Higgs Boson in RG running Inflationary Cosmology Yi-Fu Cai1, and Damien A. Easson1, ∗ † 1Department of Physics, Arizona State University, Tempe, AZ 85287 Anintriguinghypothesisisthatgravitymaybenon-perturbativelyrenormalizableviathenotion ofasymptoticsafety. WeshowthattheHiggssectoroftheSMminimallycoupledtoasymptotically safegravitycangeneratetheobservednearscale-invariantspectrumoftheCosmicMicrowaveBack- groundthroughthecurvatonmechanism. Theresultingprimordialpowerspectrumplacesanupper boundontheHiggsmass,whichforcanonicalvaluesofthecurvatonparameters,iscompatiblewith therecently released Large Hadron Collider data. PACSnumbers: 98.80.Cq 2 1 Weinberg has suggested that the effective description the Higgs boson is predicted to be mH = 126GeV with 0 of a quantum gravitational theory may be nonperturba- only severalGeV uncertainty[14]. We find a suitable in- 2 tively renormalizable through the notion of asymptotic flationarysolutioncanbeobtainedinacosmologicalsys- b safety (AS) [1]. In such a scenario the renormalization tem which contains a Higgs boson and AS gravity,along e group(RG)flowsapproachafixedpointintheultraviolet the lines of [15]. In this model, there are effectively two F (UV)limit,andafinitedimensionalcriticalsurfaceoftra- scalar degrees of freedom, one being the adiabatic mode 6 jectoriesevolvesto this pointatshortdistance scales[2]. and the other being an iso-curvature mode. We find the SuchafixedpointwasfoundintheEinstein-Hilberttrun- correspondingperturbationtheoryleads to boththe pri- ] h cation[3],andthescenariowasstudiedextensivelyinthe mordial power spectrum for the curvature perturbation t literature [4] (for recent reviews see [5]). andtheentropyperturbation. Whenthecutoffscaleruns - p lower than a critical value, inflation abruptly ends and Inflationary cosmology is the most promising candi- e the Higgs field can give rise to a reheating phase. Dur- h date theory for describing the early universe [6]. The ing this phase, the fluctuations seeded by the Higgs field [ paradigm solves the homogeneity, flatness, horizon and canbeconvertedintothecurvatureperturbationthrough unwanted relic problems. It also predicts a nearly 1 the curvaton mechanism [16, 17]. We derive a relation v scale-invariantprimordialpower spectrum, in agreement betweenthe spectralindexofthe primordialpowerspec- 5 withthedatafrommoderncosmologicalexperiments[7]. trumandtheHiggsmass. Weconfrontthisrelationwith 8 However, the model generally requires an as of yet un- 2 the latest cosmological observations and collider experi- observed scalar field. The Higgs boson is a scalar field 1 mentdata,andfindthey areconsistentunder agroupof predicted by the SM of particle physics and exciting ev- . canonical values of curvaton parameters. 1 2 idence for its existence has recently been released from 0 ConsidertheSMofparticlephysicsminimallycoupled the Large Hadron Collider (LHC) experiment [8–10]. In 2 to gravity the past, it was hoped that the Higgs field might play 1 v: a dual role as the inflaton, but the corresponding en- S = d4x√ g R−2Λ + SM . (1) ergy scale is much lower than what inflationary cosmol- Z − (cid:20) 16πG L (cid:21) i X ogy typically requires. Consequently, a model of Higgs In this AS gravity frame, the gravitational constant G r inflation was proposed in which the Higgs field is non- a and the cosmological constant Λ vary along the cutoff minimally coupled to Einstein gravity [11] (see earlier scale p. The running behaviors are approximately de- attempt in [12]). It was soon realized