Inflationary models with non-minimally derivative coupling
aa r X i v : . [ g r- q c ] S e p Inflationary models with non-minimally derivative coupling
Nan Yang, ∗ Qin Fei, † Qing Gao, ‡ and Yungui Gong
1, 3, § School of Physics, Huazhong University of Scienceand Technology, Wuhan, Hubei 430074, China School of Physical Science and Technology,Southwest University, Chongqing 400715, China CASPER, Department of Physics, Baylor University, Waco, Texas 76798, USA (Dated: October 16, 2018)
Abstract
We derive the general formulae for the the scalar and tensor spectral tilts to the second orderfor the inflationary models with non-minimally derivative coupling without taking the high frictionlimit. The non-minimally kinetic coupling to Einstein tensor brings the energy scale in the infla-tionary models down to be sub-Planckian. In the high friction limit, the Lyth bound is modifiedwith an extra suppression factor, so that the field excursion of the inflaton is sub-Planckian. Theinflationary models with non-minimally derivative coupling are more consistent with observationsin the high friction limit. In particular, with the help of the non-minimally derivative coupling,the quartic power law potential is consistent with the observational constraint at 95% CL.
PACS numbers: 98.80.Cq, 98.80.-k, 04.50.Kd ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] . INTRODUCTION Inflation successfully solves various problems in the standard big bang cosmology suchas the flatness, horizon and monopole problems, etc, and the quantum fluctuation of theinflaton seeds the formation of large-scale structure [1–4]. A scalar field with a flat potentialis usually used to model inflation. The potential of the scalar field is characterized by theslow-roll parameters, the spectral tilts and the tensor to scalar ratio are approximated by theslow-roll parameters at the horizon exit. The Planck temperature and polarization data onthe measurements of the cosmic microwave background anisotropies gives n s = 0 . ± . r . < .
11 (95% CL) [5, 6]. The results are consistent with the R inflation [1] and thenon-minimally coupled models at strong coupling limit [7–9], because both models predictthat n s = 1 − /N and r = 12 /N , where N is the number of e-folds before the end ofinflation, but the minimally coupled power law potentials with n > ξRφ , the λφ potential with λ ∼ O (1) can beconsistent with the observations [10]. Furthermore, the coupling constant ξ can even be assmall as 0 .
003 [11, 12]. If the kinetic term of the scalar field is non-minimally coupled toEinstein tensor, then the effective self-coupling λ of the Higgs boson can be lowered to bethe order of 1, and the new Higgs inflation introduces no new degree of freedom [13, 14].The non-minimal coupling can be generalized to be f ( φ ) R which is a special case of thegeneral scalar-tensor theory F ( φ, R ) [12, 15, 16], because the non-minimal coupling termcan be transformed away by a conformal transformation. If the kinetic term of the scalarfield is non-minimally coupled to curvature tensors, then conformal transformation cannottransform the model to scalar-tensor theory [17]. More general non-minimally derivativecouplings for the scalar field are discussed in [17–20]. The non-minimally derivative cou-pling usually introduces higher than second order derivatives in the field equation and moredegrees of freedom, which lead to the Boulware-Deser ghost [21]. However, Horndeski de-rived a general scalar-tensor theory with field equations which are at most of second orderin the derivatives of both the metric g µν and the scalar field φ in four dimensions [18]. InHorndeski theory, the second derivative φ ; µν couples to Einstein tensor by the general form f ( φ, X ) G µν φ ; µν , where X = g µν φ ,µ φ ,ν [18]. If we take f ( φ, X ) = φ , then we get the coupling G µν φ ,µ φ ,ν after integration by parts. The general derivative coupling which is quadratic in φ and linear in R , has the forms φ ,µ φ ,µ R , φ ,µ φ ,ν R µν , φ ✷ φR , φφ ; µν R µν , φφ ,µ R ; µ and φ ✷ R .2ue to the divergencies ( Rφ ,µ φ ) ; µ , ( R µν φφ ,µ ) ; µ and ( R ,µ φ ) ; µ , only the couplings φ ,µ φ ,µ R , φ ,µ φ ,ν R µν , and φ ✷ φR are independent. If we choose the non-minimally derivative couplingas G µν φ ,µ φ ,ν , then the field equations contain no more than second derivatives [22] and thegravitationally enhanced friction causes the scalar field to evolve more slowly. For a mass-less scalar field without the canonical kinetic term g µν φ ,µ φ ,ν , the non-minimally derivativecoupled scalar field behaves as a dark matter [23, 24]. The cosmological perturbations andthe first order approximation of the power spectrum for inflationary models with this non-minimally derivative coupling in the high friction limit were discussed in [25–28]. As weshow above, the Planck data gives n s = 0 . ± . .
