f(R) constant-roll inflation
ff ( R ) constant-roll inflation Hayato Motohashi and Alexei A. Starobinsky
2, 3 Instituto de F´ısica Corpuscular (IFIC), Universidad de Valencia-CSIC, E-46980, Valencia, Spain L. D. Landau Institute for Theoretical Physics RAS, Moscow, 119334 Russia National Research University Higher School of Economics, Moscow, 101000 Russia
The previously introduced class of two-parametric phenomenological inflationary models in Gen-eral Relativity in which the slow-roll assumption is replaced by the more general, constant-rollcondition is generalized to the case of f ( R ) gravity. A simple constant-roll condition is defined inthe original Jordan frame, and exact expressions for a scalaron potential in the Einstein frame, fora function f ( R ) (in the parametric form) and for inflationary dynamics are obtained. The regionof the model parameters permitted by the latest observational constraints on the scalar spectralindex and the tensor-to-scalar ratio of primordial metric perturbations generated during inflation isdetermined. I. INTRODUCTION
The constant-roll inflation is a two-parametric class ofphenomenological inflationary model which satisfies theassumption of constant rate of the inflaton [1–3]. Theassumption is a generalization of the standard slow-rollinflation with an approximately flat inflaton potential,and so-called ultra-slow-roll inflation [4–6], in which thepotential is constant for an extended period, and the cur-vature perturbation grows on superhorizon scales. Theattempt of such a generalization first proposed in [1], andthe inflaton potential was constructed so that it satisfiesthe constant-roll condition approximately. Later, it wasclarified in [2] that there exists a potential that satis-fies the constant-roll condition exactly. In addition, themodel possesses the exact solution that is an attractor forinflationary dynamics. It is also elucidated that the cur-vature perturbation is conserved on superhorizon scales.Not only does the constant-roll inflation serve theoret-ically interesting framework, it is also viable with themost recent observational data. In [3], we showed thatthe model can satisfy the latest observational constrainton the spectral index of the curvature power spectrumand the tensor-to-scalar ratio.This constant-roll construction refers to inflationarymodels in General Relativity (GR) where gravity is notmodified but a new scalar field has to be introduced. Onthe other hand, in the opposite limit one can constructinflationary models without new scalar fields, by chang-ing the gravity sector only, as typified by the R + R model [7] and its f ( R ) gravity modifications [8–12]. Thispurely geometrical approach is equivalent to introducinga scalar degree of freedom (dubbed a scalaron in [7]),which can be explicitly seen by performing a conformaltransformation from the Jordan frame to the Einsteinframe. Viable inflationary models in f ( R ) gravity areslow-rolling, too. Since the present level of accuracy of as-tronomical observations make it interesting to go beyondthe slow-roll approximation, in this paper we constructa new constant-roll inflationary model in the frameworkof f ( R ) gravity. In contrast to the previous works [1–3]where the constant-roll condition was effectively imposed in the Einstein frame, since inflation in GR was consid-ered, we impose a new constant-roll condition in the orig-inal Jordan frame where the form of equations is simplerin fact; see e.g. Eq. (11) below.The rest of the paper is organized as follows. In § II,we review f ( R ) gravity focusing on its Jordan and Ein-stein frame description. In § III, we introduce a novelconstant-roll condition in the Jordan frame and deriveexact solutions for the potential and the Hubble parame-ter in the Einstein frame. In § IV, we derive a parametricexpression of f ( R ), and explore the inflationary dynamicsin the Jordan frame. In § V, we consider the spectral pa-rameters for the inflationary power spectra. We use themin § VI to show the model possesses an available param-eter region. We conclude in § VII. In the Appendix, twoalternative derivations of the parametric expression forthe constant-roll f ( R ) function are presented, with thelatter of them using the Jordan frame only. II. f ( R ) GRAVITY
