Extended Slow-Roll Conditions and Rapid-Roll Conditions
aa r X i v : . [ a s t r o - ph ] O c t Extended Slow-Roll Conditions and Rapid-RollConditions
Takeshi Chiba
Department of Physics,College of Humanities and Sciences,Nihon University,Tokyo 156-8550, Japan
Masahide Yamaguchi
Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara229-8558, JapanandDepartment of Physics, Stanford University, Stanford CA 94305PACS numbers: 98.80.Cq; 98.80.Es
Abstract.
We derive slow-roll conditions for a scalar field which is non-minimallycoupled with gravity in a consistent manner and express spectral indices ofscalar/tensor perturbations in terms of the slow-roll parameters. The conformalinvariance of the curvature perturbation is proved without linear approximations.Rapid-roll conditions are also derived, and the relation with the slow-roll conditions isdiscussed. xtended Slow-Roll Conditions and Rapid-Roll Conditions
1. Introduction
In curved spacetime, a scalar field may generally couple to the scalar curvature sothat the potential of φ has an additional term ξRφ / ξ = 1 / ‡ One method to circumvent such a constraint is toadd another source of fluctuations [9]. Another interesting method is to introduce thenon-minimal coupling to the curvature, which can reduce the tensor to scalar ratio tonegligible levels [10, 11]. Then, a non-minimally coupled chaotic inflation with a quartictype of potential is still viable. Thus, it is very useful to derive generalized slow-rollconditions for such a non-minimally coupled inflaton field and provide the formulae ofthe scalar/tensor spectral indices in terms the slow-roll parameters, as in the case of aminimally coupled inflaton. However, as far as we know, such slow-roll conditions havenot been derived in a fully consistent manner. A non-minimal coupling also realizes non-slow-roll inflation. The coupling to the curvature like ξRφ / m ≃ ξH so that the inflaton cannot slow-roll with ξ = O (1). However,in the case of the conformal coupling, it has recently been shown that the inflation takesplace with the rapidly rolling inflaton [12].In this paper, we first define the slow-rolling of the scalar field in the Jordan frameand derive consistency conditions of it, which we call extended slow-roll conditions.We then apply these conditions to several examples. We also compute observationalquantities (spectral indices of scalar/tensor perturbations and the ratio of tensor toscalar perturbation) and rewrite them in terms of these slow-roll parameters. Ourformulas make it possible to calculate the observational quantities in either frame(Jordan or Einstein) using the functions appearing in the Lagrangian only. Next wedefine the rapid-rolling of the scalar field and derive rapid-roll conditions and discuss itsrelation with the slow-rolling of the scalar field and then provide several examples. InAppendix, we give several formulas in the Einstein frame which are useful for calculationsin the text and also prove the conformal invariance of the curvature perturbation. ‡ Such a chaotic inflation is predicted as a simple realization of chaotic inflation in supergravity [8]. xtended Slow-Roll Conditions and Rapid-Roll Conditions
