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arXiv:1202.1285v2 [hep-th] 7 Jan 2013 h ig edt rv inflation. drive use to to field attempt Therefore, Higgs which the models discussions). ex- for exist detailed (for challenges for fail difficult oth- would [29–35] scale; description see inflationary field ample, the effective than the larger erwise, be which to violation cou- unitarity expected for non-minimal is scale such energy that an realized to attempt leads soon earlier pling was (see non-minimally It is [21–27] [28]). gravity field in Einstein Higgs to the inflation which coupled Higgs in of proposed model was energy a Consequently, corresponding typ- a requires. the inflationary play ically what but might than lower inflaton, field much is Higgs the scale the as that role hoped dual was from In it released [18–20]. past, experiment been the (LHC) recently Collider ev- field has exciting Large scalar existence and the physics a its is for of un- idence SM yet Higgs the of The by as predicted [17]. an field. experiments requires scalar generally observed cosmological model modern the However, from with data scale- agreement nearly in the a spectrum, predicts power also primordial and invariant It horizon problems. flatness, relic homogeneity, unwanted The the [14–16]. solves universe early paradigm the describing for theory date see Einstein- reviews the recent in (for ex- [6–11] found studied 13]). literature was [12, was the scenario point the in fixed and tensively [5], a truncation scales Such distance Hilbert short at 4]. of point [3, surface this critical to dimensional ultravio- evolves finite the trajectories a in and point limit, renormalization fixed (UV) the a let scenario approach a flows such (RG) asymptotic In group of 2]. [1, notion (AS) the nonperturba- safety be through may renormalizable theory tively gravitational quantum a of † ∗ lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic nti ae,w rps htteHgsbsnmay boson Higgs the that propose we paper, this In candi- promising most the is cosmology Inflationary description effective the that suggested has Weinberg ASnmes 98.80.Cq numbers: data. PACS Collider Hadron Large released recently the with on nteHgsms,wihfor which mass, Higgs primo the resulting on The m bound mechanism. SM curvaton scale-invarian the the near of through observed ground sector the Higgs generate the can gravity that safe show We safety. asymptotic of nitiun yohssi htgaiymyb non-perturb be may gravity that is hypothesis intriguing An .INTRODUCTION I. 2 eateto hsc,MGl nvriy ot´a,Q,H Montr´eal, QC, University, McGill Physics, of Department ig oo nR unn nainr Cosmology Inflationary running RG in Boson Higgs 1 eateto hsc,AioaSaeUiest,Tme AZ Tempe, University, State Arizona Physics, of Department iF Cai Yi-Fu ,2, 1, nl tuned finely ∗ n ainA Easson A. Damien and auso h uvtnprmtr,i compatible is parameters, curvaton the of values xr em h oe sfudt ecrepnigt a to corresponding the be of to account found into is term take f model extra To the contribute term, tensor. thus extra stress and energy scale the cutoff to run- the the with con- of along gravitational ning vary the constant cosmological model, and this stant In the role explains the observation. which plays CMB spectrum the field power Higgs without primordial the gravity seeding but of AS boson, Higgs of be the frame can of solution the help inflationary in the the purely model from obtained our Different boson in [37]. [34], Higgs of of a lines idea the contains along gravity, obtained which AS be system and can cosmological solution a inflationary in suitable a find We closrain n oldreprmn aa n find and data, experiment Higgs collider cosmolog- the latest and the observations and with ical relation spectrum this spectral confront power We the primordial mass. between the relation of curvature a (see index derive the We [40] into mechanism [41–43]). converted curvaton also be re- the can a through field to perturbation Higgs rise seeded fluctuations give the the can phase, by this field During inflation Higgs phase. value, the heating critical and a curva- ends than When lower abruptly the runs perturbation. scale for entropy cutoff spectrum the the and power perturbation primordial ture the both to and mode mode. adiabatic iso-curvature the an being being other scalar one the two freedom, effectively are of there degrees consequence, a As sions). cls h aso h ig oo speitdt be to predicted is boson Higgs AS are the and there of SM if m the mass that between the argued scales scales, cosmo- been energy the has intermediate throughout It no vary thus evolution. constant and logical gravitational scale, the energy gravity, the AS of safety. equa- G RG asymptotical frame of of the conjecture form In the certain trunca- to and specific according action a tions gravity on the based experiment. of exactly any tion is in study testable current un- are The asymptotical theories safety gravity the the asymptotical of whether der proof ask to explicit deserves no it safety, is there present at h rvttoa hoyi smttclysafe. asymptotically if is universe inflationary theory early gravitational the the in role important an play ( H R efidtecrepnigprubto hoyleads theory perturbation corresponding the find We n omlgclcntn r unn ln with along running are Λ constant cosmological and oe 3](e lo[8 9 o eeaie discus- generalized for 39] [38, also (see [37] model ) pcrmo h omcMcoaeBack- Microwave Cosmic the of spectrum t 2 e ihol eea e netit [36]. uncertainty GeV several only with GeV 126 = da oe pcrmpae nupper an places spectrum power rdial tvl eomlzbevatenotion the via renormalizable atively nmlyculdt asymptotically to coupled inimally 1, † A28 Canada 2T8, 3A 85287 Although 2 they are consistent under a group of canonical values of of Ref. [37], the Ricci scalar and the cutoff scale are able curvaton parameters. to be identified as In this paper, we will work with the reduced Planck ξΛ 4GξΛ mass, Mp = 1/√8πGN , where GN is the gravitational R = 2(ΛIR )+ . (8) constant in the IR limit, and adopt the mostly-plus met- − ξGGN ξG ric sign convention ( , +, +, +). − Then we can reformulate the original theory as an f(R) model as follows, II. THE MODEL S = d4x√ g f(R)+ SM , (9) Z −  L  Consider the SM of minimally coupled to gravity with 2 [GN (R 2ΛIR)+2Z] R 2Λ f(R)= − , (10) S = d4x√ g − + SM . (1) 128πG2 Z Z −  16πG L  N where we have introduced Z = ξ /ξ . In this AS gravity frame, the gravitational constant G Λ G and the Λ vary along the cutoff scale p. The running behaviors are approximately de- III. INFLATIONARY COSMOLOGY scribed by