Let us briefly review f ( R ) gravity and the relation be-tween the Einstein and Jordan frames (see e.g. [13] fora more extensive review and the list of references). Weconsider the action S = (cid:90) d x √− g J f ( R J )2 , (1)whose gravitational field equations for the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) background withzero spatial curvature ds = − dt J + a J ( t J )( dx + dy + dz ) (2)and in the absence of other matter are given by3 F H J = 12 ( R J F − f ) − H J ˙ F , F ˙ H J = − ¨ F + H J ˙ F , (3)where F ≡ df /dR , the subscript J denotes the Jordanframe, a dot denotes a derivative with respect to the a r X i v : . [ a s t r o - ph . C O ] A ug Jordan frame time t J , and we work in the unit where M Pl = (8 πG ) − / = 1. By using the conformal transfor-mation g Eµν = F g
Jµν , we can transform the gravitationalkinetic term into the Einstein-Hilbert form. Further, wecan normalize the scalar kinetic term as S = (cid:90) d x √− g E (cid:20) R E −
12 ( ∂ µ φ ) − V ( φ ) (cid:21) , (4)where the subscript E denotes the Einstein frame, and F = e √ φ ,V ( φ ) = R J F − f F . (5)Once a functional form of f ( R ) is specified in the Jordanframe, the scalaron φ and the potential V ( φ ) in the Ein-stein frame are given by the above definition. Conversely,once the potential is specified in the Einstein frame, theRicci scalar and the function f ( R ) in the Jordan frameare given by R J = e √ φ (cid:16) √ V φ + 4 V (cid:17) ,f ( R J ) = e √ φ (cid:16) √ V φ + 2 V (cid:17) , (6)where V φ ≡ ∂V /∂φ .The time coordinate and the scale factor in the Jordanand Einstein frame are related through dt J = e − φ √ dt E ,a J = e − φ √ a E , (7)from which we obtain the relation between Jordan framequantities and the Einstein frame quantities: H J = e φ √ (cid:18) H E − √ dφdt E (cid:19) , ˙ φ = e φ √ dφdt E , ¨ φ = e √ φ (cid:34) d φdt E + 1 √ (cid:18) dφdt E (cid:19) (cid:35) . (8)The Einstein equation and the Klein-Gordon equationin the Einstein frame for the FLRW background in theabsence of the spatial curvature and other matter takethe standard form: H E = 13 (cid:34) (cid:18) dφdt E (cid:19) + V (cid:35) ,dH E dt E = − (cid:18) dφdt E (cid:19) ,d φdt E + 3 H E dφdt E + ∂V∂φ = 0 , (9)where H E ≡ a E da E dt E . It is known that these equations can be reduced toone non-linear first-order differential equation for H E ( φ )of the Hamilton-Jacobi type [14, 15]. However, for f ( R )gravity the master first-order equation for H J in the orig-inal Jordan frame considered as a function of the Ricciscalar R J has even a simpler form, which can be obtainedas follows. We represent ˙ F as˙ F = dF ( R J ) dR J dR J dH J ˙ H J = dF ( R J ) dR J dR J dH J R J − H J , (10)and plug it to the last term of the first equation of (3)to obtain dH J dR J = H J ( R J − H J )( R J − H J ) F ( R J ) − f ( R J ) dF ( R J ) dR J . (11)Note that the right-hand side can be explicitly writtendown once a functional form of f ( R J ) is specified, andhence (11) is the master first-order equation for H J as afunction of R J . III. f ( R ) CONSTANT-ROLL POTENTIAL
In the previous works [1–3], we considered the Einstein-Hilbert action with a canonical scalar field, and imposedthe constant-roll condition ¨ φ = βH ˙ φ . Now we considera natural generalization of the constant-roll condition in f ( R ) gravity: ¨ F = βH J ˙ F . (12)As we shall confirm below, the slow-roll regime amountsto β →
0, whereas a constant potential corresponds to β → −
