Beyond the Starobinsky model for inflation
YYHEP-COS20-04, CAU-THEP-2020-02, CERN-TH-2020-022
Beyond the Starobinsky model for inflation
Dhong Yeon Cheong and Seong Chan Park ∗ Department of Physics & IPAP & Lab for Dark Universe, Yonsei University, Seoul 03722 Korea
Hyun Min Lee † Department of Physics, Chung-Ang University, Seoul 06974, Korea andCERN, Theory department, 1211 Geneva 23, Switzerland
We single out the Starobinsky model and its extensions among generic f ( R ) gravity as attractorsat large field values for chaotic inflation. Treating a R curvature term as a perturbation of theStarobinsky model, we impose the phenomenological bounds on the additional term satisfying thesuccessful inflationary predictions. We find that the scalar spectral index can vary in both the redor blue tilted direction, depending on the sign of the coefficient of the R term, whereas the tensor-to-scalar ratio is less affected in the Planck-compatible region. We also discuss the role of higherorder curvature term for stability and the reheating dynamics for the unambiguous prediction forthe number of efoldings up to the R term. I. INTRODUCTION
Cosmic inflation solves various problems of standardBig Bang cosmology including the horizon problem, ho-mogeneity, structure formation, etc, and it has beentested by the measurements of Cosmic Microwave Back-ground anisotropies with unprecedented precision. Fa-vored vanilla single-field inflations consist a canonical ki-netic term, and some with monomial type potentials havenow been excluded at more than 2 σ level by the measuredscalar spectral index and the bound on the tensor-to-scalar ratio [1].The Starobinsky inflation model [2] drew new attentionfrom the fact that a successful slow-roll inflation can beobtained with a single parameter beyond the SM, namelythe coefficient of the R curvature term. The inflation-ary predictions of the Starobinsky model are well con-sistent with the Planck data. Therefore, the discussionhas been generalized to a class of Starobinsky-like modelswith common properties during inflation [3–7], includingthe Higgs inflation as a particular case [8].The unitarity issue is important in defining the validityof the semi-classical treatment of inflationary dynamics.In the case of the original Higgs inflation with a large non-minimal coupling, the unitarity problem occurs due tothe would-be Goldstone components of the Higgs field [9–12], which motivated sigma-model type extensions [13–16]. In the case of Higgs inflation at criticality where ∗ [email protected]; co-corresponding author † [email protected]; co-corresponding author both the Higgs quartic coupling and its beta functioncoefficient almost vanish [17, 18], the unitarity scale is farabove the Hubble scale during inflation, so the unitaryproblem is much milder.In the case of the Starobinsky model, the dynamicsof the dual scalar field can unitarize the Higgs inflationup to the Planck scale [6, 7]. Making an appropriatefield redefinition of the dual scalar field and transform-ing to the Einstein frame, the Starobinsky model pro-vides an appropriate coupling between the dual scalarfield and the Higgs field such that Higgs inflation is re-covered below the mass of the dual scalar field [19–22].Other theoretical issues such as fine-tuning [23], swamp-land conjecture [24] and the Palatini formulation of Higgsinflation [25, 26] are also recently addressed.It has also been shown recently that the nontrivial in-flaton trajectory in the Higgs- R inflation [19, 27] canprovide an interesting possibility that primordial blackholes can form during inflation as the dark matter can-didate [28, 29]. However, in the region of the parame-ter space where primordial