01 and the first order approximation is enough. Future experiments willmeasure n s more accurately, so it is necessary to consider the second order corrections. Inthis paper, we will derive the first order correction to the amplitude of the power spectrum,the second order corrections to both the scalar and tensor tilts and the scalar to tensor ratio r without taking the high friction limit. The cosmological consequences of the theory withnon-minimally derivative coupling were also discussed extensively [29–64].In this paper, we discuss the inflationary models with the non-minimally derivative cou-pling G µν φ ,µ φ ,ν . The paper is organized as follows. In section II, we review the cosmologicalequations and the slow-roll approximation for this theory. The general formulae for thepower spectrum and the second order corrections to the scalar and tensor spectral tilts areobtained in section III without taking the high friction limit. In section IV, we consider thepower law potential, the hilltop potential, a simple symmetry breaking potential and thenatural inflation by taking the high friction limit, and conclusions are drawn in section V. II. THE BACKGROUND EVOLUTION
The action for the scalar field with the kinetic term non-minimally coupled to Einsteintensor is S = 12 Z d x √− g (cid:20) M pl R − g µν ∂ µ φ∂ ν φ + 1 M G µν ∂ µ φ∂ ν φ − V ( φ ) (cid:21) , (1)where M pl = (8 πG ) − and M is the coupling constant with the dimension of mass. Forconvenience, we use the Arnowitt-Deser-Misner (ADM) formalism [65], and the metric is3xpressed as ds = − N dt + h ij (cid:0) dx i + N i dt (cid:1) (cid:0) dx j + N j dt (cid:1) , (2)where N , N i , h ij are the lapse function, the shift function and the metric for the threedimensional space, respectively. By using the ADM splitting of space-time, the action (1)becomes S = 12 Z dtd x ( M pl √ h " (3) R N + ˙ φ N M M pl ! + ˙ φ N M pl − N VM pl +( E ij E ij − E ) N − ˙ φ N M M pl ! , (3)where E ij = 12 (cid:16) ˙ h ij − ∇ i N j − ∇ j N i (cid:17) , (4) E = h ij E ij , the extrinsic curvature K ij = E ij /N , ˙ φ = dφ/dt , the covariant derivative iswith respect to the three dimensional spatial metric h ij , and all the spatial indices are raisedand lowered by the metric h ij . Since the lapse and shift function N and N i contain notime derivative, the variations with respect to them give the corresponding Hamiltonianand momentum constraints, (3) R N − ˙ φ M M pl ! − ( E ij E ij − E ) − φ N M M pl ! − ˙ φ M pl − N V ( φ ) M pl = 0 , (5) ∇ i " N − ˙ φ N M M pl ! ( E ij − δ ij E ) = 0 . (6)For the background with the homogeneous and isotropic flat Friedmann–Robertson–Walker (FRW) metric, N = 1, N i = 0 and h ij = a δ ij , the Hamiltonian constraint (5)gives the Friedmann equation H = (cid:18) ˙ aa (cid:19) = 13 M pl " ˙ φ (cid:18) H M (cid:19) + V ( φ ) , (7)and the momentum constraint is satisfied automatically. The equation of motion of thescalar field φ is ddt (cid:20) a ˙ φ (cid:18) H M (cid:19)(cid:21) = − a dVdφ . (8)In the cosmological background, the non-minimally derivative coupling G µν φ ,µ φ ,ν /M be-comes H ˙ φ /M which enhances the friction of the expansion. In the limit M → ∞ , the4ffect of the non-minimally derivative coupling is negligible, Eqs. (7) and (8) reduce to thestandard cosmological equations. Combining Eqs. (7) and (8), we get the Raychaudhuriequation, ˙ H − ˙ φ ¨ φM M pl H = 12 ˙ φ M M pl ˙ H −
32 ˙ φ M M pl H −
