in the (physical)Jordan frame, in which the inflaton is nonminimally cou-pled to the Ricci curvature, implies slow-roll inflation inthe (unphysical) Einstein frame, the vive versa does nothold [13]. III. FLAT POTENTIAL THROUGH QUANTUMCORRECTIONS
The Higgs potential takes the classical form V ( φ ) = λφ − µ φ , (35)however both µ and λ are subject to radiative correctionsas a function of energy. For very large values of the field φ one therefore needs to calculate the renormalized valueof these parameters at the energy scale µ ∼ φ . The run-ning of the top Yukawa coupling and the gauge couplingscannot be neglected and must be evolved simultaneously.We follow the analysis of Ref. [26], which relies upon the β -functions and improved effective potential presented inRefs. [27–29]. This involves taking the measured values ofthe gauge couplings at low energy and evolving them up-wards in energy, taking into account the thresholds wherequark species come into the running. It is neccesary tosimultaenously evolve all three gauge couplings and thetop quark Yukawa coupling in order to accurately predictthe full effect upon the Higgs self-coupling. Care mustbe taken to use the correct relationship between the polemasses and the parameters used in the running [26].At high energies the mass term is sub-dominant andone can write the effective potential as V ( φ ) = λ ( φ ) φ . (36)Then for a given mass m t of the top quark, a smallervalue of the Higgs mass will result in the quartic cou-pling being driven down at large values of φ , such that itmay develop a metastable or true vacuum at expectationvalues of φ , far in excess of that observed from StandardModel physics h φ i = 246GeV. For typical values of m t ,if this false vacuum appears at all, it will show up rel-atively close to the Planck scale. When calculating the running of λ it is in fact necessary to go to two-loop accu-racy since at one-loop this second minimum develops atscales typically far in excess of the Planck scale, wherewe would really expect higher order nonrenormalizablecontributions to the potential to become important.For each value of m t there is therefore a value ofthe Higgs mass, m h , where the effective potential is onthe verge of developing a metastable minimum at largevalues of φ and the Higgs potential is locally flattened.This is illustrated in Fig. 1. Since the region where thepotential becomes flat is narrow, slow-roll must be very slow ( i.e. , the slow-roll parameters very small), in orderto provide a sufficiently long period of quasi-exponentialexpansion. The slow-roll parameters for the top (blackcurve) potential profile of Fig. 1 are shown in Fig. 2,and one can see that the region where ǫ is extremelysmall takes the form of a narrow dip. It is there that theintegral N ∼ R ǫ − d φ can generate the required numberof e -folds. FIG. 1: Sub-Planckian flattening of the Higgs potential dueto two-loop corrections in the Standard Model ( ξ = 0). Weanalyze slow-roll for profiles just above the top (black) curve,which feature no metastable vacua. It was noted in Ref. [30] that in the minimally coupledmodel, slow-roll through this flat region will not matchthe observed amplitude of density perturbations ∆ R inthe cosmic microwave background. Inflation predicts thelatter to be related to the potential and first-slow roll pa-rameter at horizon crossing (labelled by stars). Its valueas measured by WMAP7 [31] imposes the constraint (cid:18) V ∗ ǫ ∗ (cid:19) = 2 √ πm Pl ∆ R = (2 . ± . × − m Pl , (37)where ǫ ∗ ≤
1. The mismatch arises because ǫ needs tobe extremely small in order to allow for sufficient e -foldsand the potential energy is then too large to fit the con-dition. However, even in the minimally coupled model,there remains the possibility that horizon crossing occursclose to the beginning of inflation, where ǫ is not yet sosmall, provided the flat region occurs at low enough en-ergy. Since ǫ ∗ ≤
1, the maximum potential energy athorizon crossing is 5 . × − m . We shall see in therenormalization group analysis that there exist values ofthe top quark mass for which the flattening does hap-pen at energies below this value. Furthermore, the pres-ence of nonminimal coupling has additional effects sinceit changes the potential felt by the Higgs field.When the nonminimal coupling ξ of the Higgs bosonto gravity is included in the Standard Model, it has a β -function induced by the coupling between the Higgs fieldand the matter sector whose behaviour has been ana-lyzed to one-loop [10, 11]. As previously stated, we take β ξ = 0, since the presence of a boundary value ξ = 1 / ξ = 1 /