3. Note that this condition is not conformally dualto the former one used in GR. Of course, such generaliza-tion can be produced in many ways. We have chosen justthe form (12) for the constant-roll condition in f ( R ) grav-ity from reasons of simplicity and aesthetic elegance. *1 In particular, in the case of the R + R inflationary model,it reduces to ¨ R J = βH J ˙ R J . (13)Note that, as we shall see below, while R + R modeldoes not have constant-roll solution, there exist constant-roll solutions for R p models. Also, for a generic f ( R )function, substituting the constant-roll condition (12) to(3) and integrating it, we obtain a very simple and elegantrelation which has to be satisfied for all models in thisclass at all times: F ( R J ) ∝ H / (1 − β ) J . (14) *1 When this paper was prepared for submission, a paper on thesame topic [16] has appeared in the archive. However, two dif-ferent slow-roll conditions in f ( R ) gravity proposed in that paperdiffer from the our one (12) and lead to more complicated formsof V ( φ ) and f ( R ). After obtaining an analytic solution for Hubble param-eter we can determine a proportionality constant. Weshall come back to this point soon.Let us now find the corresponding effective potentialfor the dual representation of this model in the Einsteinframe. In terms of the Einstein frame variables, the con-dition (12) reads d φdt E + 3 + β √ (cid:18) dφdt E (cid:19) − βH E dφdt E = 0 . (15)Plugging this condition to the Klein-Gordon equa-tion (9), we obtain(3 + β ) (cid:34) H E dφdt E − √ (cid:18) dφdt E (cid:19) (cid:35) + ∂V∂φ = 0 , (16)where the quadratic velocity term shows up as we imposethe constant-roll in the Jordan frame ¨ F = βH J ˙ F , ratherthan d φ/dt E = βH E dφ/dt E in the Einstein frame.Clearly, the limit β → − β →
0, we havea slow-roll equation which is approximately equivalentto the standard form as the quadratic velocity term isnegligible for slow roll.Below we shall show that one can construct an in-flationary model that satisfies the constant-roll condi-tion (15), and has an exact solution for inflationary evo-lution. Further, we shall clarify that the model has aparameter region that satisfies the latest observationalconstraint on spectral parameters of inflationary powerspectra.Following [2], we employ the Hamiltonian-Jacobi for-malism and regard H E = H E ( φ ), assuming that t E = t E ( φ ) is a single-valued function, or dφ/dt E (cid:54) = 0. When dφ/dt E = 0, the Hamiltonian-Jacobi formalism breaksdown, and the stochastic effect becomes dominant. Itshould be avoided that the inflaton passes such a pointduring inflation. If the breakdown is located before in-flation, there is no problem to rely on the Hamiltonian-Jacobi formalism. We will check this point later on.From the Einstein equation (9), we obtain dφdt E = − dH E dφ ,d φdt E = − d H E dφ dφdt E , (17)with which the condition (15) is rewritten as dH E dφ (cid:20) d H E dφ + 3 + β √ dH E dφ + β H E (cid:21) = 0 . (18)The equation allows two branches of solutions. Thefirst branch dH E /dφ = 0 gives H E = const. and V = const. in the Einstein frame, which corresponds to f ( R J ) = R J − const. In the second branch, the generalsolution is given by H E ( φ ) = M ( γe − √ φ + e − βφ √ ) , (19) and the potential is given by V ( φ ) = 3 H E − (cid:18) dH E dφ (cid:19) = 3 − β M (cid:104) γe − (3+ β ) φ √ + (3 + β ) e − √ βφ (cid:105) , (20)where we introduced two integration constants M (massdimension 1) and γ (dimensionless). Using redefinition of M and φ , we can always normalize γ . Therefore, withoutloss of generality, we consider only γ = 0 , ± M isdetermined by the CMB normalization. Below we workin the unit where M = 1.Below we shall clarify that viable parameter set is β (cid:46) γ = −
1. Unlike the constant-roll potential foundin [2] using the condition ¨ φ = βH ˙ φ , the potential (20) isnot periodic function. Its form is depicted in Fig. 1 for aspecific parameter set β = − .
02 with γ = − β and γ , thepotential can pass V = 0. We can solve V ( φ ) = 0 and thesolution is given by φ = φ c , where we define the criticalfield value as φ c ≡ √ β − (cid:12)(cid:12)(cid:12)(cid:12) β γ (cid:12)(cid:12)(cid:12)(cid:12) . (21)For instance, φ c ≈ .