12 ˙ φ M pl . (9)If the scalar field slowly rolls down the potential, we have the slow-roll conditions,12 (cid:18) H M (cid:19) ˙ φ ≪ V ( φ ) , | ¨ φ | ≪ | H ˙ φ | , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) HM + 3 H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ . (10)Under these slow-roll conditions, Eqs. (7) and (8) can be approximated as H ≈ V ( φ )3 M pl , (11)3 H ˙ φ (cid:18) H M (cid:19) ≈ − V φ , (12)where V φ = dV /dφ . To quantify those slow-roll conditions (10), we introduce the followingslow-roll parameters ǫ v = M pl (cid:18) V φ V (cid:19) F (1 + 3 F ) ,η v = M pl F V φφ V ,ξ v = M pl (1 + 3 F ) V φ V φφφ V , (13)where F = H /M . The normalization of ǫ v is chosen to recover the definition in Einstein’sgeneral relativity (GR) in the limit F ≪
1. By using Eqs. (11) and (12), we get˙ φ (1 + 9 F )2 V ( φ ) ≈ ǫ v , (14)and η H = ¨ φH ˙ φ ≈ ǫ v − η v , (15)so ǫ v ≪ | η v | ≪ ǫ H = − ˙ HH ≈ F F ǫ v (cid:18) − F + 117 F F )(1 + 3 F ) ǫ v + 2(1 + 6 F )3(1 + 3 F ) η v (cid:19) . (16)5n the GR limit, F ≪
1, we recover the result ǫ H = ǫ v (1 − ǫ v / η v / ǫ v = 2 Hǫ v (cid:20) F + 81 F (1 + 9 F ) ǫ v − η v − F + 603 F + 2538 F + 5103 F F )(1 + 9 F ) ǫ v + 2(2 + 48 F + 441 F + 1944 F + 3645 F F )(1 + 9 F ) ǫ v η v − η v (cid:21) (17)˙ η v = H (cid:18) F )1 + 9 F ǫ v η v − ξ v (cid:19) . (18)If we further take the high friction limit H ≫ M , i.e., F ≫
1, then the slow-rollparameters become ǫ v = M pl (cid:18) V φ V (cid:19) M H , (19) η v = M pl V φφ V M H , (20) ξ v = (cid:18) M H (cid:19) M pl V φ V φφφ V . (21)To be consistent with the results obtained in [25], we define ǫ = ǫ v / η = η v and ξ = ξ v in the high friction limit F ≫
1, and the different slow-roll parameters satisfy the followingrelations, ǫ H ≈ ǫ (cid:18) − ǫ + 43 η (cid:19) , η H ≈ ǫ − η. (22)In the high friction limit, Eqs. (17) and (18) become˙ ǫ ≈ Hǫ (cid:18) ǫ − η − ǫ + 103 ǫη − η (cid:19) , (23)˙ η ≈ H (4 ǫη − ξ ) . (24)Comparing with the slow-roll parameters in GR, the slow-roll parameters defined in thenon-minimally derivative coupling case have an extra factor M / (3 H ) which is small in thehigh friction limit, so more potentials can satisfy the slow-roll conditions and inflation canhappen more easily in this theory. In the high friction limit, the number of e-folds beforethe end of inflation is N ( φ ∗ ) = Z φ e φ ∗ H ˙ φ dφ = Z φ e φ ∗ √ ǫ r H M dφM pl > r H ( φ e ) M φ e − φ ∗ M pl . (25)So we get an upper bound for the field excursion,∆ φM pl < r N MH ( φ e ) . (26)6n the high friction limit, M ≪ H , the field excursion can be sub-Planckian if M/H ( φ e )
In this section, we derive the linear perturbation around the flat FRW background. Forconvenience, we choose the uniform field gauge, δφ ( x, t ) = 0 , h ij = a (cid:18) (1 + 2 ζ + 2 ζ ) δ ij + γ ij + 12 γ il γ lj (cid:19) , (29)where ζ and γ ij denote the scalar and tensor fluctuations respectively, the tensor perturbationsatisfies ∂ i γ ij = 0 and h ij γ ij = 0. Both ζ and γ ij are first order quantities and we haveexpanded ζ and h ij to the second order. Since the scalar and tensor modes are decoupled,so we consider the scalar perturbation first. A. Scalar perturbations
For the scalar perturbations, we expand the lapse and shift functions to the first order as N = 1 + N and N i = ∂ i ψ + N Ti , where ∂ i N Ti = 0. Substituting the expansion for N and7 i into Eqs. (5) and (6), we get the solutions [25, 26] N = ˙ ζ ¯ H , ¯ H = H (1 − / − Υ / , Υ = ˙ φ M M pl ,ψ = − ζ ¯ H + χ, N Ti = 0 ,∂ i χ = a Σ¯ H ˙ ζ − Υ / , Σ = ˙ φ M pl (cid:20) H (1 + 3Υ / M (1 − Υ / (cid:21) . (30)By using the above solution (30) and the background Eqs. (7)-(9), we expand the action(3) to the second order of ζ and get [26] S ζ = Z dtd x M pl a (cid:26) Σ¯ H ˙ ζ − θ s a ( ∂ i ζ ) (cid:27) , (31)where θ s = 1 a ddt (cid:20) a ¯ H (cid:18) − Υ2 (cid:19)(cid:21) − − Υ2 . (32)By using the canonically normalized field v = zζ , where z = aM pl √ H , (33)the action (31) becomes S ζ = Z d xdτ (cid:20) v ′ − c s ( ∂ i v ) + z ′′ z v (cid:21) , (34)where the conformal time τ is related to the coordinate time by dt = adτ , the prime denotesthe derivative with respect to τ , and the effective sound speed is c s = ¯ H θ s / Σ.In terms of the slow-roll parameters, we findΥ ≈ F F ǫ v (cid:20) − F )3(1 + 9 F ) ǫ v + 23 η v (cid:21) , (35)¯ H ≈ H (cid:18) − F F ǫ v (cid:19) , (36)Σ ≈ F F H ǫ v (cid:20) − F + 9 F )3(1 + 3 F )(1 + 9 F ) ǫ v + 23 η v (cid:21) , (37) θ s = 1 + 3 F F ǫ v (cid:20) − F + 42 F F )(1 + 3 F ) ǫ v + 23 η v (cid:21) , (38) c s = 1 − F (1 + 7 F )(1 + 3 F )(1 + 9 F ) ǫ v . (39)8n the GR limit, we recover the result c s = 1. In the high friction limit, we getΥ ≈ ǫ (cid:18) − ǫ + 23 η (cid:19) , (40)¯ H ≈ H − ǫ − ǫ/ ≈ H (cid:18) − ǫ (cid:19) , (41)Σ ≈ H ǫ (1 − ǫ/ η/ ǫ )1 − ǫ/ ≈ H ǫ (cid:18) − ǫ + 23 η (cid:19) , (42) θ s ≈ ǫ (cid:18) − ǫ + 23 η (cid:19) , (43) c s = ¯ H θ s Σ ≈ − ǫ. (44)Note that this result about c s is different from that in [26] because they missed the secondorder corrections due to θ s in Eq. (43) and Υ in (40). Following the standard canonicalquantization procedure, we define the operatorˆ v ( τ, ~x ) = Z d k (2 π ) h v k ( τ )ˆ a k e i~k · ~x + v ∗ k ( τ )ˆ a † k e − i~k · ~x i , (45)where the operators satisfy the standard commutation relations h ˆ a k , ˆ a † k ′ i = (2 π ) δ ( ~k − ~k ′ ) , [ˆ a k , ˆ a k ′ ] = h ˆ a † k , ˆ a † k ′ i = 0 , (46)and the mode functions obey the normalization condition v ′ k v ∗ k − v k v ∗ k ′ = − i. (47)The Bunch-Davis vacuum is defined by ˆ a k | i = 0. Varying the action (34), we obtain theMukhanov-Sasaki equation for the mode function v k ( τ ), v k ′′ + (cid:18) c s k − z ′′ z (cid:19) v k = 0 . (48)Inside the horizon in the past, as aH/k →
0, the asymptotic solution that satisfies thenormalization condition (47) is v k → √ c s k e − ic s kτ . (49)In order to solve the Mukhanov-Sasaki equation, we need to find the time derivative z ′′ /z .From the definition (33), we find z ′′ z ≈ a H (cid:18) − F F ǫ H + 3 η H (cid:19) . (50)9ince ddτ (cid:18) aH (cid:19) = − ǫ H , (51)and H and ǫ H change very slowly during inflation, so we obtain aH ≈ − − ǫ H ) τ . (52)Substituting this result into Eq. (50), we get z ′′ z ≈ τ (cid:18) F )1 + 9 F ǫ v − η v (cid:19) = ν − / τ , (53)where ν ≈
32 + 3(1 + 4 F )1 + 9 F ǫ v − η v . (54)Combining Eqs. (53) and (48), finally we obtain the equation, v k ′′ + (cid:18) c s k − ν − / τ (cid:19) v k = 0 , (55)Treating ν as a constant, the solution is v k ( τ ) = √ τ (cid:2) c H (1) ν ( − c s kτ ) + c H (2) ν ( − c s kτ ) (cid:3) , (56)where H (1) ν ( x ) and H (2) ν ( x ) are the first and second Hankel function, respectively. From theasymptotic condition (49), we obtain c = 0. Outside the horizon, the Hankel function hasthe asymptotic form, H (1) ν ( x ≪ ∼ r π e − i π ν − Γ ( ν )Γ (3 / x − ν , (57)and the mode function v k = e i ( ν − / π ν − Γ ( ν )Γ (3 /
2) 1 √ c s k ( − c s kτ ) / − ν ∝ z. (58)Thus the scalar perturbation outside the horizon is almost a constant, | ζ k | = 2 ν − Γ ( ν )Γ (3 / H (1 − ǫ H ) ν − / k / M pl √ c s θ s (cid:18) c s kaH (cid:19) / − ν . (59)The power spectrum of ζ is defined by the two-point correlation function D ˆ ζ ( τ, ~k ) ˆ ζ ( τ, ~k ′ ) E = 2 π k δ (cid:16) ~k − ~k ′ (cid:17) P ζ ( k ) , (60)10o we get the power spectrum P ζ = k π | ζ k | ≈ ν − (cid:18) Γ ( ν )Γ (3 / (cid:19) H (1 − ǫ H ) ν − π c s θ s M pl (cid:18) c s kaH (cid:19) − ν ≈ (cid:20) F F + − C ) + 3(1 − C ) F F ) ǫ v − F )(1 − C )3(1 + 3 F ) η v (cid:21) H π M pl ǫ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c s k = aH , (61)where the constant C = − γ + ln 2 ≈ − .