12 at all scales.The β -function of the quartic Higgs self-coupling changesas well due to the − ξRφ term, and this can have signif-icant effects on the remaining Standard Model parame-ters when ξ is large [32]. We have worked out how large ξ needs to be to impact the normal Standard Model run-ning by considering the two cases ξ = 1 and ξ = − V E = λ E ( φ ) φ = [ a ln ( bκφ ) + c ] φ . (38)The parameters are found to relate to the low energyvalues of m t in the following way: 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 . (39)The third parameter, c = c ( m t , m φ ), encodes the ap-pearance of an extremum ( see , Fig. 1) and depends onthe values for m t and m φ . Indeed, V E ( φ ) exhibits a sub-Planckian flat region (or local minimum) for suit-ably tuned parameters. An extremum occurs if and onlyif c/a ≤ /
16, the saturation of the bound correspond-ing to a perfectly flat region, i.e. , V ′ E ( φ ) = V ′′ E ( φ ) = 0,where φ = e − /b ( m t ) and e is Euler’s constant. Theenergy at these points is given by V E ( φ ) = a ( m t )8 eb ( m t ) κ , (40)which for 169 ≤ m t ≤
175 lies within 10 − κ − ≤ V ≤ κ − (note that V E ( φ ) increases with m t ). Thisshows that there are regions where the flattening occursat scales potentially consistent with perturbation ampli-tudes, given in Eq. (37).It is convenient to write c = [(1 + δ ) / a , where δ = 0saturates the bound below which a local minimum isformed. We restrict ourselves to δ >
0, so that thepotential contains no metastable vacua. The slow-rollconditions are met only for a narrow region, but for thepoints in parameter space which are close to δ = 0, bothslow-roll parameters vanish simultaneously and we getslow-roll inflation with extremely small ǫ . From Eq. (37) FIG. 2: Typical profiles of ǫ (blue) and η (red) with asmall sub-Planckian region of slow-roll, plotted here for m t =172GeV and δ = 0. There is a narrow region in which bothare very small. it follows that for m t > . ǫ, | η | ) ≤
1, and for inflation one should finda point in parameter space which: (i) leads to sufficient e -folds within a region [ φ end , φ ∗ ], (ii) has an ǫ ∗ whichlies within the bounds imposed by COBE normalization,and (iii) satisfies the observational constraints on n s and r . The measured value of perturbation amplitudes servesas a convenient first test of the model. For the scenarioto be viable, ǫ at horizon crossing cannot be too small.Since the requirement on a sufficient number of e -foldsrelies on a potential that has very small ǫ in a small re-gion, the problem is that the valley in ǫ is far narrowerthan that in η . As a result, within the region | η | ≤ ǫ tends to be very small. The best fit to the observedperturbation amplitude will occur for scenarios in whichhorizon crossing occurs close to the onset of inflation, i.e. , η ( φ ⋆ ) ∼
1, so that ǫ ⋆ takes its largest possible value.The corrections due to conformal coupling to thepotential in the Einstein frame are entirely embodied inthe function f ( φ ) ∼ O ( κ φ ), since the canonicalfield χ feels the potential V E /f . The value of the Higgsfield where the plateau occurs in the potential riseswith increasing top quark mass, so the greatest effectwill be at the highest top quark mass. However, thelower bound on ǫ ∗ then gets more stringent since V ∗ is larger. Due to the change in the potential, flatnessdoes not occur at δ = 0 anymore but for fixed valuesof δ depending on the value of the top quark mass.Sub-Planckian inflation is again reliant on a relationshipbetween the Higgs field and the top quark masses.The values of δ for which the potential has the rightflatness are not anymore centered around δ = 0 due tothe altered form of the potential. This has an effecton the Higgs masses where flattening occurs: for any − ≤ δ ≤
0, a given top quark mass fixes the Higgs massto a value in the range (120 − m φ /m φ ∼ − . This means that for inflation tooccur via this mechanism, the top quark mass fixes theHiggs mass extremely accurately. As an example, for m t = 171 .