57 for β = − . , γ = −
1. A neg-ative value of the potential is undesirable since it maylead to recollapse soon after the end of inflation. For thisreason, we cut the potential at some point φ = φ > φ c to realize a graceful exit from inflation.For γ = 0, the potential is given by a single exponentialfunction. On the other hand, for β ≈ − β < − γ = − IV. f ( R ) CONSTANT-ROLL DYNAMICS
As mentioned above, from the form of the potentialwe focus on β ≈ −
3. In this section we shall checkinflationary dynamics. The evolution of the inflaton isgoverned by dφdt E = − dH E dφ = 2 √ e − βφ √ (3 γE + β ) , (22)where E ( φ ) ≡ e ( β − φ/ √ . By solving this equation, weobtain t E = e √ φ γ F (cid:18) , − β , − β − β ; − βγ e (3 − β ) φ √ (cid:19) , (23)where F is the Gauss’ hypergeometric function. Thus, φ ( t E ) is obtained in terms of the inverse function of thehypergeometric function. However, without its specific - ϕ R J - - - - d ϕ / d t E - - - - - - - N J - - - V FIG. 1. The potential (20) in the Einstein frame, the phasediagram for the scalaron, the e-folds (35) and the Ricci curva-ture (26) in the Jordan frame for β = − . , γ = −
1, wherewe set M Pl = M = 1. form of the solution, we can draw interesting conclusionas follows.If β and γ have the opposite sign, there exists a solutionfor dφ/dt E ∝ γE + β = 0, which we denote φ b , φ b ≡ √ β − (cid:12)(cid:12)(cid:12)(cid:12) β γ (cid:12)(cid:12)(cid:12)(cid:12) . (24)We can show that if φ b exists, the inflaton will al-ways approach φ b spending infinite Einstein-frame timeas follows. If 3 γE + β > φ ( t E ) is increasing and3 γE ( t E ) + β is decreasing. It continues decreasing solong as 3 γE + β >
0, and thus the inflaton approaches φ = φ b . Likewise, for the opposite case with 3 γE + β < φ = φ b . In both cases, theinflaton velocity | dφ/dt E | is always decreasing, thereforeit approaches φ = φ b spending infinite Einstein-frame time.Actually, this process develops small-scale inhomo-geneity of the Universe. From the conformal invarianceof the curvature perturbation, ζ J ∼ H E dφ/dt E δφ ∼ H E dφ/dt E , (25)where the right-hand side is evaluated at the horizon exit.As we showed above, | dφ/dt E | is always decreasing in thecourse of inflation. In such a case, | ζ J | is amplified onsmall scales, and the Universe becomes inhomogeneous.This also means that the isotropic background solutioninvolved is not an attractor. Therefore, we exclude theparameter set with βγ < f ( R ): R J = 23 ( β − e − β ) φ/ √ [3 γ ( β − E + ( β − β + 3)]= ( β − (cid:20) γ ( β − F − (1+ β ) / + 23 ( β − β + 3) F − β (cid:21) ,f ( R J ) = 23 ( β − e − β ) φ/ √ [3 γ ( β + 1) E + ( β − β + 3)]= ( β − (cid:20) γ ( β + 1) F (1 − β ) / + 23 ( β − β + 3) F − β (cid:21) . (26)Here F ( φ ) = e √ φ serves as an auxiliary variable. How-ever, it is easily seen that df /dR = F , as it should be. For γ = 0 or β = −
3, we can write down f ( R J ) ∝ R pJ , with p = − β − β or p = β − β +1 , respectively. For general case with β (cid:46) , φ >
1, the R J , f ( R J ) in (26) are dominated bythe second terms. Neglecting the first terms, we obtain f ( R J ) ≈
23 ( β − β + 3)( β − × (cid:18) R J β − β + 3)( β − (cid:19) − β − β . (27)The high curvature behaviour is thus close to the R + R p model. Since it is shown in [12] that the R + R p modelpossesses a parameter region to satisfy the latest observa-tional constraint, we expect that the present case wouldalso be observationally viable. Indeed, we shall see in § VI that there exists a parameter region γ = − , − . (cid:46) β ≤ , ≤ φ ≤ .
8, which satisfies the latest obser-vational constraint on inflationary power spectra. Forthis parameter region, we confirm that the relative errorbetween the exact parametric form (26) and the approx-imated form (27) remains less than 1 . f ( R ) are depicted in Fig. 2 for thecase β = − . , γ = −
1, for which 4 ≤ φ ≤ . . ≤ R J / ≤ .