73. In the limit F = 0, we recover the standardGR result. In the high friction limit F ≫
1, the scalar power spectrum becomes P ζ ≈ (cid:20) (cid:18) − C (cid:19) ǫ − (cid:18) − C (cid:19) η (cid:21) M pl ǫ (cid:18) H π (cid:19) (cid:18) c s kaH (cid:19) − ν , (62)Note that in order to derive the second order correction to the scalar spectral tilt, we needto provide the first order correction to the amplitude of the power spectrum [67, 68].To the first order of approximation, using the relation d ln k = (1 − ǫ H ) Hdt , the scalarspectral tilt is [27] n s − − ν = 2 η v − F )1 + 9 F ǫ v . (63)In the GR limit, the standard result n s − η v − ǫ v is recovered. In the high frictionlimit, we get n s − η − ǫ . Therefore, for the same n s , the slow-roll parameter ǫ canbe smaller and the tensor-to-scalar ratio can be smaller in the high friction limit. To thesecond order, we get n s − d ln P ζ d ln k (cid:12)(cid:12)(cid:12)(cid:12) c s k = aH = − F )1 + 9 F ǫ v + 2 η v + 23 η v + (cid:18) − C (cid:19) ξ + 2[ − C + ( −
17 + 60 C ) F + 12( − C ) F ](1 + 3 F )(1 + 9 F ) ǫ v η v − C + 12(1 + 33 C ) F + 18( −
11 + 84 C ) F + 27( −
25 + 72 C ) F ]3(1 + 3 F )(1 + 9 F ) ǫ v . (64)In the high friction limit, the scalar spectral tilt is n s − ≈ − ǫ + 2 η + (cid:18) − C (cid:19) ǫ + (cid:18) C − (cid:19) ǫη + 23 η + (cid:18) − C (cid:19) ξ , (65)and the running of the scalar spectral index is n ′ s = dn s d ln k (cid:12)(cid:12)(cid:12)(cid:12) c s k = aH = − ǫ + 24 ǫη − ξ . (66)11 . Gravitational wave Now we consider the tensor perturbation. Expanding the action to the second order ofthe tensor perturbation γ ij , we obtain the quadratic action [26] S = Z d xdt M pl a (cid:20)(cid:18) − Υ2 (cid:19) ˙ γ ij − a (cid:18) (cid:19) ( ∂ l γ ij ) (cid:21) . (67)With the symmetric traceless tensor e sij which satisfies the following relation X i e sii = 0 , X i,j e sij e s ′ ij = 2 δ ss ′ , (68)the tensor perturbation can be written as γ ij = X s =+ , × e sij γ s . (69)By using the canonical variable u s = z t γ s , where z t = √ aM pl r − Υ2 , (70)the quadratic action (67) becomes S = X s =+ , × Z d xdτ (cid:20) ( u s ′ ) − c t ( ∂ i u s ) + z ′′ t z t ( u s ) (cid:21) , (71)where c t = 1 + Υ / − Υ / ≈ F F ǫ v , (72) z ′′ t z t ≈ a H (2 − ǫ H ) ≈ τ (cid:20) F )1 + 9 F ǫ v (cid:21) . (73)Due to the non-minimally derivative coupling, the speed of gravitational wave c t is a littlelarger than the speed of light. In the GR limit, c t = 1. Using the canonical quantization,ˆ u s = Z d k (2 π ) h u sk ( τ )ˆ a k e i~k · ~x + u s ∗ k ( τ )ˆ a † k e − i~k · ~x i , (74)we obtain the equation for the mode function u sk , u sk ′′ + (cid:18) c t k − µ − / τ (cid:19) u sk = 0 , (75)where µ ≈
32 + 1 + 3 F F ǫ v . (76)12he solution is u sk = e i ( µ − / π µ − Γ ( µ )Γ (3 /
2) 1 √ c t k ( − c t kτ ) / − µ . (77)The power spectrum of gravitational wave is P T = k π | γ ij | = k π X s =+ , × (cid:12)(cid:12)(cid:12)(cid:12) u sk z t (cid:12)(cid:12)(cid:12)(cid:12) ≈ µ − (cid:18) Γ( µ )Γ(3 / (cid:19) − ǫ H ) µ − c t M pl (1 + Υ / (cid:18) H π (cid:19) (cid:18) c t kaH (cid:19) − µ ≈ (cid:20) − C ) + 2(4 + 3 C ) F F ǫ v (cid:21) M pl (cid:18) H π (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c t k = aH . (78)In the limit F = 0, we recover the standard GR result. In the high friction limit, the powerspectrum of gravitational wave is P T ≈ M pl (cid:18) − ǫ − Cǫ (cid:19) (cid:18) H π (cid:19) (cid:18) c t kaH (cid:19) − µ . (79)Combining Eqs. (61) and (78), we get the tensor to scalar ratio r ≈ ǫ v (cid:20) F F − (1 + 3 F )[4(1 − C ) + 27(1 − C ) F ]3(1 + 9 F ) ǫ v + 2(1 + 3 F )(1 − C )3(1 + 9 F ) η v (cid:21) . (80)Note that there is an ambiguity in the above definition due to the difference between theeffective sound speeds and the horizon exits for the tensor and scalar modes. Because c t ≥ c s ,for the same mode k , the tensor mode exits the horizon later, so we should take c t k = aH .In the limit F = 0, we recover the standard GR result. In the high friction limit F ≫