2. Slow-Roll Inflation with a Non-minimally Coupled Scalar Field
In this section, we derive the extended slow-roll conditions of the scalar field whichcouples non-minimally to gravity and express scalar/tensor spectral indices in terms ofthese slow-roll parameters.The action in the Jordan frame metric g µν is S = Z d x √− g (cid:20) κ R − F ( φ ) R −
12 ( ∇ φ ) − V ( φ ) (cid:21) . (1)Here κ ≡ πG is the bare gravitational constant and F ( φ ) R term corresponds to thenon-minimal coupling of the scalar field to gravity.We assume that the universe is described by the flat, homogeneous, and isotropicuniverse model with the scale factor a . The field equations are then given by H ≡ (cid:18) ˙ aa (cid:19) = κ ρ φ , (2) ρ φ = 12 ˙ φ + V + 6 H ( ˙ F + HF ) , (3)˙ H = − κ ρ φ + p φ ) , (4) p φ = 12 ˙ φ − V − F − H ˙ F − F (2 ˙ H + 3 H ) , (5)¨ φ + 3 H ˙ φ + V ′ + 6 F ′ ( ˙ H + 2 H ) = 0 , (6)where the dot denotes the derivative with respect to the cosmic time and V ′ = dV /dφ .The equation of motion of the scalar field Eq. (6) is also derived from the energy-momentum conservation: ˙ ρ φ + 3 H ( ρ φ + p φ ) = 0. Introducing Ω = 1 − κ F which corresponds to a conformal factor between the Jordanframe and the Einstein frame, the equations of motions are rewritten as H Ω + H ˙Ω = κ (cid:18)
12 ˙ φ + V (cid:19) , (7)¨Ω − H ˙Ω + 2 ˙ H Ω = − κ ˙ φ , (8)¨ φ + 3 H ˙ φ + V ′ − ′ κ ( ˙ H + 2 H ) = 0 . (9)Since under the slow-roll approximations [13], the time scale of the motion of the scalarfield is assumed to be much larger than the cosmic time scale H − , as an extendedslow-rolling of the scalar field, we assume that ˙ φ ≪ V, | ˙Ω | ≪ H Ω, | ¨ φ | ≪ H | ˙ φ | and | ¨ φ | ≪ | V ′ | , then we obtain H Ω ≃ κ V, (10)3 H ˙ φ ≃ − V ′ + 6 Ω ′ κ H ≃ − Ω (cid:18) V Ω (cid:19) ′ =: − V ′ eff , (11) xtended Slow-Roll Conditions and Rapid-Roll Conditions | ˙ H/H | ≪
1, which should be checkedlater. Note that if V ∝ Ω , then V eff is flat and V ′ eff = 0 identically.In the following, we derive the consistency conditions for the extended slow-rollingof the scalar field (the extended slow-roll conditions). By computing ¨ φ from Eq. (11),we obtain ¨ φH ˙ φ ≃ − ˙ HH − V ′′ eff H . (12)Moreover, from Eqs. (10) and (11) using ˙Ω = Ω ′ ˙ φ ˙ φ V ≃ Ω V ′ eff κ V , (13)˙Ω H Ω ≃ − Ω ′ V ′ eff κ V . (14)Hence, we finally introduce the following slow-roll parameters and obtain the extendedslow-roll conditions: ǫ := Ω V ′ eff κ V ; ǫ ≪ , (15) η := Ω V ′′ eff κ V ; | η | ≪ , (16) δ := Ω ′ V ′ eff κ V ; | δ | ≪ , (17)where we have introduced a factor of 2 in the definition of ǫ so that it accords withthe standard notation of the slow-roll parameters [14, 15]. A useful bookkeeping rule isthat a term involving derivatives divided κ is to be treated as a small quantity. Theimportance of the last condition Eq. (17) in the background dynamics of the scalar fieldhas not been fully appreciated. § However, it is necessary for slow-roll inflation bothin the Jordan frame and in the Einstein frame and is