that such non- scribed by minimal coupling leads to an energy scale for unitarity violationwhichisexpectedtobelargerthantheinflation- G(p)−1 ≃G−N1+ξGp2 , Λ(p)≃ΛIR+ξΛp2 , (2) ary scale; otherwise, the effective field description would where G and Λ are the values of gravitational con- N IR fail(forexample,see[13]fordetaileddiscussions). There- stant and cosmologicalconstant in the IR limit. The co- fore, difficult challenges exist for models which attempt efficients ξ and ξ are determined by the physics near G Λ to use the Higgs field to drive inflation. the UV fixed point of RG flows in AS gravity. In this paper, we propose that the Higgs boson may The scalar sector of the SM contains the Higgs boson. play an important role in the early inflationary universe We use the unitary gauge for the Higgs boson H = h √2 if the gravitational theory is asymptotically safe. In the frameofASgravity,thegravitationalconstantGandcos- mological constant Λ are running along with the energy 1 In this Letter, we will work with the reduced Planck mass, scale, and thus vary throughout the cosmological evolu- Mp = 1/√8πGN, where GN is the gravitational constant in tion. Ithasbeenarguedthatiftherearenointermediate the IR limit, and adopt the mostly-plus metric sign convention energyscalesbetweenthe SMandASscales,themassof ( ,+,+,+). − 2 and neglect all gauge interactions for the time being. In We denote the frame proceeding the conformal trans- this case, the Lagrangianof the Higgs field is given by, formation as the Einstein frame (despite the non- canonicalformofthehkineticterm). Substitutionofthe 1 SM ∂ h∂µh V(h) , (3) flat Friedmann-Robertson-Walker (FRW) metric, ds2 = µ L ⊇−2 − dt2+a2(t)d~x2,leadstotheFriedmannequations: H2 = where V(h) is the potential of the Higgs field, which is −1 (φ˙2+e2bh˙2+V˜)andH˙ = 1 (φ˙2+e2bh˙2) ,where typically in form of λ(h2 v2)2. 3Mp2 2 2 −2Mp2 4 − wehavedefinedtheHubbleparameterH a˙ andthedot VaryingtheLagrangianwithrespecttothemetric,one ≡ a denotes the time derivative in the Einstein frame. The derives the generalized Einstein equation, coupled Klein-Gordon equations for the two scalars are: R Rg +Λg =8πG(TSM +TAS) . (4) φ¨+3Hφ˙+V˜,φ =b,φe2bh˙2,h¨+(3H+2b,φφ˙)h˙+e−2bV˜,h =0. µν − 2 µν µν µν µν To search for a successful inflationary solution, we in- troduce a series of slow roll parameters, Here the RG running of G can effectively contribute tgoµνt(cid:3)h)e(8sπtrGes)s−1en,ewrghyeretewnesohravtherionutgrohduTcµAeνSd t=he(c∇ovµa∇riνan−t ǫφ = 2Mφ˙22H2 , ǫh = 2Me2b2h˙H22 , ηIJ = 3V˜H,IJ2 . (9) derivative and the operator (cid:3) gµν . The p p µ µ ν ∇ ≡ − ∇ ∇ Higgs field h obeys the Klein-Gordon equation. Addi- The subscript “ ” denotes the derivative with respect ,I tionally, the running of cutoff scale is controlled by the to the Ith–field (with I being φ or h). During infla- Bianchiidentity,whichrequires,(R−2Λ)∇GµG+2∇µΛ= tion,these parametersarerequiredtobe lessthanunity. 0 . Consequently, the dynamics of this cosmological sys- However, the key parameters need to yield a successful tem are completely determined. inflationarybackgroundareassociatedwiththescalarφ, We now turn our attention to early universe inflation- i.e.,ǫφ andηφφ. Thisisbecausethepotentialforφisflat arysolutions. Thissystemismosteasilystudiedbymak- inthe regimeφ M andcorrespondinglythe param- p ≪− ing a conformal transformation, eters related to the Higgs boson h are suppressed by the small-valued factor e2b. G g˜ =Ω2g , Ω2 = N , (5) Thebackgrounddynamicsisdeterminedbythefollow- µν µν G ing solutions (under slow roll approximation), where Ω2 is the conformal factor. We also introduce a U e2bV U new scalar field φ, defined by φ˙ ,φ , h˙ ,h , H2 . (10) ≃−3H ≃− 3H ≃ 3M2 p √6M G p N φ ln , (6) Inflationendswhenǫ =1. Combingthisconditionwith ≡− 2 G φ the background solution for φ˙ in (10), we find the value where Mp = √8π1GN. ofφattheendofinflation: φf ≃−0.56Mp. Thenumber The original system, therefore, is equivalently de- of e-folding of inflation is given by = f Udφ , so scribed in terms of two scalar fields minimally coupled N − i Mp2U,φ that R to Einstein gravity without RG running, 3 (φ) e 2b(φ)+3b(φ) 1.68 . (11) = R˜ 1(˜φ)2 e2b(φ)(˜h)2 V˜(φ,h) ,(7) N ≃ 2 − − L 16πG − 2 ∇ − 2 ∇ − N To obtain = 60, we require the initial inflaton value N where the factor b(φ) φ and V˜(φ,h) = to be, φi ≃ −4.7Mp, which lies in the regime where RG ≡ √6Mp flows of AS gravityhaveapproachedthe UV fixed point. U(φ)+e4bV(h), with U(φ) 8πM4 ξΛ 1 e2b(φ) + Eventually,the slow roll conditions are violated, when ≃ p(cid:20)ξG(cid:18) − (cid:19) φ reaches φ . Consequently, φ enters a period of fast f G Λ e2b(φ) . Note that, the form of φ’s potential is roll, and finally approaches φ = 0 at which point the N IR (cid:21) AS gravity reduces to traditional Einstein gravity. We derivedfromthe RGrunningofthe ASgravity[15]. The suggest two possible reheating mechanism. One is that lasttermofU(φ)isproportionaltoGNΛIR. Substituting the inflaton φ decays to radiation, driving the universe the observedvalues of G and Λ , we find this term is N IR to a phase of thermal expansion directly; the other pos- insignificantthroughoutthe pastcosmologicalevolution. sibility is that the universe reheats only after the energy Hence, we make the approximation scale drops sufficiently so that the SM Higgs boson is responsible for the reheating process. Based on the as- ξ U(φ)≃8πMp4ξGΛ h1−e2b(φ)i , (8) sisumappptriooxnimofaitnesltyangtivreenhebaytiTnrge,≃the(πr22eλghde)a14thinrgetwehmepreerharteuries which is a sufficiently flat inflationary potential in the the value of h at the reheating surface, and g 106.75 d ≃ regime where b(φ) 1. is the number of degrees of freedom of the SM. After ≪− 3 inflation but before reheating, h oscillates along the po- distribution at the Hubble exit. In general, the Hubble- tentialV λh4. Thusthevalueofhcanberestrictedto crossing value of the Higgs boson h can be related to ∼ ∗ v < h < h with h being the value of the Higgs bo- the initial amplitude of curvaton oscillation h through re o | | ∗ ∗ son at the moment of Hubble crossing which we discuss a model-dependent function h = g(h ). For example, o ∗ below. in the present model, if the curvaton starts to oscillate During inflation, the backgrounddynamics are not af- immediately after inflation, ho h ; however, if there ≃ ∗ fected by the Higgs field, however its quantum fluctua- is a short slow rolling behavior for h following inflation, tions are able to source a nearly scale-invariant entropy ho ≃ 1121h∗. In this case, the curvature perturbation of perturbation. In a two field inflationary model, we de- the Higgs field in the oscillating phase is given by, compose the field variables into a background part and δρ δh h fluctuations: φ φ+δφandh h+δh. Thefieldfluctu- ζh = qh ∗ , (13) ationscombine→togiveadiabati→candiso-curvaturemodes 3(1+wh)ρh ≃ h∗ aswisni:thθδσt=h=eetbcrh˙oasj[e1θc8δt]φo.r+yTaosinntgaθlkeeebbδeihni,ntogδsda=ecficon−uendsitnbtyθhδceφoms+θect=oriscφσθ˙˙eflabunδchd- wopcliictfihuerdqshaas≡tqehhnh2oeo−≃rhvg∗2yhh.