70 GeV and δ = − . m φ = 125 . N = 62 of e -foldsbetween κφ = 0 . κφ end = 0 . e -folds are indeed generated provided a suitablytuned relationship between m t and m φ holds. Numeri-cal integration needs to be performed carefully since theslow-roll approximation implies a strongly peaked inte-grand in the number of e -folds. Using a Runge-Kuttaintegrator in FORTRAN we identify the curve in param-eter space along which sufficient expansion occurs duringalmost perfect de Sitter inflation, both for minimal andconformal couplings.The next step is a comparison with astrophysical mea-surements. To probe the parameter space more finelywe use a Monte-Carlo chain. It turns out that in boththe minimal and conformal cases, the perturbation am-plitudes are too large — the best fit to the ratio ( V ⋆ /ǫ ⋆ ) is still too large by two orders of magnitude. Small posi-tive nonminimal couplings such as ξ = 1 /
12 improve thefit only minimally. It should be noted also that whenperturbation amplitudes are too large, scenarios whereperturbations are generated by a curvaton are in turnruled out as well, because the quantum fluctuations ofthe inflaton are already too large.In Fig. 3 we show the best fit, i.e. , the scenario with thelargest possible values of ǫ ∗ , the first slow-roll parameterat horizon crossing, for a given top quark mass alongwith the potential energy V ∗ , at horizon crossing. Theresulting ratio of perturbation amplitudes is too large forany value of m t .On a side note, let us mention that the renormalizationof Standard Model parameters is generally performed inMinkowski space-time, while inflationary perturbationsare calculated on a general de Sitter background. The FIG. 3: The value of the potential (solid) in units of κ − andthe maximum value of the first slow-roll parameter (dashed)at horizon crossing for minimal ξ = 0 (black) and conformal ξ = 1 /
12 (blue). The striped area represents the region of thetop mass excluded by Eq. (37) from the height of the plateauin the potential. The inset shows the ratio ( V ⋆ /ǫ ⋆ ) in bothcases and WMAP7 observations (red region). The calculatedvalue of perturbation amplitudes is off by several orders ofmagnitude and the improvement at conformal coupling min-imal. conditions for the geralization of a slow-roll inflationaryera should of course be studied in a de Sitter space andthe Coleman-Weinberg result should then be recoveredas a limit to the flat Minkowski space-time. This analy-sis has been performed in a recent study [21], where theone-loop improved potential for the nonminimally cou-pled scalar λφ theory in de Sitter space was calculated.Their analysis poses a stringent constraint on the cou-pling parameter ξ . The assumption | ˙ H | < H along withthe requirement that f ( φ ) in the equations of motion re-main nonsingular, f ( φ ) < ∞ , implies116 ˜ N ≪ | ξ | ≪ , (41)where ˜ N ≈ N +1 − ξ . This rules out most values of ξ usedin literature. However, | ˙ H | < H is in fact a strongercondition than the condition − ˙ H < H for inflationaryexpansion. The latter implies the former only when ˙ H is negative. The stronger condition | ˙ H | < H could becircumvented in an inflationary universe where ˙ H is largeand positive. In the minimally coupled case this is clearlynot possible, since ˙ H = − ˙ φ /
2. For nonzero ξ we havehowever˙ H = φ ˙ φf ( φ ) " −
12 (1 − ξ ) ˙ φφ − ξH + 2 ξ ¨ φ ˙ φ , (42)wich for the slow-roll conditions, Eq. (9), reduces to˙ H = − ξφ ˙ φf ( φ ) H . (43)Since the field rolls down the potential, sign( φ ) = − sign( ˙ φ ) and ˙ H is indeed positive when the nonmini-mal coupling is positive ( e.g. , conformal) in our notation.This means that the above constraint does not apply inthe conformally coupled case. However, it should be men-tioned that for negative choices of ξ , popular due to theirpromise in achieving Higgs driven inflation, ˙ H <