essential to relate these slow-rollparameters to the slow-roll parameters in the Einstein frame, which are discussed inAppendix A.We also need to make sure that | ¨ φ | ≪ | V ′ | is satisfied in deriving Eq. (11). FromEq. (11), we have¨ φV ′ ≃ − ˙ HH ˙ φV ′ − V ′′ eff H ˙ φV ′ ≃ ˙ H H V ′ eff V ′ + V ′′ eff H V ′ eff V ′ . (18)Therefore, comparing Eqs. (16) and (12), | ¨ φ | ≪ | V ′ | is satisfied as long as (cid:12)(cid:12)(cid:12)(cid:12) V ′ eff V ′ (cid:12)(cid:12)(cid:12)(cid:12) = O (1) . (19)Note that since from Eq. (8) | ˙ H/H | is approximated as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ HH (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≃ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙Ω2 H Ω − φ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≃ (cid:12)(cid:12)(cid:12)(cid:12) Ω ′ V ′ eff κ V + Ω V ′ eff κ V (cid:12)(cid:12)(cid:12)(cid:12) , (20) § In [17, 11], a similar slow-roll parameter is introduced in the context of a scalar perturbation equation. xtended Slow-Roll Conditions and Rapid-Roll Conditions | ˙ H/H | ≪ k As an example of extended slow-roll inflation, we consider a non-minimally coupledscalar field for chaotic inflation with: V ( φ ) = λn φ n , F ( φ ) = 12 ξφ , (21)where ξ is a dimensionless coupling parameter and ξ = 1 / | ξ | κ φ ≫
1, from Eq. (10) ξ < ≃ − ξκ φ , Ω ′ ≃− ξκ φ, f ≃ − ξ and V ′ eff ≃ ( n − V / ((1 − ξ ) φ ) for n = 4. Hence slow-rollparameters become ǫ = − ( n − ξ − ξ ) , η = − ( n − n − ξ − ξ , δ = − n − ξ − ξ , (22)while a subsidiary condition becomes | V ′ eff /V ′ | = 1 / | − ξ || ( n − /n | . Therefore, forgeneral n , | ξ | ≪ n = 4. In this case from Eq. (11)for | ξ | κ φ ≫ V ′ eff ≃ − λφ/ ((1 − ξ ) ξκ ) = − V / ((1 − ξ ) ξκ φ ) and V ′ eff becomesvanishingly small, which corresponds to the flat plateau in the Einstein frame found byFutamase and Maeda [1]. The slow-roll parameters are ǫ = − − ξ ) ξκ φ , η = 4(1 − ξ ) κ φ , δ = 8(1 − ξ ) κ φ . (23)Hence, for n = 4 the slow-roll conditions are automatically satisfied irrespective of ξ aslong as ξ <
0, which again coincides with the results in [1].On the other hand for | ξ | κ φ ≪
1, Ω ≃ , Ω ′ ≃ − ξκ φ, f ≃ V ′ eff ≃ nV /φ .So slow-roll parameters become ǫ = n κ φ , η = n ( n − κ φ , δ = − ξn, (24)and V ′ eff /V ′ ≃
1. Hence | ξ | ≪ κ φ ≫ | ξ | κ φ ≪ In Appendix, it is shown that the gauge invariant curvature perturbation R is invariantunder the conformal transformation into the Einstein frame. Then we can calculate thepower spectrum P R ( k ) [16, 17, 11], P / R ( k ) = b H π | d b φ/d b t | = H πf / | ˙ φ | = 3 H πf / | V ′ eff | ≃ κ V / √ π Ω / | V ′ eff | , (25) k It is to be noted that the subsidiary condition is a sufficient condition for slow-roll and a necessarycondition is that Eq. (19) multiplied by ǫ, η or δ is sufficiently small. xtended Slow-Roll Conditions and Rapid-Roll Conditions g µν Einstein frame b g µν = Ω g µν slow-roll parameters ǫ, η, δ b ǫ = ǫ, b η = η − δ scalar spectral index n S − ǫ + 2 η − δ − b ǫ + 2 b η tensor spectral index n T − ǫ − b ǫ tensor/scalar ratio r ǫ = − n T b ǫ = − c n T Table 1.