6. The relative error increases as φ or R J decreases, and reaches 5 ,
10% at R J / − = 1 . .
88, respectively.Since in the inflationary regime Ricci curvature in (26)should be positive, we are interested in the field region R J / f / FIG. 2. The function f ( R J ) for β = − . , γ = − that satisfies( β − γ ( β − E + ( β + 3)( β − > . (28)However, for the case β < − γ = −
1, Ricci cur-vature is always negative, which is another reason whywe do not consider this parameter set, in addition to thenegative potential mentioned above. For other parame-ter sets, the Ricci curvature can pass R J = 0 and changethe sign at φ = φ r , where φ r ≡ √ β − (cid:12)(cid:12)(cid:12)(cid:12) ( β + 3)( β − γ ( β − (cid:12)(cid:12)(cid:12)(cid:12) . (29)For instance, φ r ≈ − .
55 for β = − . , γ = −
1. WhileRicci curvature is negative for φ < φ r , in this case it doesnot occur during inflation as we cut the potential at somepoint φ = φ > φ c ≈ .
57 to realize a graceful exit frominflation. For later convenience, it is worthwhile to notethat the tensor-to-scalar ratio r ( φ ) at φ = φ r does notdepend on β nor γ and takes very large value which isunacceptable from observational point of view: r | R J =0 = 643 ≈ . . (30)In addition, we require that the Ricci curvature is de-creasing during inflation, namely, dR J dt E = − dR J dφ dH E dφ = − e (2 − β ) φ √ (3 − β )(1 − β )( β + 3 γE ) × [2( − β )(3 + β ) + 3(1 + β ) γE ] , (31)should be negative. We use these expressions in § V toconstrain the parameter space. On the other hand, Hubble parameter in the Jordanframe is given by H J = e φ √ (cid:18) H E + 2 √ dH E dφ (cid:19) = 3 − β e (1 − β ) φ √ . (32)Using the definition F = e √ φ we obtain F = (cid:18) H J − β (cid:19) / (1 − β ) , (33)which precisely reproduces (14). Also, using the relation H J = dN J dt J = dt E dt J dφdt E dN J dφ , (34)we obtain e-folds in the Jordan frame as N J = 1 β log (cid:12)(cid:12)(cid:12)(cid:12) β γ e (3 − β ) φ √ (cid:12)(cid:12)(cid:12)(cid:12) . (35) V. INFLATIONARY POWER SPECTRA
Now we check the spectral parameters of inflationarypower spectra and compare them with observational con-straint to find viable parameter set ( β, γ ). First, thepower spectrum of scalar (curvature) and tensor pertur-bations can be calculated in the Jordan frame directly,e.g. as was quantitatively correctly done in [17] for themodel [7] using the δN formalism. Second, the calcula-tion in the Einstein frame leads to the same result sincethe constant modes of scalar (curvature) and tensor per-turbations are not affected by a generic (inhomogeneous)conformal transformation after the end of inflation; seee.g. [18, 19] for more details, and [20] for more generalinvariance under disformal transformation. The subtlepoint is that though the value of the power spectrum isthe same in both frames, it refers to slightly differentinverse scales k E and k J . However, corrections to thepower spectra of scalar and tensor perturbations follow-ing from this difference are proportional to | n s − | and | n t | correspondingly. In particular, they would be absentfor the exactly scale-invariant spectra. Thus, they canbe neglected in the leading order of the slow-roll approx-imation.We evaluate the spectral parameters by exploiting theslow-roll parameters for the inflaton potential in the Ein-stein frame, which are given by (cid:15) ≡ (cid:18) V (cid:48) V (cid:19) = (3 + β ) ( β + 3 γE ) β + 6 γE ) ,η ≡ V (cid:48)(cid:48) V = (3 + β )[2 β + 3(3 + β ) γE ]3(3 + β + 6 γE ) , (36) ξ ≡ V (cid:48) V (cid:48)(cid:48)(cid:48) V = (3 + β ) ( β + 3 γE )[4 β + 3(3 + β ) γE ]9(3 + β + 6 γE ) , where a prime denotes derivative with respect to φ . Weare interested in regime where these slow-roll parameters TABLE I. Reasons why parameter regions except β (cid:46) , γ = − γ = +1 γ = − γ = 0 β (cid:46) − R, V < r = 8(1 − n s ) β (cid:38) − r ≥ . R J ≥ r = 8(1 − n s ) β (cid:46) r = 8(1 − n s ) β (cid:38) r ≥ .