1 1 2 A. Background Dynamics G(p)− G− + ξGp , (2) ≃ N 2 Λ(p) ΛIR + ξ p , (3) ≃ Λ We now turn our attention to early universe inflation- ary solutions. This system is most easily studied by mak- where G and Λ are the values of gravitational con- N IR ing a conformal transformation, stant and cosmological constant in the IR limit. The co- efficients ξG and ξΛ are determined by the physics near G g˜ =Ω2g , Ω2 = N , (11) the UV fixed point of RG flows in AS gravity. µν µν G The scalar sector of the SM contains the . We use the unitary gauge for the Higgs boson H = h where Ω2 is the conformal factor with a new scalar field √2 and neglect all gauge interactions for the time being. In √6Mp GN this case, the Lagrangian of the Higgs field is given by, φ ln , (12) ≡− 2 G SM 1 µ being introduced. ∂µh∂ h V (h) , (4) L ⊇−2 − The original system, therefore, is equivalently de- scribed in terms of two scalar fields minimally coupled where V (h) is the potential of the Higgs field, which is to Einstein gravity without RG running, typically in form of λ (h2 v2)2. 4 − Varying the Lagrangian with respect to the metric, one R˜ 1 e2b(φ) derives the generalized Einstein equation, = ( ˜ φ)2 ( ˜ h)2 V˜ (φ, h) , (13) L 16πGN − 2 ∇ − 2 ∇ − R SM AS φ ˜ Rµν gµν +Λgµν =8πG(Tµν + Tµν ) . (5) where the factor b(φ) and V (φ, h) = U(φ) + − 2 ≡ √6Mp e4bV (h), with Here the RG running of G can effectively contribute to the stress energy tensor through 4 ξΛ 2b(φ) 2b(φ) U(φ) 8πM 1 e + GN ΛIR e . (14) ≃ p ξ  −   AS  1 G Tµν = ( µ ν gµν )(8πG)− , (6) ∇ ∇ − Note that, the form of φ’s potential is derived from the where we have introduced the covariant derivative µ RG running of the AS gravity [37]. The last term of U(φ) µν ∇ and the operator  g µ ν . The Higgs field h is proportional to G Λ . Substituting the observed val- ≡ − ∇ ∇ N IR obeys the Klein-Gordon equation. Additionally, the run- ues of GN and ΛIR, we find this term is insignificant ning of cutoff scale is controlled by the Bianchi identity, throughout the past cosmological evolution. Hence, we which requires, make the approximation