Slow-roll parameters and inflationary observables in Jordan/Einstein frame where k is a comoving wavenumber at the horizon exit ( k = aH ), the hatted variablesare those in the Einstein frame and we have used Eqs.(A.5) and (A.14) and Eq. (A.15).In the last equality we have assumed the slow-roll approximation. Using d ln k = d ln aH ≃ Hdφ/ ˙ φ ≃ − κ V Ω V ′ eff dφ, (26)to the first order in the slow-roll parameters, ¶ the spectral index of scalar perturbationis then given by n S − ≡ d ln P R d ln k = − ǫ + 2 η − δ. (27)Moreover, using the relation Eqs.(A.11) and (A.12), we finally obtain the simple formula n S − − ǫ + 2 η − δ = − b ǫ + 2 b η = c n S − , (28)which proves the invariance of the spectral index under the conformal transformation.Tensor perturbations are also invariant under the conformal transformation intothe Einstein frame. Then we can calculate the tensor power spectrum P h ( k ) P / h ( k ) = 2 κ b H √ π = 2 κH √ π ≃ κ V / √ π Ω , (29)where in the last equality the slow-roll approximation is assumed. Then the tensorspectral index is given by n T ≡ d ln P h d ln k = − ǫ = − b ǫ = c n T , (30)which is also conformally invariant. The tensor to scalar ratio r is also calculated as r ≡ P h P R = 16 ǫ = 16 b ǫ = b r. (31)Again this is also conformally invariant. Therefore, the consistency relation for a singlescalar field inflation r = − n T (32)is conformally invariant [11]. These results are summarized in Table 1.It should be noted that the invariance refers to the equality between the quantitiescalculated using V ( φ ) with g µν and those using b V ( b φ ) with b g µν , not V ( b φ ) with b g µν . ¶ Note that d ln k ≃ − κ b V d b φ/ ( d b V /d b φ ) under the same approximations. xtended Slow-Roll Conditions and Rapid-Roll Conditions V ( φ ) with non-minimal couplingare different from those by V ( φ ) with minimal coupling.To demonstrate this, as an example, we calculate n S , n T and r with Eq. (21). For | ξ | κ φ ≫
1, from Eq. (22), they are calculated for n = 4 (note ξ < | ξ | ≪ n S − n T ≃ ( n − ξ, r ≃ − n − ξ. (33)Here we note that these are independent of the e-folding number. This feature can beeasily understood by calculating them in the Einstein frame. From Eqs. (A.5) and (A.6),the canonical scalar field ˆ φ and the potential ˆ V ( ˆ φ ) in the Einstein frame are given byˆ φ ≃ s − ξ | ξ | κ log φφ , (34)ˆ V ( ˆ φ ) ≃ ˆ V exp "s | ξ | − ξ ( n − κ ˆ φ , (35)where φ is a constant field value which yields the origin of ˆ φ and ˆ V = λφ n − / ( nξ κ ).Thus, the inflation becomes the power-law type with the exponential potential in theEinstein frame. In fact, the slow-roll parameters in the Einstein frame are given byˆ ǫ ≃ − ( n − ξ/ η ≃ − ( n − ξ , and inserting them into the formulas (28), (30)and (31) yields the same values as those calculated in the Jordan frame.For n = 4 from Eq. (23), ǫ = − / (8 ξN ) , η = 1 / N and δ = 1 /N , where N is thee-folding number until the end of inflation and is written for n = 4 as N = Z t e t Hdt ≃ (1 − ξ )8 κ φ , (36)so that n S − ξN − N , n T = 14 ξN , r = − ξN . (37)On the other hand, for a minimally coupled ( F = 0) inflaton with the samepotential, n S , n T and r are calculated in the standard manner [15] n S − − n + 22 N , n T = − n N , r = 4 nN , (38)which clearly shows that both (Eq. (33) or Eq. (37) vs. Eq. (38)) are different. Inparticular, although the tensor-scalar ratio r for a minimally coupled inflaton is generallylarge r ≃ . n/ /N ) and it is so for a non-minimal coupling with n = 4, r ≃ . − n S ) / . n = 4 it can besmall, r ≃ . . / | ξ | )(60 /N ) , due to the extreme flatness of the effective potential V eff for n = 4.