∗osTcwahhleeescnohe|ihgffiohc|eir≫entthvaqnhifctcahunervbeaneteofrnugryrtehsheceraalsteiimnog-f σ˙ tuation (the gravitational potential Φ) we introduce the the SM. canonical perturbation variables, vσ = a(δσ+ Hσ˙ Φ) and Wenowneedtorelateζh toζ. Inthesuddendecayap- vs = aδs, which characterize gauge-invariant adiabatic proximation, the relation can be computed analytically. and iso-curvature perturbations. Up to leading order in Consider the case that the Higgs boson decays on a uni- theslowrollapproximations,thesetwovariablesobeythe form total density hypersurface. On this slice we have perturbation equations [19]: vσ′′(s)+(k2− aa′′)vσ(s) ≃0 , ρh+ρr =ρT where ρr and ρT denote the energydensity where the prime denotes differentiation with respect to of radiation and that of the total system, respectively. conformal time, τ dt. Solving this equation in the Makinguseofthe expressionforthe curvatureperturba- ≡ a inflationary phase, weRcan obtain nearly scale-invariant tion on a uniform density slice, we find ρr = ρ¯re4(ζr−ζ) primordialpowerspectra for adiabatic andiso-curvature and ρh = ρ¯he3(1+wh)(ζh−ζ) during curvaton oscillation. perturbations, and their corresponding amplitudes are As a consequence, ζ and ζ are related on the reheating h δσ δs H∗ at the Hubble crossing moment t . hypersurface as follows, | | ≃ | | ≃ 2π ∗ Hence, the amplitudes of the field fluctuations are, (1 Ωh)e4(ζr−ζ)+Ωhe3(1+wh)(ζh−ζ) =1 , (14) − H H |δφ∗|≃ 2π∗ , |δh∗|≃ 2πe∗b , (12) where Ωh = ρh/ρT is the dimensionless density param- eter for the curvaton. For the curvaton mechanics to ∗ at the moment of Hubble crossing. succeed, we must assume that the fluctuation ζr seeded by the inflaton field is negligible. We will address this Wheninflationends,φrapidlyapproachestheIRlimit concern below and now turn our attention to the Higgs of the AS gravity. Because φ generally couples to other matterfieldsthroughtheconformalfactorΩ2,weexpect curvaton. Therefore, we have radiationtobeproducedfollowingtheinflationaryphase. 3(1+w )Ω h h Recallthat, the Higgsbosoncansurviveduring inflation ζ =qTζh , qT = , (15) 4 (1 3w )Ω h h due to the slow roll conditions and then starts to oscil- − − late along the λh4 potential. Consequently, the universe and in our explicit case, qT = Ωh at the curvaton decay is dominated by both the radiationand the Higgs boson surface. after inflation. This process is analogous to the famil- Combining Eqs. (13) and (15) and the field fluctua- iar curvaton scenario. Instead of a matter-like curvaton tion (12), we obtain the primordial power spectrum of oscillation with w = 0 as studied in [16, 20], a λh4 po- curvature perturbation seeded by the Higgs boson, tential yields an effective equation of state for the Higgs q2q2 H2 boson, which is the same as radiation with wh = 13 [21]. Pζ = 4πh2eT2b∗ h2∗ . (16) Thus, this generalized curvaton mechanism [22] may be ∗ used to generate the primordial curvature perturbation We see from (16), that the final curvature perturbation in agreement with the current CMB measurements [7]. depends on the five parameters: q , q , H , h , and h T Webeginbywritingdowntherelationbetweenthecur- eb∗. Compared with the usual curvaton mech∗anism∗ , our vaton fluctuation δh and its curvature perturbation ζ . model contains a new parameter eb∗ due to the confor- h Choosing the spatially flat slice for the Higgs curvaton, mal transformationmade in Eq. (5). However,since the one finds ρh = ρ¯he3(1+wh)ζh in the neighborhood of the background dynamics of inflation are driven by the RG curvaton reheating