0. Theconstraint in Eq. (41) is then valid and seems to be incontradiction with large | ξ | . IV. NONCOMMUTATIVE SPECTRAL ACTIONAND INFLATION
Using the language of noncommutative geometry andspectral triples, Connes and collaborators have reformu-lated the Standard Model in terms of purely geometricdata [14]. Based on spectral triples, A. Connes [33] hasdeveloped a new calculus that deals not with the under-lying spaces, but with the algebra of functions definedupon them instead. This reformulation allows a natu-ral generalization of the differential calculus on Rieman-nian manifolds to a wider class of geometric structures, i.e. , noncommutative spaces. It is the geometry of thesespaces that encodes not only space-time and gravity, butalso the matter content of the Standard Model.In NCG, the fundamental particles and interactionsderive from the spectral data of an action functional de-fined on noncommutative spaces, the Spectral Action.The Standard Model emerges as the asymptotic expan-sion of this action at an energy Λ below the Planck scale,at which the fundamental noncommutative space is ap-proximated by an almost-commutative space. This spaceis assumed to be the simplest noncommutative extensionof the smooth four-dimensional space-time manifold, andis obtained by taking its tensor product with a finite non-commutative space. Having recovered low energy physicsin the framework of NCG, the next step will be to findthe true geometry at Planckian energies, for which thisproduct is a low energy limit. We consider here the ef-fective action functional at the scale Λ.In this section, we will first highlight the main prin-ciples of the noncommutative geometry approach andwe will then investigate possible inflationary mechanismsdriven by one of the available scalar fields.
A. Elements of NCG spectral action
Within General Relativity, the group of symmetries ofgravity is given by diffeomorphism of the underlying dif-ferentiable manifold of space-time; a key ingredient thatone would like to extend to the theory of elementary par-ticles. To achieve such a geometrization of the StandardModel coupled to gravity, one should turn the SM cou-pled to gravity into pure gravity on a preferred space,whose group of diffeomorphisms is given by the semidi-rect product of the group of maps from the background manifold to the gauge group of the SM, with the group ofdiffeomorphisms of the background manifold. Such pre-ferred space cannot be obtained however within ordinaryspaces, while noncommutative spaces can easily lead tothe desired answer. This is the main reason for extend-ing the framework of geometry to spaces whose algebraof coordinates is noncommutative.To extend the Riemaniann paradigm of geometry tothe notion of metric on a noncommutative space, thelatter should contain the Riemaniann manifold with themetric tensor (as a special case), allow for departuresfrom commutativity of coordinates as well as for quan-tum corrections of geometry, contain spaces of complexdimension, and offer the means of expressing the Stan-dard Model coupled to Einstein gravity as pure gravity ona suitable geometry. A metric NCG is given by a spectraltriple ( A , H , D ), in the sense that we will discuss below.Thus, within NCG, geometric spaces emerge naturallyfrom purely spectral data. The fermions of the StandardModel provide the Hilbert space H of a spectral triplefor a suitable algebra A , and the bosons arise naturallyas inner fluctuations of the corresponding Dirac operator D . To study the implications of this noncommutativeapproach coupled to gravity for the cosmological modelsof the early universe, we will only consider the bosonicpart of the action; the fermionic part is however crucialfor the particle physics phenomenology of the model.More precisely, let us consider a geometric space de-fined by the product of a continuum compact Riemaniannmanifold, M , and a tiny discrete finite