The Reconstruction of Non-Minimal Derivative Coupling Inflationary Potentials
uuniverse
Communication
The Reconstruction of Non-Minimal DerivativeCoupling Inflationary Potentials
Qin Fei , Zhu Yi * and Yingjie Yang School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, China; [email protected] Department of Astronomy, Beijing Normal University, Beijing 100087, China School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China; [email protected] * Correspondence: [email protected]: date; Accepted: date; Published: date (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:1) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Abstract:
We derive the reconstruction formulae for the inflation model with the non-minimal derivativecoupling term. If reconstructing the potential from the tensor-to-scalar ratio r , we could obtain thepotential without using the high friction limit. As an example, we reconstruct the potential from theparameterization r = α / ( N + β ) γ , which is a general form of the α -attractor. The reconstructed potentialhas the same asymptotic behavior as the T- and E-model if we choose γ = α (cid:28)
1. We alsodiscuss the constraints from the reheating phase by assuming the parameter w re of state equation duringreheating is a constant. The scale of big-bang nucleosynthesis could put an upper limit on n s if w re = n s if w re = Keywords: reconstruction; non-minimal derivative coupling inflation; reheating
1. Introduction
In the standard big-bang cosmology, inflation has successfully solved various problems, such asthe flatness, horizon and monopole problems. Besides, its quantum fluctuation can produce the seedof the formation of large-scale structure [1–4]. A scalar field with a flat potential is usually chosen toinvestigate inflation. The most economical and fundamental candidate for the inflaton is therefore theStandard Model Higgs boson. However, the Higgs boson is disfavored by the observational data [3,5]when minimally coupled to gravity due to its large tensor-to-scalar ratio. If the kinetic term of the scalarfield is non-minimal coupled to Einstein tensor, the tensor-to-scalar ratio r could be reduced to beingconsistent with the observational data, and the effective Higgs self-coupling λ could be the order of 1 [6,7].This inflation model with non-minimal derivative coupling belongs to the subclass of the Horndeskitheory [8], which is a general scalar–tensor theory, with field equations that are at most of the second-orderderivatives of both the metric g µν and scalar field φ in four dimensions [9]. Therefore, the non-minimalderivative coupling inflation model could save the Higgs model without introducing a new degree offreedom. For more about the non-minimal derivative coupling inflation model, refer to [10–17].The most important observables of inflation are the spectral tilt n s and the tensor-to-scalar ratio r . Tobe compared with the observational data easily, they are usually expressed by the e -folding number N before the end of inflation at the horizon exit of the pivotal scale. Among them, one of the predictionsthat is greatly favored by the observational data may be the α -attractors, n s = − N and r = α / N .Numerous inflation models make the α -attractors prediction, for example the Starobinsky model [1],the Higgs inflation with a non-minimal coupling ξφ R in the strong coupling limit ξ (cid:29) Universe , a r X i v : . [ g r- q c ] N ov niverse , , 5 2 of 13 pole inflation with the kinetic term being ( ∂φ ) / ( − φ /6 α ) [20] and the T/E model [21,22]. It istherefore worth studying whether there are still other models that can make the prediction of α -attractors.In this paper, we consider the non-minimal derivative coupling inflation models to investigate this α -attractors issue by reconstructing the potential. Starting from the observational data and parameterizingthe observable with N , using the relationships between the observable and the potential, we can thenreconstruct the potential [23,24]. By this reconstruction, the model parameters can be constrained easilyand the reconstructed potential would always be consistent with the observational data[24–47].After the inflation, it is followed by the reheating phase, which may give additional constraints on theinflation phase [46,48]. Assuming that the effective parameter w re of state equation during reheating isa constant and the entropy is a conserved quantity, we can relate the e -folding number and the energyscale during reheating to those during inflation [48–54]. From these relations, the constraints on the energyscale during reheating would transfer to the constraints on the inflation model.In this paper, we reconstruct the inflationary potentials of the non-minimal coupling inflation modelsand research the additional constraints from the reheating phase. The paper is organized as follows. InSection 2, we give a brief review about the inflation model with the non-minimal derivative coupling termand the reconstruction method. In Section 3, we reconstruct the potential from the parameterization oftensor-to-scalar ratio r . We discuss the constraints from the reheating in Section 4, and give the conclusionin Section 5.