48 for dR J dt E ≤ r = 8(1 − n s ) are sufficiently small to obtain a nearly scale-invariantspectrum. By virtue of the conformal invariance of thecurvature and tensor perturbations, we can use the stan-dard slow-roll expansion of the spectral parameters n s − − (cid:15) + 2 η = − β )3(3 + β + 6 γE ) [ β (3 + β )+ 3( − β + β ) γE + 9(3 + β ) γ E ] , (37) r = 16 (cid:15) = 16(3 + β ) ( β + 3 γE ) β + 6 γE ) , (38) dn s d ln k = 16 (cid:15)η − (cid:15) − ξ = 2(9 − β ) γE ( β + 3 γE )[( − β )(3 + β ) + 12 γE ](3 + β + 6 γE ) . (39)For γ = 0, they read n s − → − β , r → β , dn s d ln k → . (40)We thus obtain a consistency relation r = 8(1 − n s ), forwhich it is impossible to satisfy the observational con-straint. For instance, r = 0 . , .
24 for n s = 0 . , . γ = ± r along with the evolution of R J . As we mentioned in § IV, we require R J > dR J /dt E < β (cid:38) − γ = − R J ≥ φ ≥ φ r where φ r is defined by (29). Thenit can be shown by checking dr/dφ and d r/dφ that forthe region φ ≥ φ r the minimum value of r is given at φ = φ r which is given by (30). Therefore, so long as weconsider the region where R J ≥
0, we have r ≥ . β (cid:38) γ > dR J /dt E ≤ φ ≤ √ β − log (cid:104) − β )(3+ β )3 γ (1+ β ) (cid:105) , and for this field position, r ≥ (4 + β ) ≥ .
48, which is also not acceptable.We thus find that only allowed possibility is β (cid:46) γ = −
1. Indeed, parameter set − . (cid:46) β < , γ = − n s , r ). Otherparameter regions are not feasible for various reasons,which are summarized in Table I. N J = N J = N J = r = �� - � - - - - - - β ϕ i / M P l FIG. 3. Observational constraint from CMB by Planck andBICEP2/Keck Array [21] on parameter space ( β, φ i /M Pl ) forthe case β (cid:46) , γ = −
1. The 68% and 95% confidence regions(blue), the Jordan frame e-folds N J counted back from φ c (green, dashed), and r = 10 − (purple, dotted). VI. OBSERVATIONAL CONSTRAINTS
As shown in Table I, in the previous sections wechecked that the parameter regions other than β (cid:46) , γ = − β (cid:46) , γ = − V ∼− e − φ + const so long as | βφ | (cid:46)
1. The inflaton rolls onthe plateau of the potential at positive φ region towardsnegative direction with dφ/dt E <
0. Before the inflatonreaches to φ = φ c where V = 0, we need to cut thepotential at some point φ = φ > φ c to realize a gracefulexit from inflation.Indeed, this form of the potential with a long plateauis favored by the observational data. During the infla-tion on the plateau, V and R J remain positive, and theplateau is sufficiently long to produce a large number ofe-folds N J ∼
50 (see Fig. 1).Furthermore, we compare our model with the latestobservational constraint on ( n s , r ) by Planck and BI-CEP2/Keck Array (see Fig. 7 in [21]). Figure 3 depictsthe allowed parameter region. We find that the parame-ter set − . (cid:46) β < , γ = − r ∼ .