1, weget r = P T P ζ ≈ ǫ (cid:20) − (3 − C ) ǫ + (cid:18) − C (cid:19) η (cid:21) ≈ − n T . (81)To the first order of approximation, the tensor spectral tilt is n T = − F )1 + 9 F ǫ v . (82)In both the GR and the high friction limits, we get n T ≈ − ǫ to the first order of approxi-mation. To the second order of approximation, the tensor spectral index is n T = d ln P T d ln k (cid:12)(cid:12)(cid:12)(cid:12) c t k = aH ≈ − F )1 + 9 F ǫ v (cid:20) C + 6(7 + 9 C ) F F ) ǫ v − C )3 η v (cid:21) . (83)13 = = n s r Planck TT + lowPPlanck TT,TE,EE + lowP F ` F p FIG. 1. The theoretical results for the quartic potential V ( φ ) = λφ and the marginalized joint68% and 95% CL regions for n s and r . from Planck 2015 [6]. The solid black lines show theeffect of F . In the high friction limit, the tensor spectral tilt is n T ≈ − ǫ − (cid:18)
283 + 12 C (cid:19) ǫ + (cid:18)
83 + 4 C (cid:19) ǫη, (84)and the running of the tensor spectral index is n ′ T = dn T d ln k (cid:12)(cid:12)(cid:12)(cid:12) c t k = aH ≈ − ǫ + 4 ǫη. (85) IV. INFLATIONARY MODELS
Now let us apply the general results obtained in the previous section to the power lawpotential, the double well potential, the hilltop inflation and the natural inflation.We use the quartic potential λφ as an example to consider the effect of F first. Theresults are shown in Fig. 1. As F increases, the tensor to scalar ratio r becomes smaller.From Fig. 1, we see that inflation with the quartic potential in the GR limit F ≪ σ level in the high friction limit F ≫
1. In the following discussion, we considerinflationary models in the high friction limit only.14 . Power Law Potential
For the power-law potential V ( φ ) = λM pl (cid:18) φM pl (cid:19) n , (86)the slow-roll parameters are ǫ ( φ ) = n λ M M npl φ n +2 ,η ( φ ) = n ( n − λ M M npl φ n +2 ,ξ ( φ ) = n ( n − n − λ M M npl φ n +4 . (87)These formulae are also valid for the inverse power law potential. For the inverse power lawcase with n = −
2, all the above slow-roll parameters are constants. However, inflation doesnot end for the intermediate inflation with inverse power law potential [69, 70].For 0 < n <
2, inflation ends when ǫ ( φ e ) = 1, so φ e = (cid:18) n λ (cid:19) / ( n +2) ( M M npl ) / ( n +2) . (88)The number of e-folds before the end of inflation is N ∗ = λφ n +2 ∗ n ( n + 2) M M npl − n n + 2) . (89)For n ≥
2, inflation ends when η ( φ e ) = 1, so φ e = (cid:18) n ( n − λ (cid:19) / ( n +2) ( M M npl ) / ( n +2) , (90)and the number of e-folds before the end of inflation is N ∗ = λφ n +2 ∗ n ( n + 2) M M npl − n − n + 2 . (91)So the value of scalar field at the horizon exit is φ ∗ = n ( n + 2) ˜ Nλ ! / ( n +2) ( M M npl ) / ( n +2) , (92)where ˜ N = N ∗ + n/ n +2) for 0 < n < N = N ∗ +( n − / ( n +2) for n ≥
2. In order toavoid quantum gravity, we require that H ≪ M pl , this can be guaranteed if φ ≪ M pl during15nflation. The sub-Planckian field excursion is possible for the slow-roll inflation with thepower-law potential because of the high friction condition H ≫ M , and the high frictionlimit is satisfied if φ e ≪ M pl . For the Higgs inflation with n = 4, the coupling constant λ ≈ .
13 at the energy scale around 100 GeV [71], if we ignore the running of the couplingconstant, and take N ∗ = 60 and M = 1 . × − M pl , then we find that φ e = 0 . M pl , H ( φ e ) = 5 . × − M pl , φ ∗ = 0 . M pl and the field excursion ∆ φ = 0 . M pl which issub-Planckian. If λ is larger, then the field excursion will be even smaller.In terms of N ∗ , we get the value of the slow-roll parameters at the horizon exit φ ∗ , ǫ = nn + 2 12 ˜ N , η = n − n + 2 1˜ N , ξ = ( n − n − n + 2) N . (93)The values of the slow-roll parameters at the horizon exit do not depend on the modelparameters λ and M explicitly, and the results are a factor of ( n + 2) / ξ is much smaller thanthe first order parameters, we consider the first order correction only. The scalar spectraltilt is [27] n s − − n + 1)( n + 2) ˜ N . (94)The running of the scalar spectral index is n ′ s = − n + 1)( n + 2) ˜ N . (95)The tensor spectral tilt is n T = − ǫ = − n/ ( n + 2) ˜ N . The running of the tensor spectralindex is n ′ T = − n ( n + 2) ˜ N . (96)The tensor to scalar ratio is r = 8 n ( n + 2) ˜ N . (97)The n s − r and n s − n ′ s results for n = 2 and n = 4, along with the Planck 2015 constraints[6] are shown in Figs. 2 and 3. For comparison, we also plot the GR results [72]. B. Hilltop models