2. The Relations
In this section, we develop the formulae for the reconstruction of the inflationary potential with thekinetic term non-minimal coupled to Einstein tensor. We start from the action S = (cid:90) d x (cid:112) − g (cid:20) R − g µν ∂ µ φ∂ ν φ + M G µν ∂ µ φ∂ ν φ − V ( φ ) (cid:21) , (1)where we choose the unit c = M pl = ( π G ) = M is a constant with thedimension of mass. For the homogeneous and isotropic Universe with the Robertson–Walker metric ds = − dt + a ( t ) (cid:20) dr − Kr + r (cid:16) d θ + sin θ d φ (cid:17)(cid:21) , (2)where K = S = (cid:90) d x (cid:112) − g (cid:20) R + (cid:18) + H M (cid:19) ˙ φ − V ( φ ) (cid:21) . (3)The kinetic term of this model is ( + H M ) ˙ φ >
0, (4)so there are no ghosts in this model. The scale range of the parameter M is very broad. If M is extremelylarger than the Hubble parameter, M (cid:29) H , the non-minimal derivative coupling term can be neglectedand the model reduces to the canonical case. If M is extremely smaller than the Hubble parameter, M (cid:28) H , the non-minimal derivative coupling term dominates the inflation, and may make some newpredictions different from the canonical case. niverse , , 5 3 of 13 The Friedmann equation is H = (cid:18) ˙ aa (cid:19) = (cid:20) ˙ φ ( + F ) + V ( φ ) (cid:21) , (5)where F = H / M is the friction parameter. The equation of motion for the scalar field φ is ddt (cid:104) a ˙ φ ( + F ) (cid:105) = − a dVd φ . (6)For the slow-roll inflation, the slow-roll conditions are12 ( + F ) ˙ φ (cid:28) V ( φ ) , (cid:12)(cid:12) ¨ φ (cid:12)(cid:12) (cid:28) (cid:12)(cid:12) H ˙ φ (cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) HM + H (cid:12)(cid:12)(cid:12)(cid:12) (cid:28)
1. (7)Under these slow-roll conditions, the background Equations (5) and (6) become H ≈ V ( φ ) H ˙ φ ( + F ) ≈ − V φ , (9)where V φ = dV / d φ . With Equation (8), the friction parameter becomes F ≈ V ( φ ) M . (10)The corresponding slow-roll parameters are (cid:101) V = (cid:18) V φ V (cid:19) + F ( + F ) , (11) η V = + F V φφ V . (12)Using Equations (8), (9) and (11), we obtain3 ˙ φ ( + F ) V ( φ ) ≈ (cid:101) V . (13)The derivative of (cid:101) V with respect to t is [10]˙ (cid:101) V = H (cid:101) V (cid:20) + F + F ( + F ) (cid:101) V − η V − η V − + F + F + F + F ( + F )( + F ) (cid:101) V + ( + F + F + F + F ) ( + F )( + F ) (cid:101) V η V (cid:21) . (14) niverse , , 5 4 of 13 By using the relation dN = − Hdt , to the first order of slow-roll parameters, Equation (14) becomes d ln (cid:101) V dN = (cid:20) η V − + F + F ( + F ) (cid:101) V (cid:21) , (15)where N is the e-folding number before the end of inflation at the horizon exit. The power spectrum forthe scalar perturbation is [10] P ζ ≈ + F + F × H π (cid:101) V . (16)The power spectrum for the tensor perturbation is [10] P T ≈ H π . (17)The scalar tilt n s and the tensor-to-scalar ratio r are [10,55] n s − = η V − ( + F ) + F (cid:101) V , (18) r = ( + F ) + F (cid:101) V . (19)From Equations (15) and (18), we obtain the relation between n s and (cid:101) V , n s − = d ln (cid:101) V dN − + F + F ( + F ) (cid:101) V . (20)From Equations (5) and (13), we obtain the relation between φ and N , d φ = ± (cid:114) (cid:101) V + F dN , (21)where the sign ± depends on the sign of dV / d φ . Without loss of generality, in this paper, we only researchthe ‘ + ’ case. Combining Equations (11) and (21), we get the relation between the potential and the slow-rollparameter, (cid:101) V = + F + F ( ln V ) , N . (22)By using Equations (10) and (19), Equations (16), (20) and (22) become P ζ = H π r , (23) n s − = d ln rdN − r r = ( ln V ) , N . (25)These relations (23), (24) and (25) do not contain the friction parameter F , thus it is possible to reconstructthe potential from the tensor-to-scalar ratio without using the high friction limit. In the following sections,we discuss this issue. niverse , , 5 5 of 13