noncommutativespace, F , composed of only two points. The productgeometry M × F has the same dimension as the ordi-nary space-time manifold, namely 4. Hence, the non-commutative space F has zero metric dimension. Thespace F represents the geometric origin of the StandardModel and it is specified in terms of a real spectral triple( A , H , D ), where A is a noncommutative ⋆ -algebra, H is a Hilbert space on which A is realized as an algebraof bounded operators, and D is a suitably defined Diracoperator on H . The Dirac operator can be seen as the in-verse of the Euclidean propagator of fermions. Since theaction functional only depends on the spectrum of theline element, it is a purely gravitational action. In otherwords, the physical Lagrangian is entirely determined bythe geometric input, which implies that the physical im-plications are closely dependent on the underlying chosengeometry, see , Ref. [14].By assuming that the algebra constructed in M × F is symplectic-unitary , the algebra A is restricted to be ofthe form A = M a ( H ) ⊕ M k ( C ) , (44)where k = 2 a and H is the algebra of quaternions. Thechoice k = 4 is the first value that produces the correctnumber ( k = 16) of fermions in each of the three gen-erations [34]. The Dirac operator D connects M and F via the spectral action functional on the spectral triple.It is defined as Tr ( f ( D/ Λ)), where f > (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) , (45)where f k = R ∞ f ( v ) v k − d v are the momenta of thefunction f , the noncommutative integration is defined interms of residues of zeta functions, and the sum is overpoints in the dimension spectrum of the spectral triple.The test function enters through its momenta f , f , f ;these three additional real parameters are physically re-lated to the coupling constants at unification, the gravi-tational constant and the cosmological constant. In thefour-dimensional case, the term in Λ in the spectral ac-tion, Eq. (45), gives a cosmological term, the term inΛ gives the Einstein-Hilbert action functional with thephysical sign for the Euclidean functional integral (pro-vided f > M × F gives abosonic functional S which includes cosmological terms,Riemannian curvature terms, Higgs minimal coupling,Higgs mass terms, Higgs quartic potential and Yang-Millsterms. Moreover, one can introduce a relation betweenthe parameters of the model, namely a relation betweenthe coupling constants at unification. More precisely, weimpose the relation g f π = 14 and g = g = 53 g , (46)between the coefficient f and the coupling constants g , g , g , which is dictated by the normalization of thekinetic terms. This condition means that the so-obtainedspectral action has to be considered as the bare action at unification scale Λ, where one supposes the mergingof the coupling constants to take place. The gravitational terms in the spectral action, in Eu-clidean signature, are of the form S Egrav = Z (cid:18) κ R + α C µνρσ C µνρσ + τ R ⋆ R ⋆ − ξ R | H | (cid:1) √ g d x . (47)Note that H is a rescaling H = ( √ af /π ) φ of the Higgsfield φ to normalize the kinetic energy; the momentum f is physically related to the coupling constants at unifi-cation and the coefficient a is related to the fermion andlepton masses and lepton mixing. In the above action,Eq. (47), the first two terms only depend upon the Rie-mann curvature tensor; the first is the Einstein-Hilbertterm with the second one being the Weyl curvature term.The third term R ⋆ R ⋆ = 14 ǫ µνρσ ǫ αβγδ R αβµν R γδρσ , is the topological term that integrates to the Euler char-acteristic and hence is nondynamical. Notice the absenceof quadratic terms in the curvature; there is only the termquadratic in the Weyl curvature and topological term R ⋆ R ⋆ . In a cosmological setting namely for Friedmann-Lemaˆıtre-Robertson-Walker geometries, the Weyl termvanishes. The spectral action contains one more termthat couples gravity with the SM, namely the last termin Eq. (47), which should always be present when oneconsiders gravity coupled to scalar fields. B. Higgs field inflation