For the hilltop models with the potential [73] V ( φ ) = Λ (cid:18) − φ p µ p (cid:19) , (98)16 .94 0.95 0.96 0.97 0.98 0.990.000.050.100.150.200.250.300.35 n s r Planck TT + lowPPlanck TT,TE,EE + lowP --- GRderivative coupling V µΦ V µΦ Natural inflationDouble wellHilltop quadraticHilltop quarticN = = FIG. 2. The theoretical results for some inflationary models and the marginalized joint 68% and95% CL regions for n s and r . from Planck 2015 [6]. The solid lines are for the inflationary modelswith non-minimally derivative coupling and the dashed lines are for the inflationary models in GR. the slow-roll parameters are ǫ ( φ ) = p M M pl ( φ/µ ) p − µ [1 − ( φ/µ ) p ] ,η ( φ ) = − p ( p − M M pl ( φ/µ ) p − Λ µ [1 − ( φ/µ ) p ] = − p − p − ( φ/µ ) p ( φ/µ ) p ǫ ( φ ) ,ξ ( φ ) = p ( p − p − M M pl ( φ/µ ) p − Λ µ [1 − ( φ/µ ) p ] . (99)The high friction condition requires that M M pl ≪ Λ . (100)The end of inflation is determined bymax { ǫ ( φ e ) , | η ( φ e ) |} = 1 . (101)For p ≥ ǫ ( φ ) is smaller than η ( φ ) by a factor ( φ/µ ) p . So the end of inflation is determinedby | η ( φ e ) | = 1. If φ e ≪ µ or M M pl ≫ Λ µ , we get (cid:18) φ e µ (cid:19) p − ≈ p ( p −
1) Λ M M pl (cid:18) µM pl (cid:19) , (102) M M pl ≪ Λ ≪ M M pl /µ . (103)17 .94 0.95 0.96 0.97 0.98 0.99 - - - n s n s ' Planck TT + lowPPlanck TT,TE,EE + lowP --- GRderivative coupling V µΦ V µΦ N = = FIG. 3. The theoretical results for the power law potentials with n = 2 and n = 4 and themarginalized joint 68% and 95% CL regions for n s and n ′ s from Planck 2015 [6]. The solid linesare for the non-minimally derivative coupling and the dashed lines are for GR. For the case p = 2, η ( φ ) is almost a constant when φ ≪ µ . Contrary to the cases with p ≥ M M pl ≪ Λ µ and inflation ends when φ e ∼ µ .Because of the high friction condition, the second order slow-roll parameter ξ is smallerthan the first order parameters ǫ and η by a small factor M M pl / Λ , so we neglect the secondorder contribution. The scalar spectral index is n s − − pM M pl x p − ∗ [( p −
1) + ( p + 1) x p ∗ ]Λ µ [1 − x p ∗ ] , (104)where x ∗ = φ ∗ /µ . The tensor to scalar ratio is r = 8 p M M pl x p − ∗ Λ µ (1 − x p ∗ ) . (105)The number of e-folds before the end of inflation is N ∗ = f ( φ e /µ ) − f ( φ ∗ /µ ), where f ( x ) = Λ µ pM M pl (cid:18) x − p − p + x p p + 2 − (cid:19) x , (106)for p = 2. If φ ≪ µ , we get N ∗ ≈ Λ µ p ( p − M M pl (cid:18) µφ ∗ (cid:19) p − − p − p − , (107)18herefore, the scalar spectral index can be written as n s − ≈ − p − p − N ∗ + ( p − / ( p − , (108)and the scalar to tensor ratio is r ≈ px p ∗ ( p − N ∗ + ( p − / ( p − ≪ . (109)If we take p = 4, Λ = µ = 0 . M pl and M = 0 . µ , then we get φ e = 0 . µ , φ ∗ = 0 . µ , n s = 0 . n ′ s = − . r = 1 . × − for N ∗ = 60.For p = 2, the function f ( x ) is f ( x ) = Λ µ M M pl (cid:18) ln x + x − x (cid:19) , (110)and φ ∗ can be obtained from N ∗ and φ e by using the above function (110). For example,if we take Λ = 10 − M pl , µ = 0 . M pl and M = 10 − M pl , we get φ e = 0 . µ by setting ǫ ( φ e ) = 1. For N ∗ = 60, we find that φ ∗ = 0 . µ by using the function (110), so ∆ φ =0 . µ = 0 . M pl , n s = 0 . n ′ s = − . r = 0 . n s − r results for p = 2 and p = 4 are shown in Fig. 2. In plotting the resultsfor p = 4, we don’t use the approximate relation (108) and (109), φ e and φ ∗ are solvednumerically instead. C. A simple symmetry breaking potential
For the symmetry breaking potential [74] V = Λ (cid:18) − φ µ (cid:19) , (111)the slow-roll parameters are ǫ ( φ ) = 8 M M pl ( φ/µ ) Λ µ (1 − φ /µ ) ,η ( φ ) = − M M pl (1 − φ /µ )Λ µ (1 − φ /µ ) = − (cid:18) µ φ − (cid:19) ǫ ( φ ) ,ξ ( φ ) = − M M pl ( φ/µ ) Λ µ (1 − φ /µ ) . (112)The potential is also called the double well potential. In the region φ ≫ µ , the above doublewell potential becomes the power law potential with n = 4. In the region φ ≪ µ , the above19otential becomes the hilltop potential with p = 2. Here we consider the intermediate region φ ∼ µ , this requires M M pl ≪ Λ µ . (113)In this region, | η ( φ ) | > ǫ ( φ ). Inflation ends when(1 − x e ) x e − M M pl Λ µ , (114)where x e = φ e /µ . Note that the high friction condition requires thatΛ (cid:18) − φ µ (cid:19) ≫ M M pl . (115)The number of e-folds before the end of inflation is N ∗ = f ( x e ) − f ( x ∗ ) ,f ( x ) = − Λ µ M M pl (cid:18) x − x x − log( x ) (cid:19) , (116)where x ∗ = φ ∗ /µ . The scalar spectral tilt is n s − − M M pl (1 + x ∗ )Λ µ (1 − x ∗ ) . (117)The tensor to scalar ratio is r = 128 M M pl x ∗ Λ µ (1 − x ∗ ) . (118)For a given N ∗ , we can determine φ ∗ from Eqs. (114) and (116). If we take Λ = 10 − M pl , µ = 0 . M pl and M = 10 − M pl , we get x e = 0 .