3. Rapid-Roll Inflation
Rapid-roll inflation is a novel type of inflation with a non-minimal coupling in whichinflation occurs even without slow-roll for the conformal coupling. In this section, we xtended Slow-Roll Conditions and Rapid-Roll Conditions | ˙ φ | ∼ Hφ . Nevertheless, as shown in [12], for the conformalcoupling, such a rapid motion of the scalar field does not affect the expansion rate of theuniverse and the universe inflates. In the following, we first derive necessary conditionsof the coupling F ( φ ) for de Sitter expansion for constant V ( φ ) = V , and then deriverapid-roll conditions for general V ( φ ). The equation of motions are given by H = κ (cid:18)
12 ˙ φ + V + 6 H ( ˙ F + HF ) (cid:19) , (39)¨ φ + 3 H ˙ φ + V ′ + 6 F ′ ( ˙ H + 2 H ) = 0 . (40)Then, by defining π ≡ ˙ φ + 6 HF ′ , this can be written as H = κ (cid:18) π + V + 6 H ( F − F ′ ) (cid:19) , (41)˙ π + 2 Hπ + V ′ + (1 − F ′′ ) H ˙ φ = 0 . (42)Therefore, for constant V = V if the following conditions are satisfied, F ′′ = 16 , F = 3 F ′ , (43)then from Eq. (42) π decays as π ∝ a − and thus from Eq. (41) H becomes constant:de Sitter expansion. The conditions Eq. (43) can be integrated to give F ( φ ) = 112 ( φ − v ) , (44)where v is a constant, which may be called a ’shifted conformal coupling’. Thus if theconditions Eq. (43) are satisfied, even if φ itself does not move slowly, inflation occurs,which is called rapid-roll inflation. Next, we consider a general V ( φ ) and F ( φ ) and derive rapid-roll conditions in terms of V and F . We assume that the time scale of ˙ π is also determined by the Hubble scale,so that ˙ π ≃ cHπ and that the equations of motions are approximated by H ≃ κ V, (45)( c + 2) Hπ ≃ − V ′ , (46)where c is a proportionality constant at most of the order O (1) and will be explicitlygiven later which is not given in [12]. Note that the left hand side of Eq. (45) does not xtended Slow-Roll Conditions and Rapid-Roll Conditions (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ HH (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ . (47)In the following, we shall derive the consistency conditions for rapid-roll inflationfollowing [12]. In Eq. (45), the first and the last terms of the right hand side in Eq. (41)are neglected. Since π V + 6 H V ( F − F ′ ) ≃ V ′ c + 2) κ V + 2 κ ( F − F ′ ) , (48)this is consistent if ǫ c := V ′ κ V + 2( c + 2) κ ( F − F ′ ); | ǫ c | ≪ , (49)Moreover, from the time derivative of Eq. (46)˙ π ( c + 2) Hπ ≃ − ˙ H ( c + 2) H − V ′′ ( c + 2) κ V − F ′ V ′′ ( c + 2) V ′ . (50)Therefore, neglecting ˙ π − cHπ + H (1 − F ′′ )( π − HF ′ ) in Eq. (46) is consistent if η c := V ′′ κ V + 2( c + 2) F ′ V ′′ V ′ + c ( c + 2)3 − c + 23 (1 − F ′′ ) − c + 2) κ (1 − F ′′ ) F ′ VV ′ (51) | η c | ≪ , (52)where we have assumed | ˙ H | /H ≪
1, which also should be checked. From the timederivative of Eq. (45) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ HH (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≃ (cid:12)(cid:12)(cid:12)(cid:12) c + 2) V ′ κ V + 3 F ′ V ′ V (cid:12)(cid:12)(cid:12)(cid:12) . (53)Therefore assuming | ˙ H | /H ≪ δ c := V ′ c + 2) κ V + F ′ V ′ V ; | δ c | ≪ . (54)To sum, the rapid roll conditions consist of three conditions Eqs. (49), (52) and (54).The constant c may be expressed in terms of the potential V and the couplingfunction F . The time derivative of Eq.(45) and Eq.(46) yields a quadratic equation for c , and the solutions of it are given by c = − (cid:18) F ′ V ′′ V ′ − F ′ V ′ V (cid:19) ± s(cid:18) − F ′ V ′′ V ′ + 32 F ′ V ′ V (cid:19) − V ′′ κ V + 32 V ′ κ V . (55)For F satisfying Eq. (43), using the rapid-roll conditions Eq. (49) and Eq.(54), it canbe approximated as (note that this can also be derived by setting η c ≃ c ≃ − (cid:18) F ′ V ′′ V ′ (cid:19) ± s(cid:18) − F ′ V ′′ V ′ (cid:19) − V ′′ κ V , (56) xtended Slow-Roll Conditions and Rapid-Roll Conditions | c | ≪ c ≃ − V ′′ κ V (cid:18) F ′ V ′′ V ′ (cid:19) − . (57)Hence | c | ≪ | V ′′ /κ V | ≪