µG 4 ξΛ 2b(φ) (R 2Λ)∇ +2 µΛ=0 . (7) U(φ) 8πMp 1 e , (15) − G ∇ ≃ ξG h − i Consequently, the dynamics of this cosmological system which is a sufficiently flat inflationary potential in the are completely determined. Explicitly, from the analysis regime where b(φ) 1. ≪− 3

We denote the frame proceeding the conformal trans- C. Graceful exit formation as the Einstein frame (despite the non- canonical form of the h kinetic term). Substitution Eventually, the slow roll conditions are violated, when of the flat Friedmann-Robertson-Walker (FRW) metric, φ reaches φf . Consequently, φ enters a period of fast ds2 = dt2 +a2(t)d~x2, leads to the Friedmann equations: − roll, and finally approaches φ = 0 at which point the AS gravity reduces to traditional Einstein gravity. We ˙2 2b ˙ 2 2 1 φ e h ˜ suggest two possible reheating mechanism. One is that H = 2 ( + + V ) and (16) 3Mp 2 2 the inflaton φ decays to radiation, driving the universe 1 to a phase of thermal expansion directly; the other pos- H˙ = (φ˙2 + e2bh˙ 2) , (17) sibility is that the universe reheats only after the energy 2M 2 − p scale drops sufficiently so that the SM Higgs boson is responsible for the reheating process. Note that, in the where we have defined the Hubble parameter H a˙ and ≡ a latter case, after inflation the inflaton would experience the dot denotes the time derivative in the Einstein frame. a period of fast roll during which its effective equation The coupled Klein-Gordon equations for the two scalars of state approximately equals to 1. However, the energy are: density of the Higgs field scales as radiation due to the λh4 potential. As a consequence, the relative fraction of ¨ ˙ 2b ˙ 2 φ +3Hφ + V˜,φ = b,φe h , (18) Higgs boson could grow after inflation even in the case 2b that the Higgs has large coupling to other h¨ + (3H +2b,φφ˙)h˙ + e− V˜,h =0 . (19) . The decay of the inflaton φ into radiation strongly de- pends on the interaction between them. The detailed B. Inflationary Solution reheating process could be realized by the first mecha- nism or the second, or even the combination of these To search for a successful inflationary solution, we in- two. However, in the present paper we will not address troduce a series of slow roll parameters, on the details but simply consider the occupation of the Higgs boson at the last reheating surface as a free pa- φ˙2 e2bh˙ 2 V˜ rameter. Assuming that the Higgs boson would decay ǫ = , ǫ = , η = ,IJ . (20) φ 2M 2H2 h 2M 2H2 IJ 3H2 into radiation instantly at the last reheating surface (re- p p gardless which mechanism), the reheating temperature 1 2λ 4 is approximately given by Tre ( 2 ) hre where hre is The subscript “,I ” denotes the derivative with respect ≃ π gd the value of h at the reheating surface, and gd 106.75 to the Ith–field (with I being φ or h). During infla- ≃ tion, these parameters are required to be less than unity. is the number of degrees of freedom of the SM. After However, the key parameters need to yield a successful inflation but before reheating, h oscillates along the po- tential V λh4. Thus the value of h can be restricted inflationary background are associated with the scalar φ, ∼ i.e., ǫ and η . This is because the potential for φ is to v < hre < h with h being the value of the Higgs φ φφ | | ∗ ∗ flat in the regime φ Mp and correspondingly the pa- boson at the moment of Hubble-exit. rameters related to h≪−are suppressed by the small-valued factor e2b. The background dynamics is determined by the follow- IV. HIGGS CURVATON ing solutions (under slow roll approximation), A. Field fluctuations U e2bV U φ˙ ,φ , h˙ ,h , H2 . (21) 2 During inflation, the background dynamics are not af- ≃− 3H ≃− 3H ≃ 3Mp fected by the Higgs field, however its quantum fluctua- tions are able to source a nearly scale-invariant entropy Inflation ends when ǫφ = 1. Combing this condition with perturbation. In a two field inflationary model, we de- the background solution for φ˙ in (21), we find the value compose the field variables into a background part and of φ at the end of inflation: φf 0.56Mp. The number ≃− f Udφ fluctuations: φ φ + δφ and h h + δh. The field of e-folding of inflation is given by = M 2U , so → → N − i p ,φ fluctuations combine to give adiabatic and iso-curvature that R modes as:

b 3 2b(φ) δσ = cos θδφ + sin θe δh , (23) (φ) e− +3b(φ) 1.68 . (22) N ≃ 2 − δs = sin θδφ + cos θebδh , (24) − To obtain = 60, we require the initial inflaton value ˙ N with the trajectory angle being defined by cos θ = φ to be, φi 4.7Mp, which lies in the regime where RG σ˙ ≃ − ebh˙ flows of AS gravity have approached the UV fixed point. and sin θ = σ˙ [44]. To take into account the metric 4

fluctuation (the gravitational potential Φ) we introduce Hubble-exit value of the Higgs boson h can be related to ∗ the canonical perturbation variables, the initial amplitude of curvaton oscillation ho through a model-dependent function ho = g(h ). For example, σ˙ in the present model, if the curvaton starts∗ to oscillate vσ = a(δσ + Φ) , vs = aδs , (25) H immediately after inflation, ho h ; however, if there is a short slow rolling behavior for≃ h∗ following inflation, which characterize gauge-invariant adiabatic and iso- h 11 h . In this case, the curvature perturbation of curvature perturbations. Up to leading order in the slow o 12 the≃ Higgs∗ field in the oscillating phase is given by, roll approximations, these two variables obey the pertur- bation equations [45]: δρh δh ζh = qh ∗ , (28) 3(1 + wh)ρh ≃ h 2 a′′ ∗ vσ′′(s) + (k )vσ(s) 0 , (26) − a ≃ hoh∗ with qh h2 v2 . The coefficient qh can be further sim- ≡ o− where the prime denotes differentiation with respect to plified as q h∗ when h v if curvaton reheating h ho o conformal time, τ dt . Solving this equation in the occurs at energy≃ scales higher| | ≫ than the SM scale. ≡ a inflationary phase, weR can obtain nearly scale-invariant We now need to relate ζh to ζ. In the sudden decay ap- primordial power spectra for adiabatic and iso-curvature proximation, the relation can be computed analytically. perturbations, and their corresponding amplitudes are Consider the case that the Higgs boson decays on a uni- H∗ δσ δs at the Hubble-exit moment t . Hence, form total density hypersurface. On this slice we have | | ≃ | | ≃ 2π ∗ the amplitudes of the field fluctuations at this moment ρh + ρr = ρT where ρr and ρT denote the energy density are, of radiation and that of the total system, respectively. Making use of the expression for the curvature perturba- H H 4(ζr ζ) ∗ ∗ tion on a uniform density slice, we find ρr =ρ ¯re − δφ , δh b . (27) ∗ ∗ 3(1+wh)(ζh ζ) | |≃ 2π | |≃ 2πe and ρh =ρ ¯he during curvaton oscillation. ∗ − Thus, ζ and ζh are related on the reheating hypersurface as follows, B. The curvaton mechanism