hypersurface. Consider the curvaton running of AS gravity, we find eb∗ 0.15 for observable ≃ perturbation generated from vacuum fluctuations inside perturbation modes at Hubble exit. Moreover, we have the Hubble radius. Thesefluctuations satisfyaGaussian specified the curvaton to be the Higgs boson and hence, 4 the potential is of an explicit form, and thus q 1. intermediate energy scales between the SM and the AS h ∼ Since the latest CMB data reveals P 2.4 10 9, we scales, the mass of the Higgs boson is determined by a ζ − candeducetheusefulrelationH 4.5≃10 5×h∗. Subse- fixed point and is approximately 126GeV. This predic- ∗ ≃ × − qT quently,wearelefttoconstrainq ,H andh byvarious tioniscoincidentwiththerecentresultsannouncedfrom T ∗ ∗ theoreticalandobservationalrequirements. Inthefollow- the ATLAS and CMS experiments, indicating that the ing, we calculate the tensor-to-scalar ratio, the spectral Higgs mass is in the range 116 131GeV (ATLAS[9]) or − index, and the reheating temperature, respectively, and 115 127GeV(CMS[10]), with othermassesexcludedat − then constrain the remaining parameters. the 95% confidence level. We have demonstrated that The calculation of primordial tensor perturbations is a sufficiently long inflationary solution can be obtained identical to that of ordinary inflationary models, and in this scenario due to the RG running of gravitational thus the tensor powerspectrumis nearly scale-invariant, andcosmologicalconstants,wherethe Higgsbosonplays given by P = 2H∗2 . The tensor-to-scalar ratio: r the roleofacurvatonwhichis responsiblefor generating t π2Mp2 ≡ the primordial curvature perturbation. From the analy- PPζt = q8h2eq2Tb2∗Mh2∗p2 . Since eb∗ ≃ 0.15 and qh ∼ 1, we ob- sis of linear perturbations, we find this model favors low tain r 0.2h2∗. According to the latest CMB data, r is energy scale inflation. Furthermore, if the occupation of ≃ qT2Mp2 curvaton density at the reheating surface is fixed, then required to be less than 0.36 [7], so that h < 2q M . T p observationalconstraints on the spectral index imply an ∗ A further constraint comes from the curvaton condition upper bound on the Higgs mass. After a fine tuning of thatthe contributionofinflatonfluctuationto curvature the curvaton parameters, the model is consistent with perturbation should be negligible. This condition re- recent LHC data. A complete data fitting of our model quiresq ζ (1 q )ζ . Sinceζ correspondstothera- T h ≫ − T r r will appear in the forthcoming work [23]. diationperturbation,inheritedfrominflatonfluctuation, ζ H∗ . Choosingagroupofcanonicalparameter We are grateful to R. Brandenberger for discussions. r ≃ 2π√2ǫ∗Mp The work of Y.F.C. and D.A.E is supported in part by values, this condition requires h <q M which is close T p ∗ the Cosmology Initiative at Arizona State University. to the observational bound provided by the tensor-to- scalar ratio. Combining this inequality and the expres- sion of H derived from P , one obtains a constraint on ζ the inflati∗onary Hubble parameter H <4.5 10 5M . − p ∗ × During inflation the Hubble parameter is approxi- ∗ [email protected] mately constant and the field fluctuations are approx- † [email protected] imately conserved after Hubble exit. The spectral tilt [1] S. Weinberg, in Understanding the Fundamental Con- nmhom≡en1t+ofddhlnlonPrkihzoonfctrhoesspinrgimiso:rdial perturbations at the sYtoitruke,n1t9s7o7f);Matter, ed. A. Zichichi (Plenum Press, New in General Relativity, ed. S. W. Hawking and W. 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