3. The Reconstruction
In this section, we reconstruct the potential from the tensor-to-scalar ratio r . The observational datafavor small r , and the α -attractor gives r = α / N , which is small enough to be consistent with theobservational data when α (cid:28)
1. In this section, we discuss a general parameterization of the α -attractor r = α ( N + β ) γ , (26)where γ >
1, and β accounts for the contribution from the scalar field φ e at the end of the inflation. Fromthe relation (24), we obtain the spectral tilt n s − = − γ N + β − α ( N + β ) γ . (27)With the help of relation (25), we obtain the potential V = V exp (cid:20) − α ( γ − )( N + β ) γ − (cid:21) . (28)Combining the slow-roll Friedmann Equation (8) and the power spectrum in Equation (23), we relate theamplitude of the power spectrum A s to the potential, A s = V π r . (29)Substituting the reconstructed potential (28) into relation (29) and using the parameterization (26),we obtain V = π A s r exp (cid:34) αγ − (cid:16) r α (cid:17) γ − γ (cid:35) . (30)Combining Equation (28) and (22), we get the slow-roll parameter (cid:101) V = + F exp (cid:2) α ( − γ ) − ( N + β ) − γ (cid:3) + F exp [ α ( − γ ) − ( N + β ) − γ ] α ( N + β ) γ , (31)where the amplitude of the friction parameter F = V / M . From the condition of the end of inflation, (cid:101) V ( ) =
1, we obtain the relation among α , β and γ + F exp (cid:2) α ( − γ ) − β − γ (cid:3) + F exp [ α ( − γ ) − β − γ ] × αβ γ =
1. (32)Under the GR limit F (cid:28)
1, relation (32) reduces to α = β γ ; under the high friction limit F (cid:29)
1, relation(32) reduces to α = β γ /3. From Equation (26), the tensor-to-scalar ratio r under the high friction limit istherefore smaller than that under the GR limit when β and γ is unchanged. Substituting Equation (31) intoEquation (21), we get the relation between φ and N , d φ = (cid:118)(cid:117)(cid:117)(cid:116) r (cid:32) + F exp (cid:104) α ( N + β ) − γ ( − γ ) (cid:105)(cid:33) − dN . (33) niverse , , 5 6 of 13 Combining it with Equation (26), the relation becomes d φ = (cid:118)(cid:117)(cid:117)(cid:116) r (cid:32) + F exp (cid:104) α ( α / r ) ( − γ ) / γ ( − γ ) (cid:105)(cid:33) − dN . (34)To the first order of tensor-to-scalar ratio r , it becomes d φ = (cid:114) r + F dN , (35)and the solution is φ − φ = − γ (cid:114) α + F ( N + β ) − γ , γ (cid:54) = (cid:114) α + F ln ( N + β ) , γ =
2, (36)where φ is the integration constant. Substituting Equation (36) into Equation (28), we get the reconstructedpotential V ( φ ) = V exp (cid:34) − λ (cid:16)(cid:112) + F φ − (cid:112) + F φ (cid:17) γ − γ − (cid:35) , γ (cid:54) = V exp (cid:104) − α e −√ + F ( φ − φ ) / √ α (cid:105) , γ =
2, (37)where λ = αγ − (cid:18) γ − √ α (cid:19) γ − γ − . (38)Therefore, we reconstruct the potential from the parameterization (26) without using the high friction limit.Furthermore, the potential (37) and parameter (38) show that the effect of the no-minimally derivativecoupling term is the rescaling of the inflaton field by a factor √ + F . For the α -attractors parameterization γ =
2, under the GR limit F (cid:28)
1, the potential reduces to[39] V ( φ ) = V exp (cid:104) − α e − ( φ − φ ) / √ α (cid:105) . (39)If α (cid:28)
1, this potential reduces to V ( φ ) = V (cid:104) − α e − ( φ − φ ) / √ α (cid:105) , (40)which is asymptotic behavior of the T-model and E-model.Taking N =
60 and F (cid:29)
1, and comparing the theoretical predictions (26) and (27) with the Planck2018 data[5], we obtain the constraints on the parameters β and γ shown in Figure 1. Taking γ = β = N =
60, the theoretical predictions are n s = r = niverse , , 5 7 of 13 Planck TT,TE,EE + lowE + lensing + BK14 + BAO n s r . β γ Figure 1.