The asymptotic expansion of the Spectral Action, pro-posed in Ref. [14], gives rise to the following Gravity-Higgs sector L GH ⊂ L NCG : S GH = Z d x √− g (cid:26) − κ ξ H κ R −
12 ( ∇ H ) − V ( H ) (cid:27) , (48)where V ( H ) = λ H − µ H . In the derivation ofthe Standard Model from the Spectral Action princi-ple, the metric carries Euclidean signature. The discus-sion of phenomenological aspects of the theory relies ona Wick rotation to imaginary time, into the standard(Lorentzian) signature. While sensible from the phe-nomenological point of view, there exists as yet no justi-fication on the level of the underlying theory.To discuss the phenomenology of the aspects of thecut-off scale Λ, the Spectral Action principle leads toa number of boundary conditions on the parameters ofthe Lagrangian. These conditions encode the geometricorigin of the Standard Model parameters. Normalizationof the kinetic terms in the action implies the following1relations: κ = 12 π f Λ − f c ,ξ = 112 ,λ = π b f a ,µ = 2Λ f f . (49)We emphasize that the action, Eq. (48), has to be takenas the bare action at some cutoff scale Λ. The renormal-ized action will have the same form but with the barequantities κ, µ, λ and the three gauge couplings g , g , g replaced with physical quantities.The factor f is fixed by the canonical normalizationof the Yang-Mills terms (not included here) in terms ofthe common value of the gauge coupling constants g atunification, f = π / (2 g ). The value of g at the uni-fication scale is determined by standard renormalizationgroup flow, i.e. , it is given a value which reproduces thecorrect observed coupling at low energies. Note that itis not unique since the gauge couplings fail to meet ex-actly in the nonsupersymmetric Standard Model (or itsextension by right-handed neutrinos). The coefficients a , b , c are the Yukawa and Majorana parameters subjectto renormalization group flow, see e.g. Ref. [14]. Theparameter f is a priori unconstrained in R ∗ + .Assuming the big desert hypothesis, we can connectthe physics at low energies with those at E = Λ throughthe standard renormalization procedure. This was car-ried out at one loop in Ref. [14], and more recently inRef. [16] where Majorana mass terms for right-handedneutrinos were included and the see-saw mechanism wastaken into account. In our renormalization group analy-sis of the Higgs potential, following Ref. [26], the choiceof boundary conditions is the standard one motivated byparticle physics considerations. The focus here has ofcourse been on the different boundary conditions at lowenergies for which a flat section develops in the Higgspotential.The relations above rely on the validity of the asymp-totic expansion at Λ, and are therefore tied intimately tothe scale at which the expansion is performed. There isno a priori reason for the constraints to hold at scales be-low Λ — they represent mere boundary conditions. Theconstraint ξ (Λ) = 1 /
12 by itself therefore does not re-quire the coupling to remain conformal all the way downto present energy scales, or even during an inflationaryepoch, since it may run with the energy scale. However,we will assume no running in ξ as the arguments laid out( see , discussion in Section IIA) above still apply.As we can see from the results presented above, theconformally coupled Higgs field in the Spectral ActionStandard Model is not a viable candidate for inflaton ifthe coupling remains conformal at all scales. However, at present it is still unclear whether conformal invari-ance and ξ = 1 /
12 is a generic feature of models fromnoncommutative geometry. If it turns out not to be, onecan proceed along the line of the analyses presented inRefs. [15, 16].