4(ζr ζ) 3(1+wh)(ζh ζ) (1 Ωh)e − +Ωhe − =1 , (29) When inflation ends, φ rapidly approaches the IR limit − of the AS gravity. Because φ generally couples to other where Ωh = ρh/ρT is the dimensionless density param- fields through the conformal factor Ω2, we expect eter for the curvaton. For the curvaton mechanics to radiation to be produced following the inflationary phase. succeed, we must assume that the fluctuation ζr seeded Recall that, the Higgs boson can survive during inflation by the inflaton field is negligible. We will address this due to the slow roll conditions and then starts to oscil- concern below and now turn our attention to the Higgs late along the λh4 potential. Consequently, the universe curvaton. Therefore, we have is dominated by both the radiation and the Higgs boson 1 3(1 + wh)Ωh after inflation . This process is analogous to the familiar ζ = qT ζh , qT = , (30) curvaton scenario. Instead of a matter-like curvaton os- 4 (1 3wh)Ωh − − cillation with w = 0 as studied in [40, 46], a λh4 potential yields an effective equation of state for the Higgs boson, and in our explicit case, qT = Ωh at the curvaton decay 1 surface. which is the same as radiation with wh = 3 [47, 48]. Thus, this generalized curvaton mechanism [49] may be Combining Eqs. (28) and (30) and the field fluctua- used to generate the primordial curvature perturbation tion (27), we obtain the primordial power spectrum of in agreement with the current CMB measurements. curvature perturbation seeded by the Higgs boson, We begin by writing down the relation between the 2 2 2 qhqT H curvaton fluctuation δh and its curvature perturbation Pζ = ∗ . (31) 4π2e2b∗ h2 ζh. Choosing the spatially flat slice for the Higgs curva- ∗ ton, one finds ρ =ρ ¯ e3(1+wh)ζh in the neighborhood of h h We see from (31), that the final curvature perturba- the curvaton reheating hypersurface. Consider the cur- tion depends on five parameters: qh, qT , H , h , and vaton perturbation generated from vacuum fluctuations ∗ ∗ eb∗ . Compared with the usual curvaton mechanism, our inside the Hubble radius. These fluctuations satisfy a model contains a new parameter eb∗ due to the confor- Gaussian distribution at the Hubble-exit. In general, the mal transformation made in Eq. (11). However, since the background dynamics of inflation are driven by the RG running of AS gravity, we find eb∗ 0.15 for observable perturbation modes at Hubble-exit.≃ Moreover, we have 1 To be precise, there exists a short period during which the uni- specified the curvaton to be the Higgs boson and hence, verse is dominated by a fast-roll φ with wφ ≃ 1. However, as the contribution of φ would dilute out soon, only radiation and the the potential is of an explicit form, and thus qh 1. 9∼ Higgs boson could be left after the fast-roll phase. Since the latest CMB data reveals Pζ 2.4 10− , we ≃ × 5