1. Since Ω = 1 − κ φ / O (1), together with theconditions Eq.(49) and Eq.(54), this implies that the rapid-roll conditions are reducedto the slow-roll conditions, which is also seen from | ˙ π | ≃ cH | π | ≪ H | π | . Thus therelation between rapid-roll and slow-roll is clarified: c = O (1) for rapid-roll; | c | ≪ As an example of rapid-roll inflation, we consider a scalar field with the conformalcoupling F ( φ ) = φ /
12 and the potential V ( φ ) = v ± m φ , (58)where v is the typical energy scale of inflation and m is the inflaton mass. This typeof potential often appears in hybrid inflation (plus sign) or new inflation (minus sign).Firstly, we derive the rapid-roll conditions for this potential and then confirm the relation˙ π ≃ cHπ by directly solving the equation of motion.The rapid-roll parameters are estimated as ǫ c = m φ κ V , η c = ± m κ V + ( c + 2)3 + c ( c + 2)3 , δ c = m φ c + 2) κ V ± m φ V . (59)Hence from | δ c | ≪ ǫ c ≪ m φ ≪ v , so that V ≃ v . From ǫ c ≪ m φ ≪ κ v . c is determined by solving η c ≃ c ≃ − ± s ∓ m H , (60)where H ≡ κv / √
3. Then the scalar field moves according to ˙ π ≃ cHπ with π = ˙ φ + Hφ . Since c is always negative as long as the determinant is positive, π → m ≫ H , so that one of the usual slow-roll condition, | η | = | V ′′ /κ V | ≪
1, is violated, which is related to the situation in so-called fast-rollinflation [18].We can now confirm the assumption ˙ π ≃ cHπ by solving the equation of motionfor the scalar field φ directly. For m φ ≪ v , the equation of motion of φ is given by¨ φ + 3 H ˙ φ + (2 H ± m ) φ ≃ . (61)Inserting φ = φ e ωt into the above equation yields ω + 3 H ω + 2 H ± m = 0 . (62)Then, ω is given by ω = − H ± q H ∓ m = cH . (63) xtended Slow-Roll Conditions and Rapid-Roll Conditions π = ˙ φ + Hφ = ( ω + H ) φ = ( c + 1) H φ , and so indeed ˙ π = c ( c + 1) H φ = cH π .Finally, we would like to point out that for the polynomial potential V ( φ ) ∝ φ n ,which often appears in chaotic inflation, one of the rapid roll parameters δ c ≃ n/ O (1) violates the rapid-roll condition. Therefore, the rapid roll inflation does not occurfor this type of potential.
4. Summary
In this paper, we have derived the slow-roll conditions for the scalar field non-minimallycoupled to gravity in consistent manner, which was made possible by rewriting theequation of motion using the conformal factor Ω and by introducing the effectivepotential V eff . The slow-roll conditions consist of three main conditions Eqs. (15), (16)and (17) and one subsidiary condition Eq. (19). The third condition | δ | ≪ ǫ, η, δ ) to those in the Einstein frame ( b ǫ, b η ), so that observationalquantities can be calculated in either frame and the conformal invariance of them canbe proved. We have also derived the rapid-roll conditions by slightly generalizing thosein [12]. The rapid-roll conditions consist of three conditions Eqs. (49), (52) and (54).We also discussed the relation between rapid-roll and slow-roll.Our formulae, Eqs. (25), (28), (29) and (30), allow us to calculate the observationalquantities in either frame using the functions appearing in the Lagrangian only.Although we have shown the conformal invariance of the curvature perturbation beyondlinear perturbation, the conformal invariance of the spectral indices of scalar/tensorperturbations is proved only in the first order in the slow-roll parameters. It would beinteresting to extend the invariance to higher orders. Acknowledgments
The authors would like to thank the participants of Summer Institute 2008 (Fujiyoshida,Japan, August 3-13, 2008), especially Hideo Kodama, Shinji Mukohyama and MisaoSasaki for useful comments. This work was supported in part by Grant-in-Aid forScientific Research from JSPS (No. 17204018 and No. 20540280 (T.C.), No. 18740157and No. 19340054 (M.Y.)) and from MEXT (No. 20040006 (T.C.)) and in part by NihonUniversity.