C. Inflation through the massless scalar field
The spectral action gives rise to an additional masslessscalar field [17], denoted by σ . Including this field, thecosmologically relevant terms in the Wick rotated actionread S = Z d x √− g (cid:26) κ R − ξ H R H − ξ σ Rσ −
12 ( ∇ H ) −
12 ( ∇ σ ) − V ( H , σ ) (cid:27) (50)where V ( H , σ ) = λ H H − µ H H + λ σ σ + λ H σ | H | σ . (51)The constants are related to the underlying parametersas follows : ξ H = 112 , ξ σ = 112 (52) λ H = π b f a , λ σ = π d f c (53) µ H = 2Λ f f , λ H σ = 2 π e a c f . (54)This action also admits a rescaling of the metric whichtransforms it to the Einstein frame. The rescaled metricˆ g µν = f ( H , σ ) g µν with f ( H , σ ) = 1 − ξ H H − ξ σ σ is The coupling term between the Higgs field and the Riccicurvature, appearing in the spectral action functional, is − f / (12 π ) aR | φ | , which after rescaling H = ( √ af /π ) φ , leadsto the term − R | H | /
12. This indeed shows the conformal cou-pling between the background and the Higgs field. The field σ is unlike all other fields in the theory, such as theHiggs field and gauge fields. Usually one starts with a param-eter in the Dirac operator of the discrete space, and then innerfluctuations of the product space would generate the dynamicalfields. The only exception being the matrix entry that gives massto the right-handed neutrinos, where the parameter can eitherremain as such, or one can use the freedom to make it a dy-namical field, which a priori may lead to important cosmologicalconsequences [ ? ]. Note that the σ field was not considered inthe original noncommutative geometry spectral action analysispresented in Ref. [14], where the authors were mainly interestedin recovering the Standard Model. Note that a similar action has been studied in Ref [32], but intheir analysis the additional scalar field has a nonzero mass andthe nonminimal couplings are studied in the previously men-tioned large negative ξ regime, which flattens the classical quarticpotential in the Einstein frame. χ H and χ σ relatedto the Jordan frame fields by dχ H d H = p − ξ H (1 − ξ H ) H f ( H , σ ) CC −−→ f ( H , σ ) , (55) dχ σ dσ = p − ξ σ (1 − ξ σ ) σ f ( H , σ ) CC −−→ f ( H , σ ) . (56)The Einstein frame Lagrangian reads S = Z d x p − ˆ g (cid:26) κ ˆ R −
12 ( ˆ ∇ χ H ) −
12 ( ˆ ∇ χ σ ) − P ( χ H , χ σ ) ˆ ∇ µ χ H ˆ ∇ µ χ σ − V ( χ H , χ σ ) (cid:27) , (57)with ˆ V ( χ H , χ σ ) = V ( H , σ ) f ( H , σ ) , (58)and a novel coupling P ( χ H , χ σ ) = 24 κ ξ H ξ σ f ( H , σ ) d H dχ H dσdχ σ H σ CC −−→ κ σ H . (59)Note that there exists no conformal transformation whichgets rid of both the nonminimal coupling to gravity andthe cross-term P ( χ H , χ σ ) [35]. However, at conformalcoupling P ( χ H , χ σ ) can be neglected as long as σ H ≪ κ . We are then left with a minimally coupled theoryof two scalar fields with potential ˆ V ( χ H , χ σ ). When themass term is negligible the theory is symmetric in thetwo fields.Consider the first slow-roll parameter for the σ -field,defined asˆ ǫ σ = 12 κ V ∂ ˆ V∂σ ! (cid:18) ∂χ σ ∂σ (cid:19) − (60)= 12 κ [ λ H H + λ σ σ + λ H σ | H | σ ] − × σ (cid:20)(cid:0) λ σ σ + 2 λ H σ H (cid:1) f ( H , σ )+ 23 κ (cid:0) λ σ σ + λ H H + λ H σ H σ (cid:1)(cid:21) . (61)For H = 0 this reduces to the earlier case and one getsan insufficient numebr of e -folds below the Planck scale.If we have a nonzero H however, say H ∼ κ − close tothe Planck mass, then the situation changes somewhat. Due to the additional terms in ˆ ǫ σ , the coupling constantsdo not fall out of the expression, and they can thereforeinfluence the magnitude of the integrand in the numberof e -folds. For this effect to take place however, it is nec-essary that the assisting field maintains a relatively largevalue throughout the inflationary era driven by the infla-ton. This in turn requires the curvature of the potentialto be much less in the direction of the constant field.Since only quartic terms arise in the model, the quarticself-coupling of the assisting field is then required to bemuch lower than that of the inflaton. But in that case,the new terms due to the assisting field are not largeenough to enable the inflaton to generate sufficient num-ber of e -folds. The situation is of course entirely sym-metric in the two fields (except for the nonzero H -masswhich is negligible at high energies), so the rˆoles of thetwo fields may well be interchanged depending on whichconstraints lay on the respective coupling constants. V. RUNNING OF THE GRAVITATIONALCONSTANT