5 h∗ can deduce the useful relation H 4.5 10− . Subse- VI. CONCLUSIONS AND DISCUSSION ∗ ≃ × qT quently, we are left to constrain qT , H and h by various ∗ ∗ theoretical and observational requirements. In the follow- In this paper, we proposed a new inflation model in ing, we calculate the tensor-to-scalar ratio, the spectral the context of an asymptotically safe gravitational cos- index, and the reheating temperature, respectively, and mology with renormalization group running gravitational then constrain the remaining parameters. constant. The theory is minimally coupled to the Stan- dard Model sector. As observed in [36], if there are no intermediate energy scales between the SM and the AS V. CONSTRAINTS scales, the mass of the Higgs boson is determined by a fixed point and is approximately 126GeV. This predic- The calculation of primordial tensor perturbations is tion is coincident with the recent results announced from identical to that of ordinary inflationary models, and thus the ATLAS and CMS experiments, indicating that the the tensor power spectrum is nearly scale-invariant, given 2 Higgs mass is in the range 116 131GeV (ATLAS[19]) 2H∗ or 115 127GeV (CMS[20]), with− other masses excluded by Pt = π2M 2 . The tensor-to-scalar ratio: p at the− 95% confidence level. In the frame of the AS in- 2b∗ 2 flation model, we introduce the Higgs boson to play the Pt 8e h r = ∗ . (32) role of a curvaton which is responsible for generating the ≡ P q2q2 M 2 ζ h T p primordial curvature perturbation. Furthermore, if the 2 occupation of curvaton density at the reheating surface b∗ 0.2h∗ Since e 0.15 and qh 1, we obtain r q2 M 2 . Ac- is fixed, then observational constraints on the spectral ≃ ∼ ≃ T p cording to the latest CMB data, r is required to be less index imply an upper bound on the Higgs mass. After a than 0.36 [17], so that h < 2qT Mp. A further constraint fine tuning of the curvaton parameters, the model is con- comes from the curvaton∗ condition that the contribution sistent with recent LHC data. A complete data fitting of of inflaton fluctuation to curvature perturbation should our model will appear in the forthcoming work [50]. be negligible. This condition requires qT ζh (1 qT )ζr. We conclude with a few remarks concerning the details ≫ − Since ζr corresponds to the radiation perturbation, inher- of the model as follows. H∗ ited from inflaton fluctuation, ζr . Choosing The first involves the estimate of the values of ξ and ≃ 2π√2ǫ∗Mp Λ a group of canonical parameter values, this condition re- ξG from the AS gravity. In a general case, AS gravity quires h < qT Mp which is close to the observational implies this parameter is typically less than unity but bound provided∗ by the tensor-to-scalar ratio. Combin- strongly depends on the detailed parametrization of RG ing this inequality and the expression of H derived from flows. Namely, in the simplest parametrization in which ∗ Pζ , one obtains a constraint on the inflationary Hubble the UV and IR behaviors are connected in a linear ap- 5 parameter H < 4.5 10− Mp. proximation, the ratio of ξΛ and ξG often leads to infla- During inflation∗ the× Hubble parameter is approxi- tionary scale which is much higher than the GUT scale, mately constant and the field fluctuations are approx- and thus the inflation model based on this example suf- imately conserved after Hubble exit. The spectral tilt fers from a fine tuning problem [37]. However, one may d ln Ph nh 1+ d ln k of the primordial perturbations at the consider RG flows which are more general than the linear moment≡ of horizon crossing is: approximation. For example, one can parameterize the

1 RG flows by a log approximation as suggested in Ref. 2 2 2 2 2ǫ qhqT mH [38]. In this case, the parameter space is significantly n 1 ∗ 2ǫ + , (33) h 2 2 larger and a very small value of ξ /ξG is allowed. ≃ − √3 − ∗ 4π v Pζ Λ Our second remark concerns the estimate of the values where in the r.h.s. we have used the expression for the of the fields φ and h at the beginning of inflation. In prin- primordial power spectrum. The Higgs mass is deter- ciple, they may be completely determined by numerical mined by mH = √2λv. In this model the inflaton po- calculations. We may choose the inflaton φ to initially be 4 tential is explicitly defined, and yields ǫ 1.6 10− at about a few times the Planck mass in order to attain a the beginning of inflation. Thus, the spectral≃ × index can sufficiently long inflationary period. Interestingly, these 2 2 2 qhqT mH be simplified as nh 0.985 + 2 2 for perturbation values are also consistent with observational constraints ≃ 4π v Pζ 2 modes which exit the Hubble radius during the first sev- given in the present paper . eral efolds. As a consequence, the Higgs mass acquires The third is on the relation between the spectral index an upper bound from the CMB measurement: of the primordial power spectrum and the Higgs mass obtained in Eq. (33). We should be aware of that this 5 v mH < 3 10− . (34) × qT

For the SM Higgs, v 246GeV, and if qT is smaller than 2 Note that the scenario of Higgs curvaton is recently analyzed in 5 ≃ 5.7 10− , we conclude that the Higgs mass has to be × [51] and some stringent bound on primordial non-Gaussianity is less than 129GeV. achieved. 6 relation is only derived at tree level in the present pa- ments and introducing relevant references on the non- per. However, one would expect the relation to be altered minimally coupled Higgs inflation model and [51]. The when radiative corrections are taken into account. This work of Y.F.C. and D.A.E is supported in part by the is an interesting topic which we would like to address in DOE and the Cosmology Initiative at Arizona State Uni- the following-up project. versity.

Acknowledgments

We are grateful to R. Brandenberger for discussions. We also thank the anonymous referee for valuable com-

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