The constraints on n s and r from Planck data [5] and the theoretical predictions for theparameterization (26) in the high friction limit. The Planck constraints on n s and r are displayed in the leftpanel and the constraints on β and γ for N =
60 are displayed in the right panel. The red and blue regionsdenote the 68% and 95% confidence level, respectively. r = α /( N + β ) γ F + ( ϕ - ϕ e ) V / V Figure 2.
The reconstructed potentials are normalized with V from Equation (30), and the inflaton field isnormalized with 1/ √ F +
1. We choose the value of φ that could make φ e =
4. Reheating
The inflation ends when the inflaton rolls down to the minimum of the potential; around the minimum,the inflaton field will oscillate to reheat the cold universe. Because the inflation phase is followed bythe reheating phase, these two phases may constrain each other, so the reheating phase may give otherconstraints on the inflation phase. In this section, we research the constraint from the reheating phase onthe reconstructed model under the high friction limit F (cid:29) F (cid:28) k ∗ = − and the present Hubble parameter is k ∗ a H = a ∗ H ∗ a H = a ∗ a e a e a re a re a H ∗ H = e − N − N re a re a H ∗ H , (41)where N re is the e -folding number during reheating, a re is the scale factor at the end of reheating, and weassume the radiation domination phase follows the reheating phase immediately and the reheating phase niverse , , 5 8 of 13 follows inflation phase immediately. Because the physics of the reheating is still unknown, for simplicity,we assume a constant parameter w re of state equation during reheating, and we get N re = ( + w re ) ln ρ e ρ re , (42)where the relation between ρ re and the temperature T re is ρ re = π g re T re , (43)with g re denoting the effective number of relativistic species at reheating phase. By using the condition ofthe entropy conservation, we get the relation between temperature T re and the present cosmic microwavebackground temperature T = K , a re g s , re T re = a (cid:18) T + × T ν (cid:19) , (44)where g s , re denotes the effective number of relativistic species for entropy, and T ν = ( ) T is thepresent neutrino temperature. By using the above relations, we obtain [48,49] N re = − w re (cid:20) − N − ln ρ e H ∗ +
13 ln 4311 g s , re +
14 ln π g re − ln k ∗ a T (cid:21) , (45) T re = exp (cid:20) − N re ( + w re ) (cid:21) (cid:20) ρ e π g re (cid:21) . (46)The relations (45) and (46) show that N re and T re depend on g re and g s , re logarithmically, thus we choose g re = g s , re = (cid:101) V ≈
1; from Equation (13), we obtain the relation˙ φ = V e / ( F ) , so we have ρ e = V e /3. By using the observational value of the amplitude of the powerspectrum [5], from Equation (16), we have A s = H ∗ / ( π (cid:101) V ∗ ) = × − , (47)and Equations (45) and (46) become N re = − w re (cid:18) − N − ln V e + ln (cid:101) V ∗ (cid:19) , (48) T re = exp (cid:20) − N re ( + w re ) (cid:21) (cid:20) V e π (cid:21) . (49)By using Equations (28) and (31), under the high friction limit F (cid:29)
1, we obtain the constraint from thereheating process on the model parameters, N re = − w re (cid:20) + α ( γ − ) β γ − +
14 ln α − N − γ ( N + β ) − α ( γ − )( N + β ) γ − (cid:21) , (50) T re = α ( N + β ) γ /4 exp (cid:20) − α ( γ − ) β γ − + α ( γ − )( N + β ) γ − − N re ( + w re ) (cid:21) , (51) niverse , , 5 9 of 13 where α = β γ /3. Under the GR limit F (cid:28)
1, the relations are N re = − w re (cid:20) + α ( γ − ) β γ − +