black holes saturate the relicdensity, the resulting spectral index of the curvature per-turbations is slightly more red-tilted as compared to thebest-fit value of the Planck data at 1 σ level [28, 29].In this article, we discuss the Starobinsky inflationmodel among general f ( R ) gravity models from the pointof attractors at large fields for chaotic inflation. Extend-ing the Starobinsky model with a cubic R curvatureterm, we impose the conditions on the cubic curvatureterm for maintaining a successful inflation and identifyhow the inflationary predictions of the Starobinsky modelcan be modified. We also briefly discuss the potential in- a r X i v : . [ h e p - ph ] A p r stability of the cubic term and the effects of even higherorder curvature term on this issue. The reheating dynam-ics up to R term is also dealt with for completeness.The article is organized as follows. We begin with aconnection between a generic f ( R ) gravity and its scalardual theory. Then, we show the criteria for f ( R ) grav-ity to give successful predictions for inflation. Next, weextend the Starobinsky model with a cubic R term andderive the inflationary observables as compared to thoseof the Starobinsky model. We go on to discuss the re-heating dynamics up to R correction and show the un-ambiguous prediction for the number of efoldings in thiscase. We also discuss the roles of the dual scalar field inthe extended Starobinsky model for unitarizing the Higgsinflation with a non-minimal coupling and curing the vac-uum instability problem in the SM. Finally, conclusionsare drawn. II. THE DUAL SCALAR THEORY OF f ( R ) We can connect a generic f ( R ) gravity to a correspond-ing scalar-tensor theory by Legendre transformation: S = 12 (cid:90) d x √− g f ( R ) (1) → S = 12 (cid:90) d x √− g [ f ( φ ) + f (cid:48) ( φ )( R − φ )] (2) ≡ (cid:90) d x √− g (cid:20)
12 Ω R − V ( φ ) (cid:21) (3)where the frame function and the potential are respec-tively given as Ω ( φ ) = f (cid:48) ( φ ) , (4) V ( φ ) = 12 [ φf (cid:48) ( φ ) − f ( φ )] . (5)One notes that the variation δφ of the second equationrecovers the original action.The action in the Einstein frame can be obtained byWeyl transformation g Eµν = Ω g µν : S E = (cid:90) d x √− g E (cid:20) R E − g µνE ∂ µ s∂ ν s − V E ( s ) (cid:21) , (6)where the canonical field s and the potential in the Ein- stein frame is V E are respectively given as s ( φ ) = (cid:114)
32 ln Ω ( φ ) = (cid:114)
32 ln [ f (cid:48) ( φ )] , (7) V E ( s ) = V ( φ ( s ))Ω( φ ( s )) = φf (cid:48) ( φ ) − f ( φ )2 f (cid:48) ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) φ = φ ( s ) , (8)where φ ( s ) can be obtained by inverting s ( φ ).We note that the chaotic inflation constrains theasymptotic form of f ( φ ): for instance, a monomial func-tion f ( φ ) ∼ φ n +1 leads to V E ∼ φ n +1 φ n ∼ φ − n (9)such that n = 1 gives a flat potential for inflation. III. SELECTION RULES FOR INFLATION
We consider a general form of the higher curvaturecorrection to Einstein gravity by taking f ( R ) = aR + bR n +1 with n ≥
1, and discuss the selection rules for asuccessful slow-roll inflation.Putting a = 1 and b ≡ β/ ( n + 1), f (cid:48) ( φ ) = 1 + βR n , s ( φ ) = (cid:114)