886 from Eq. (114). For N ∗ = 60, Eq. (116)gives x ∗ = 0 . φ = ( x e − x ∗ ) µ = 0 . M pl , n s = 0 . n ′ s = − . r = 0 . n s − r results in the intermediate region φ ∼ µ are shown in Fig. 2. D. Natural inflation
For the natural inflation with the potential [75] V ( φ ) = Λ (cid:20) (cid:18) φf (cid:19)(cid:21) , (119)the slow-roll parameters are ǫ = M M pl sin ( φ/f )2 f Λ [1 + cos( φ/f )] ,η = − M M pl cos( φ/f ) f Λ [1 + cos( φ/f )] ,ξ = − M M pl sin ( φ/f ) f Λ [cos( φ/f ) + 1] . (120)20he high friction condition requires thatΛ [1 + cos( φ/f )] ≫ M M pl . (121)So the second order slow-roll correction can be neglected. The scalar spectral index is n s − M M pl [cos( φ ∗ /f ) −
2] sec ( φ ∗ / f )2 f Λ . (122)the horizon exit is determined by the number of e-folds before the end of inflation, N ∗ = f ( x e ) − f ( x ∗ ) , (123)where f ( x ) = f Λ M M pl (cid:16) cos x + 4 ln h sin (cid:16) x (cid:17)i(cid:17) , (124) x = φ/f , and the end of inflation is determined bysin x e (1 + cos x e ) = 2 f Λ M M pl . (125)If f Λ / ( M M pl ) ≫
1, then x e ∼ π and we get [27]( π − x e ) ≈ M M pl f Λ . (126)Since inflation happens around φ/f ∼ π , the potential behaves like the quadratic potential.By solving Eqs. (123) and (124), we obtain( π − x ∗ ) ≈ M M pl f Λ (cid:18) N ∗ + 14 (cid:19) . (127)As we discussed above, the result is the same as the power law potential with n = 2.Substituting the above result (127) into Eq. (122), we get [27] n s − ≈ − N ∗ + 1 , (128) r ≈ N ∗ + 1 . (129)Note that the high friction condition (121) requires that Λ ≫ f M . If we take Λ = 0 . M pl , f = 0 . M pl and M = 10 − M pl , then we find that φ e = 0 . M pl , φ ∗ = 0 . M pl , n s = 0 . n ′ s = − . r = 0 .
05 for N ∗ = 60.If f Λ ≪ M M pl , then either ǫ ( φ ) or η ( φ ) is big for 0 < φ/f < π , so slow-roll inflationcan not happen for this choice of model parameters. The n s − r results along with the GRresults [72, 76] are shown in Fig. 2. In plotting the results, we don’t use the approximaterelation (128) and (129), φ e and φ ∗ are solved numerically.21 . CONCLUSIONS By introducing the slow-roll parameters defined in Eq. (13), we obtain the general expres-sions for the scalar and tensor spectral tilts to the second order for the inflationary modelswith non-minimally derivative coupling. The results can recover the well known GR resultsin the limit H ≪ M . Furthermore, we extend the results of the scalar and tensor spectraltilts to the second order in slow-roll parameters in the high friction limit H ≫ M . Thenon-minimal coupling of the kinetic term to Einstein tensor leads to enhanced friction forthe scalar field so that inflation happens more easily. The Lyth bound is modified with anextra suppression factor M/H so that the field excursion of the inflaton is sub-Planckian.For the power law potential V ( φ ) ∼ φ n , due to the non-minimally derivative coupling,the field excursion of the inflaton is sub-Planckian. The tensor to scalar ratio r is a factorof ( n + 2) / µ can be smaller than the Planck energy. For the case p = 2, inflation ends when φ ∼ µ . For p >
2, small field inflation is realized, the tensor toscalar ratio r is negligibly small, and an approximate relation between n s and N ∗ is derivedin Eq. (108).For the double well potential, the potential behaves like power law potential with n = 4in the regime φ ≫ µ and the hilltop potential with p = 2 in the regime φ ≪ µ . In theintermediate regime φ ∼ µ , we obtain ∆ φ = 0 . M pl , n s = 0 .
975 and r = 0 . φ/f ∼ π , andthe behaviour is similar to the quadratic potential V ( φ ) ∼ φ . Due to the suppression ofthe non-minimally kinetic coupling, the symmetry breaking scale f can be smaller than thePlanck energy.In conclusion, the non-minimally kinetic coupling to Einstein tensor brings the energyscale appeared in the hilltop inflation, double well potential and natural inflation down tosub-Planckian scale, and the field excursion of the inflaton becomes sub-Planckian. Themodel is more consistent with the observations in the high friction limit.22 CKNOWLEDGMENTS
This research was supported in part by the Natural Science Foundation of China underGrants No. 11175270, and No. 11475065; the Program for New Century Excellent Talentsin University under Grant No. NCET-12-0205; and the Fundamental Research Funds forthe Central Universities under Grant No. 2013YQ055. [1] Starobinsky A A 1980
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