14 ln α − N − γ ( N + β ) − α ( γ − )( N + β ) γ − (cid:21) , (52) T re = α ( N + β ) γ /4 exp (cid:20) − α ( γ − ) β γ − + α ( γ − )( N + β ) γ − − N re ( + w re ) (cid:21) , (53)where α = β γ . These two situations make almost the same constraint except the 0.5 e -folding difference in N re and the different relations of α . Therefore, the friction parameter F has little influence on the reheatingphase, and we just consider the high friction limit situation in the following.For different kinds of β , γ , N and w re , by using Equations (27), (50) and (51), we calculate thecorresponding spectral tilt n s , reheating e -folds N re and reheating temperature T re , and the results aredisplayed in Figure 3. niverse , , 5 10 of 13 N * = N * = N r e γ = β = n s l og ( T r e / G ev ) N * = N * = N r e γ = β = n s l og ( T r e / G ev ) N * = N * = N r e γ = β = n s l og ( T r e / G ev ) N * = N * = N r e γ = β = n s l og ( T r e / G ev ) Figure 3. (Top) The relations between N re and n s ; and (Bottom) the relations between T re and n s .The corresponding values of β and γ for each model are indicated in each panel. The 1 σ Planck constraint n s = ± σ Planck constraint on the e -folds N is alsoindicated. The black, red, blue and green lines correspond to the reheating models with w re = − N enlargement. The horizontal graysolid and dashed lines in the bottom panels denote the electroweak scale T EW ∼
100 GeV and the big bangnucleosynthesis scale T BBN ∼
10 MeV, respectively.
The figures show that different model parameters β and γ and the value of w re provide differentconstraints on the reheating e -folds N re and the reheating temperature T re , while the parameter β almostdoes not affect the reheating process. For larger spectral tilt n s , the allowed reheating e -folding number N re with w re = − e -folding number N re with w re = n s if w re = n s if w re =
5. Conclusions
The non-minimal derivative coupling term in the inflation model could reduce the tensor-to-scalarratio, which can make the large tensor-to-scalar ratio models, such as the Higgs inflation, be consistent niverse , , 5 11 of 13 with the observations. We derive the reconstruction formulae of the inflation model with non-minimalderivative coupling. To reconstruct the potential without using the high friction limit, we consider theparameterization of the tensor to scalar ratio r = α / ( N + β ) γ inspired from the α -attractor. For γ = α attractor, we get the same potential as obtained in [39], in the GR limit F (cid:28)
1. When α (cid:28) γ (cid:54) =
2, the potential is theexponential form. The observational constraints on the parameters are 1.2 < γ < β < α attractor case with γ ∼ n s from the Planck data could provide constraints on the reheatingprocess. Different model parameters provide different constraints on reheating e -folds N re , reheatingtemperature T re and reheating state equation w re . For larger spectral tilt n s , the allowed reheating e -foldingnumber N re with w re = − e -foldingnumber N re with w re = γ = β = w re = n s < γ = β = w re = n s > Author Contributions:
Conceptualization, Z.Y.; investigation, Q.F.; data curation, Q.F.; writing—original draftpreparation, Z.Y. and Y.Y.; and writing—review and editing, Z.Y. All authors have read and agreed to the publishedversion of the manuscript.
Funding:
This research was supported in part by the National Natural Science Foundation of China under Grant No.11947138, the Postdoctoral Science Foundation of China under Grant No. 2019M660514, the Hubei College Students’innovation and entrepreneurship training program under Grant No. S201910920050 and the Talent-IntroductionProgram of Hubei Polytechnic University under Grant No.19xjk25R.
Acknowledgments:
The authors thank Yungui Gong from Huazhong University of Science and Technology.
Conflicts of Interest:
The authors declare no conflict of interest.
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