32 ln [1 + βφ n ] , (10) σ ( s ) ≡ e √ s = 1 + βφ n . (11)The equation is easily solved and we obtain φ ( s ): φ ( s ) = (cid:18) σ ( s ) − β (cid:19) n . (12)The potential in Einstein frame is V E ( s ) = φ ( s ) σ ( s ) − f ( φ ( s ))2 σ ( s ) (13)= n n + 1) β /n ( σ − n +1 n σ , (14)where f ( φ ( s )) = (cid:16) σ ( s ) − β (cid:17) n + b (cid:16) σ ( s ) − β (cid:17) n +1 n is alreadytaken into account. When n = 1, we recover theStarobinsky’s inflaton potential V E ( s ) = β (1 − e − √ s ) .Indeed, the case n = 1 is special: when we considerthe large field limit, s (cid:29) σ ( s ) (cid:29) s →∞ V E = n n + 1) β /n e (cid:114) (cid:16) − nn (cid:17) s , (15) ¯ g ab = e ψ g ab gives √− ¯ g = e Dψ √− g and ¯ R = e − ψ (cid:2) R − D − ∇ ψ − ( D − D − g ab ∂ a ψ∂ b ψ (cid:3) in D -dimensions. which approaches constant if n = 1 so that we can real-ize a large field inflation scenario as Starobinsky pointedout [2].By expanding the potential the nave cutoff scale of thetheory near s ∼ V E = α n (cid:88) k =1 ,(cid:96) =0 ( − (cid:96) (2 / ( k + (cid:96) ) / k ! (cid:96) ! s k + (cid:96) (16) ≡ (cid:88) k =1 ,(cid:96) =0 s k + (cid:96) Λ k + (cid:96) − , (17)where the cutoff scales for operators with mass dimension D = k + (cid:96) > D = (cid:20) k ! (cid:96) ! α n ( − (cid:96) (2 / k + (cid:96) (cid:21) k + (cid:96) − (18)where α n = nβ − n n +1) . Now requesting Λ D >
1, we find thelower bound on β as β > (cid:20) n (cid:96) + k n + 1)3 k + (cid:96) k ! (cid:96) ! (cid:21) n , k + (cid:96) > . (19)As the number in the parentheses is smaller than unityin the region of our interest, the theory setup does notsuffer from unitarity issues below the Planck scale as longas the condition in Eq. (19) is satisfied. IV. EXTENSION OF THE STAROBINSKYMODEL
Given that the Starobinsky model is selected for in-flation as an appropriate extension of the Einstein grav-ity, we introduce a cubic curvature term as the exten-sion of the Starobinsky model, namely, take f ( R ) = aR + bR + cR . Then, we present the modified pre-dictions for inflation in this case. Taking a = 1 , b = β/ , and c = γ/
3, we get the framefunction in the dual scalar theory as f (cid:48) ( φ ) = 1+ βφ + γφ ,and s ( φ ) = (cid:114)
32 ln (cid:2) βφ + γφ (cid:3) , (20) σ ( s ) ≡ e √ s = 1 + βφ + γφ . (21)The quadratic equation is easily solved and we get φ ( s ): φ ( s ) = β γ (cid:18)(cid:114) γβ ( σ ( s ) − − (cid:19) . (22) We note other extensions of the Starobinsky model were alsostudied with different perspectives [30–33]. If γ is small ( γ (cid:28) β ) and φ ∼
1, we may treat the γ term as a small perturbation in σ ( s ), so that we find aconvenient approximation βφ ( s ) + 1 = σ ( s ) − γβ ( σ ( s ) − + · · · , or φ ( s ) = σ ( s ) − β (cid:34) − γβ (cid:18) σ ( s ) − β (cid:19) + O (cid:18) γβ (cid:19) (cid:35) . (23)The potential in Einstein frame is V E ( s ) = βφ ( s ) (1 + γ β φ ( s ))4 (cid:16) βφ ( s )(1 + γβ φ ( s )) (cid:17) , (24) ≈ V ( s ) (cid:20) − γβ (cid:18) σ ( s ) − β (cid:19) + · · · (cid:21) (25)where V ( s ) = β (1 − σ ) = β (1 − e − √ s ) is the po-tential for γ = 0. As the potential is expanded by powersof (cid:112) / s , this setup is free from unitarity issues. A. Inflation
The slow-roll parameters are (cid:15) = 12 (cid:18) V (cid:48) E V E (cid:19) = (cid:15) + γβ ∆ (cid:15), (26) η = V (cid:48)(cid:48) E V E = η + γβ ∆ η (27)where (cid:15) and η are the slow roll parameters when γ = 0and the corrections are perturbatively calculated as: (cid:15) = 43( σ ( s ) − , (28) η = − σ ( s ) − σ ( s ) − , (29)∆ (cid:15) = − σ ( s )9 β ( σ ( s ) −
1) + O (cid:18) γβ (cid:19) , (30)∆ η = − σ ( σ + 3)9 β ( σ −
1) + O (cid:18) γβ (cid:19) . (31)The number of efoldings from the start ( s ) to the end( s e < s ) of inflation is calculated N e ( s ) = (cid:90) ss e ds √ (cid:15) (32)= N e + ∆ N e , (33)where N e for γ = 0 and the correction term ∆ N e are N e = (cid:90) ss e ds √ (cid:15) = (cid:34) σ ( s ) − √ s (cid:35) s s e , (34) ≈ σ ( s ) , ( s (cid:29) s e ∼ n s , r ) for N e = 60(blue), N e = 56 . N e = 55(purple) efoldings with δ = [ − . , . × − vsPlanck2018 1 σ (Yellow) and 2 σ (Green) constraints [1].and ∆ N e = − (cid:18) γβ (cid:19) (cid:90) ss e ∆ (cid:15) d | s | / (cid:15) / (36) ≈ (cid:18) γβ (cid:19) σ ( s ) β , ( s (cid:29) s e ∼ s e requesting Min( (cid:15), | η ) | ) = 1 and s ∗ requesting60-efoldings: N e ( s ∗ ) = 34 σ ( s ∗ ) + (cid:18) γβ (cid:19) σ ( s ∗ ) β = 60 . (38)Finally, from the COBE normalization [1], V ∗ (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) s ∗ ≈ σ ( s ∗ )16 β + (cid:18) γβ (cid:19) σ ( s ∗ )8 β = 0 . . (39)where V ∗ is the inflaton vacuum energy at horizon exist,and correspondingly determine the R coupling as β ≈ . × (cid:16) N (cid:17) . (40)Having two conditions from Eqs. (38) and (39), we nowtry to make the predictions for cosmological observations.Here we consider the spectral index and the tensor-to-scalar ratio taking σ ( s ∗ ) ≈ N e − δ N e where δ ≡ γ/β (cid:28) n s = 1 − (cid:15) ( s ∗ ) + 2 η ( s ∗ ) (41) ≈ − N e − N e − δ N e , (42)and r = 16 (cid:15) ( s ∗ ) ≈ N e − δ . (43) FIG. 2: The bound on δ for varying N e ∈ (50 ,
65) fromPlanck2018 1 σ (Yellow) and 2 σ (Green) constraints [1]. N e = 56 . δ = 0, we recover the well-known relations in R inflation and consequently Higgs inflation with non-minimal coupling [34] and the small δ -corrections giveadditional contributions to observables so that we canset the bounds on the size of δ [28, 30].In Fig. 1, we show the effect of the R correction with δ = γ/β ∼ − in comparison with the Planck con-straints in ( n s − r ) plane [1]. The blue, red and purplelines from bottom to top correspond for N e = 60 , . N e = 56 .
9, is theefolding number required for solving the horizon problemobtained by considering reheating, which we discuss indetail in the next section. Due to the negative correctionto n s and the positive correction to r from the positive δ ,the prediction moves from right-up to the left-down when δ changes from − . × − to 2 . × − . The middlepoint is for δ = 0 corresponding to the Starobinsky limit(or the Higgs inflation limit).In Fig. 2, we show the bound on δ for different choicesof N e = 50 −
65 taking the Planck 2018 data into account.The vertical dotted line depicts the case N e = 56 . dn s d log k = − ξ + 16 (cid:15)η − (cid:15) (44)= − N e + 12881 δ (cid:18) − N e (cid:19) (45)where ξ = V (cid:48) E ( s ) V (cid:48)(cid:48)(cid:48) E ( s ) V E ( s ) . The TT,TE,EE+ lowE+lensingconstraint from Planck 2018 [1] is dn s d log k = − . ± . . (46)That leads − . < δ | N e =60 < . R corrections. In par-ticular, for a negative value of γ , there is a potentialinstability developing at large inflaton field values. Fromeq. (8), the Einstein frame potential with the R termincluded is given explicitly as a function of φ by V E ( φ ) = 14 βφ + γφ (1 + βφ + γφ ) . (47)The field φ is not a canonical field due to the modifiedkinetic term, but it is sufficient to take the above poten-tial for the analysis of the initial condition for inflation.Then, we find that there exists a maximum of the po-tential at φ c = − δβ > β > δ <
0, but itis located far beyond the regime of the slow-roll infla-tion near φ e ∼ β N e , that is, φ c (cid:29) φ e for | δ | ∼ − .Nonetheless, there might be a concern on the correct ini-tial condition for the slow-roll inflation, φ i , because theinflaton could have rolled down to a wrong minimum for φ i > φ c . Therefore, we restrict ourselves to the infla-ton field values satisfying φ i < φ c , such that the initialcondition for the slow-roll inflaton is set for our previousdiscussion to hold.We remark that even higher order curvature correc-tions such as κR can be included, but their effects aresubdominant compared to the contributions up to R term, as far as the coefficient of the new correction termis small enough. In particular, the dual scalar theory forthe extension with R gives rise to a quartic potential,as f ( φ ) = φ + βφ + γφ + κφ , thus stabilizing thescalar potential for κ >