At the scale Λ, the gravitational constant is related tothe geometric parameters of the theory by κ = 12 π f Λ − f c ; (62) f is fixed by one of the unification conditions f = π g , f = π g , f = π g ,f is an unconstrained parameter in R ∗ + , and c is deter-mined by the renormalization group equations. Note thatthis value of the gravitational constant does not need tobe the same as its present value, κ = [2 . × GeV] − ,since the gravitational constant may run.Indeed, such a running has been suggested [16] due tothe relation between κ and c = Tr( M M † ); M stands forthe Majorana mass term. The coefficient c is a functionof the neutrino mass matrix subject to running with therenormalization group equations dictated by the particlephysics content of the model, in this case the StandardModel with additional right-handed neutrinos with Ma-jorana mass terms. Since the renormalization group flowruns between a unification energy Λ, taken to be of theorder of 2 × GeV, down to the electroweak scale of100 GeV, the parameter c runs as a function of Λ, with as-signed initial conditions at the preferential energy scaleof unification. One may thus deduce that through therunning of κ , the number N of e -folds may increase.However, since at conformal coupling N is a logarithmicfunction of κ , the gravitational constant would have tochange drastically, in order for N to have an interestingchange. As previously mentioned, we need a very sig-nificant running of the gravitational constant in order toget inflation — a local kind of inflation where the flat3region is confined to a small range of energies where thepotential is flattened.In conclusion, unless modification of the spectraltriple allows for a nonconformal boundary value of ξ ,there seems to be no viable slow-roll scenario for any ofthe two scalars. Furthermore, if one assumes the validityof the suggestion in Ref. [16] in relating the running of c and κ , the only situation in which this could triggerinflation (with conformal coupling) would be one inwhich the running changes drastically, e.g. , through thesee-saw mechanism. However, the inevitable lack ofdifferentiability of the renormalized couplings at see-sawscales [16, 36] makes such a scenario very unlikely andalso inaccessible to slow-roll analysis. VI. CONCLUSIONS
In many realistic cosmological models, the nonminimalcoupling of the scalar field to the Ricci curvature cannotbe avoided. In particular, there are arguments requir-ing a conformal coupling between the scalar field andthe background curvature. The existence of such a termwill generically lead to difficulties in achieving a slow-roll inflationary era. In this paper, we have investigatedwhether two-loop corrections to the Higgs potential couldlead to a slow-roll inflationary period in agreement withthe constraints imposed by the CMB measurements. Ourfindings do not favor the realization of such an era. Moreprecisely, even though slow-roll inflation can be realized,we cannot satisfy the COBE normalization constraint for any values of the top quark and Higgs masses allowedfrom current experimental data.We have in particular investigated Higgs inflation inthe context of the Noncommutative Geometry SpectralAction, which provides an elegant explanation for thephenomenology of the Standard Model. Within this con-text, a conformal coupling arises naturally between theHiggs field and the Ricci curvature. It is also impor-tant to note that once conformal coupling is set at thepreferential (boundary) energy scale of the spectral ac-tion model, then it will remain conformal at all scales.Running of the gravitational constant and corrections byconsidering the more appropriate de Sitter, instead of aMinkowski, background do not favor the realization ofa successful inflationary era. The NCG Spectral Actionprovides in addition to the Higgs field, another (massless)scalar field which exhibits no coupling to the matter sec-tor. Our analysis has shown that neither this field canlead to a successful slow-roll inflationary era if the cou-pling values are conformal. One may be able to improveupon this (negative) conclusion, if important deviationsof ξ from its conformal value can be allowed; the value ξ = 1 /
12 may turn out that is not a generic feature ofNCG models.
VII. ACKNOWLEDGMENTS
This work is partially supported by the EuropeanUnion through the Marie Curie Research and TrainingNetwork
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