Appendix A. Slow-Roll Conditions and Perturbations in Einstein Frame
In this appendix, we perform the conformal transformation to the Einstein frameand introduce slow-roll parameters and give their relations with those in the Jordan xtended Slow-Roll Conditions and Rapid-Roll Conditions φ ) = 1 − κ F ( φ ), the action Eq. (1) is S = Z d x √− g (cid:20) Ω( φ )2 κ R −
12 ( ∇ φ ) − V ( φ ) (cid:21) . (A.1)Introducing the Einstein metric b g µν by the conformal transformation b g µν = Ω( φ ) g µν , (A.2)the action becomes that of a scalar field minimally coupled to Einstein gravity S = Z d x p − b g (cid:20) κ b R − (cid:18) ′ κ Ω (cid:19) ( b ∇ φ ) − V Ω (cid:21) (A.3)= Z d x p − b g (cid:20) κ b R −
12 ( b ∇ b φ ) − b V ( b φ ) (cid:21) , (A.4)where in the second line we have introduced a canonical scalar field b φ with a potential b V d b φ = 1Ω( φ ) (cid:18) ′ ( φ ) κ Ω( φ ) (cid:19) =: f ( φ )Ω( φ ) dφ , (A.5) b V ( b φ ) = V ( φ )Ω( φ ) . (A.6) Appendix A.1. Slow-Roll Conditions
The slow-roll conditions in the Einstein frame are simply given by b ǫ := 12 κ b V d b Vd b φ ! ; b ǫ ≪ , (A.7) b η := 1 κ b V d b Vd b φ ; | b η | ≪ . (A.8)In terms of φ , using Eq. (A.5) and Eq. (A.6), these parameters are rewritten as b ǫ = Ω V ′ eff κ f V , (A.9) b η = Ω / κ f / V (cid:18) V ′ eff f / Ω / (cid:19) ′ , (A.10)where V ′ eff is defined in Eq. (11). If we assign O ( ε ) to the slow-roll parameters ( ǫ, η, δ ), + then f = 1 + O ( ε ), and f ≃ b ǫ ≃ ǫ, (A.11) b η ≃ Ω V ′′ eff κ V −
32 Ω ′ V ′ eff κ V = η − δ. (A.12) + We adopt the standard mathematics notation, according to which ǫ = O ( ε ) means that ǫ falls like ε or faster as ε → xtended Slow-Roll Conditions and Rapid-Roll Conditions Appendix A.2. FRW metric and Perturbations
From Eq. (A.2), the line element in the Einstein frame is d b s = Ω ds . (A.13)So, the cosmic time b t and the scale factor b a in the Einstein frame become d b t = √ Ω dt, b a ( b t ) = p Ω( t ) a ( t ) . (A.14)Hence the Hubble parameter b H and the acceleration in the Einstein frame become[17, 11, 19] b H = d b a/d b t b a = 1 √ Ω H + ˙Ω2Ω ! , (A.15) d b a/d b t b a = 1Ω ¨ aa + H ˙Ω2Ω + 12 ˙ΩΩ ! . ! . (A.16)These show that the acceleration in the Einstein frame does not immediately imply theacceleration in the Jordan frame. For the latter, | ˙Ω | ≪ H Ω is required, which is realizedby | δ | ≪