0. For a small κ coupling, therecan be a new minimum sufficiently far away from the in-flationary regime, nevertheless the inflation can roll downto a correct vacuum after inflation, being consistent withthe perturbativity of the R term. Several studies inthe literature deal with the curvature terms beyond theStarobinsky inflation model [30–33] and inflation withhigher curvature terms in four or higher dimensions [35–38]. V. REHEATING
In this section, we discuss the reheating dynamics inthe Starobinsky model via the minimal gravitational in-teractions and the impact on the precise determinationof the number of efoldings. The interaction Lagrangian between the inflaton andthe SM in Einstein frame is given in terms of the traceof the energy-momentum tensor [16, 39], as follows, L int √− g = − f (cid:48) ( φ ) T µµ = − e − √ s T µµ (48)with T µµ = − ( ∂ µ h ) + 4 V E + m f v h ¯ f f − δ V m V v h V µ V µ + T µµ, loops . (49)Here, h is the Higgs boson, f denotes the SM fermions, V = W, Z with δ V = 1 ,
2, respectively, and T µµ, loops cor-respond to the loop corrections due to trace anomalies[16]. Expanding the inflaton near the minimum of theinflaton potential, we identify the inflaton coupling as L int = √ s T µµ . Then, assuming that electroweak sym-metry is already broken at the time of reheating, the totaldecay rate of the inflaton with m s (cid:29) m h , m V is domi-nated by the inflaton decay modes into the electroweaksector [16], given approximately byΓ s ≈ m s πM P . (50)Here, from Eq. (40), the inflaton mass is given by m s = M P √ β = 2 . × GeV (cid:16) N e (cid:17) . (51)As a result, using Eq. (50) with Eq. (51), the reheatingtemperature is determined from the perturbative decayof the inflaton as T RH = (cid:18) π g ∗ (cid:19) / (cid:112) M P Γ s = (cid:18) g ∗ (cid:19) / (cid:18) N e (cid:19) / × (4 . × GeV) . (52)It is known that the number of efoldings required tosolve the horizon problem depends on the reheating tem-perature T RH and the equation of state w during reheat-ing [40], as follows, N e = 61 . w − w ) ln (cid:18) V ∗ π g ∗ T (cid:19) − ln (cid:18) V / ∗ H ∗ (cid:19) . (53)In our model, the universe is dominated by matter duringinflation, i.e. w = 0. Therefore, using the results ineqs. (52) and (39), we determine the number of efoldingsas N e = 56 . . (54)Consequently, from Fig. 1, we can make a definite predic-tion for the spectral index and the tensor-to-scalar ratioup to R corrections. VI. UNITARIZING HIGGS INFLATIONBEYOND THE STAROBINSKY MODEL
In this section we discuss the roles of the dual scalarfield for unitarizing the Higgs inflation beyond theStarobinsky model and solving the vacuum instabilityproblem in the SM.In the extended Starobinsky model with f ( R ) = R + βR + γR , discussed in the previous sections, we in-clude a non-minimal coupling ξ for the Higgs field h inunitary gauge. Then, in the dual scalar theory, the framefunction in Eq. (4) becomesΩ ( φ ) = 1 + βφ + γφ + ξh . (55)Moreover, we also add the Higgs potential in Jordanframe to get V ( φ, h ) = 14 βφ + 13 γφ + 14 λ ( h − v ) . (56)Then, similarly as in Eq. (21), we make the field defini-tion by β ˆ σ = 1 + βφ + γφ + ξh . (57)From this, taking the R curvature term as perturba-tions, the approximate solution for φ to the above equa-tion is given in terms of ˆ σ and h by φ (ˆ σ, h ) = ˆ σ − β − ξβ h − γβ (cid:16) ˆ σ − β − ξβ h (cid:17) , (58)in turn, leading to the Jordan frame action in a simpleform, S = (cid:90) d x √− g (cid:20) β ˆ σR −
12 ( ∂ µ h ) − β (cid:16) ˆ σ − β − ξβ h (cid:17) + 16 γ (cid:16) ˆ σ − β − ξβ h (cid:17) − λ ( h − v ) (cid:21) . (59)This is nothing but the induced gravity model, unita-rizing the Higgs inflation [7, 13, 19, 20]. By using theequation of motion for ˆ σ with ˆ σ = β + ξβ h , we can inte-grate out the ˆ σ field to get precisely the effective actionfor the Higgs inflation [7, 13]. In this process, the R cur-vature term maintains the same equation of motion forthe ˆ σ field as in the Starobinsky model. In this regard,we can take the extended Starobinsky model as an UVcompletion of the Higgs inflation up to the Planck scale. As discussed in the previous sections, the robustness ofthe Starobinsky model for a successful inflation can beensured in the presence of small higher curvature terms.Finally, we remark that the approximate potential inEinstein frame can be obtained from V E = V / Ω at thelinear order in γ , as follows, V E (cid:39) β ˆ σ (cid:20) β (cid:16) ˆ σ − β − ξβ h (cid:17) − γ (cid:16) ˆ σ − β − ξβ h (cid:17) + 14 λ ( h − v ) (cid:21) . (60)As a result, for (cid:104) ˆ σ (cid:105) (cid:39) β , we find that the running Higgsquartic coupling is given by λ h = λ + ξ β , (61)which amounts to a positive tree-level shift for β >
0, en-suring the vacuum stability in the SM for a given value λ , inferred from the Higgs mass [41, 42], as far as theperturbativity constraint on the running Higgs quarticcoupling, i.e. ξ /β (cid:46)
1, is satisfied. Furthermore, the R curvature term leads to a suppressed dimension-6 op-erator, L D = − c H h with c H = γ ξ /β = δ ξ /β (cid:46) δ β / (cid:46) . N/ /M P where we used ξ /β (cid:46) | δ | (cid:46) − and Eq. (40). VII. CONCLUSION
We considered an f ( R ) = R + βR / γR / R term asperturbations, we identified the modifications to the in-flationary parameters of the original Starobinsky model.We also showed that the dual scalar theory is well de-fined without issues regarding unitarity below the Planckscale. The analytic expressions for the scalar spectral in-dex ( n s ) and the tensor-to-scalar ratio ( r ) were derivedand compared with the Planck 2018 results. We foundthat the ratio of the coefficient of R ( γ ) and that of R ( β ) is constrained as | γ/β | < . × − at 2 σ level or0 . × − at 1 σ level, which is consistent with the treat-ment of δ = γ/β as small perturbations in our analy-sis. As an important consequence of this study, we foundthat a slight negative R correction to the Higgs- R in-flation may provide a better fit in n s − r plane when theprimordial black hole production is significant [29] as no-ticed earlier by other authors [28]. Lastly we showed thatthe dual scalar field in the extended Starobinsky modelis responsible for unitarizing the Higgs inflation in thepresence of the non-minimal coupling for the Higgs field. Acknowledgments — We thank Shi Pi, Misao Sasakiand Qing-Guo Huang for helpful discussions and com-ments. The work was initiated during the CERN-CKCTheory Institute on New Physics in Low-Energy Preci-sion Frontier in 2020. The work is supported in part by Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistry of Education, Science and Technology (NRF-2018R1A4A1025334, NRF-2019R1A2C2003738 (HML),NRF-2019R1A2C1089334 (SCP)). [1] Y. Akrami et al. (Planck), (2018), arXiv:1807.06211[astro-ph.CO].[2] A. A. Starobinsky, Phys. Lett. , 99 (1980),[,771(1980)].[3] R. Kallosh and A. Linde, JCAP , 002 (2013),arXiv:1306.5220 [hep-th].[4] R. Kallosh, A. Linde, and D. Roest, Phys. Rev. Lett. , 011303 (2014), arXiv:1310.3950 [hep-th].[5] R. Kallosh, A. Linde, and D. Roest, JHEP , 198(2013), arXiv:1311.0472 [hep-th].[6] A. Kehagias, A. Moradinezhad Dizgah, and A. Riotto,Phys. Rev. D89 , 043527 (2014), arXiv:1312.1155 [hep-th].[7] G. F. Giudice and H. M. Lee, Phys. Lett.
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