1, thismodel has small number of e-folds N J ∼ VII. CONCLUSION
We have constructed a simple and natural generaliza-tion of the class of constant-roll inflationary models inGR to the case of f ( R ) gravity. The constant-roll con-dition (12) is introduced in the original Jordan frame.Using it, we derived the exact solutions for the Einstein-frame potential in (20), the parametric expression of f ( R J ) in (26), as well as the inflationary evolution inthe Einstein and Jordan frames. The functional form of f ( R J ) is expressed parametrically, while for some specialparameter values it is possible to write down f ( R J ) ex-plicitly as a function of R J . We showed that the modelhas an interesting parameter region − . (cid:46) β < , γ = − n s , r ) obtained by the Planck and BICEP2/Keck ArrayCollaborations. ACKNOWLEDGMENTS
H.M. thanks the Research Center for the Early Uni-verse (RESCEU), where part of this work was com-pleted. He was also supported in part by MINECOGrant SEV-2014-0398, PROMETEO II/2014/050, Span-ish Grant FPA2014-57816-P of the MINECO, and Eu-ropean Unions Horizon 2020 research and innovationprogramme under the Marie Sk(cid:32)lodowska-Curie grantagreements No. 690575 and 674896. A.S. acknowledgesRESCEU for hospitality as a visiting professor. He wasalso partially supported by the grant RFBR 17-02-01008and by the Scientific Programme P-7 (sub-programme7B) of the Presidium of the Russian Academy of Sci-ences.
Appendix A: Alternative derivations of f ( R ) In this appendix we present two alternative derivationsfor the parametric form of f ( R ) in (26), with the latterof them using the Jordan frame only. First, we show analternative derivation based on the results obtained inthe Einstein frame. By plugging (20) to the definition(5), we obtain a differential equation for f ( R ) as f = Rf (cid:48) + 23 ( β − f (cid:48) (cid:104) γf (cid:48)− β + (3 + β ) f (cid:48)− β (cid:105) , (A1)where a prime denotes derivative with respect to R . Thisequation is known as Clairaut’s equation.In general, Clairaut’s equation is defined as y ( x ) = xy (cid:48) + g ( y (cid:48) ) , (A2) where a prime denotes derivative with respect to x . Bytaking a derivative of the equation, we obtain( x + g (cid:48) ) y (cid:48)(cid:48) = 0 , (A3)which clearly has two branches of solutions. The firstbranch y (cid:48)(cid:48) = 0 yields general solution y = cx + g ( c ) , (A4)where c is integration constant. The second branch x + g (cid:48) = 0 yields singular solution x = − g (cid:48) ( c ) , y = − cg (cid:48) ( c ) + g ( c ) . (A5)This solution defines an envelope of the general solutionsand thus does not involve any integration constant. Thus c plays a role of parameter and we have dydx = c. (A6)For some special case, y can be written down explicitlyas a function of x . For instance, if g ( c ) = c , the singularsolution yields x = − c and y = − c = − x / g ( c ) = 23 ( β − (cid:104) γc − β + (3 + β ) c − β (cid:105) , (A7) g (cid:48) ( c ) = 23 ( β − (cid:104) − β ) γc − β + (3 + β )(2 − β ) c − β (cid:105) , and the singular solution R = − g (cid:48) ( c ) , f = − cg (cid:48) ( c ) + g ( c ) , (A8)precisely reproduces (26) with c = e √ φ = df /dR , asexpected from (A6).Now let us show how to derive Eq. (26) for theconstant-roll f ( R ) function working in the Jordan frameonly. Let us first denote the arbitrary constant appearingin (14) as A : F ( R ) = AH / (1 − β ) . (A9)Introducing (A9) into (11), we get the following equation f = β + 1 β − RF − β + 3) β − F − β A − β , (A10)which is a particular case of d’Alembert’s differentialequation (also known as Lagrange-d’Alembert’s equa-tion) y ( x ) = xh ( y (cid:48) ) + g ( y (cid:48) ) , (A11)where the prime means derivative with respect to x .The d’Alembert’s equation is more general than theClairaut’s equation considered above, and it can besolved using the same trick: considering y (cid:48) ≡ z as anindependent variable and differentiating (A11) with re-spect to z leads to the linear differential equation for x = x ( z ): dxdz − xh (cid:48) ( z ) z − h ( z ) = g (cid:48) ( z ) z − h ( z ) . (A12)Now the prime means derivative of the functions f and g with respect to their argument z . Its general solution is x ( z ) = exp (cid:18)(cid:90) z h (cid:48) ( u ) u − h ( u ) du (cid:19) (A13) × (cid:20) B + (cid:90) z g (cid:48) ( u ) u − h ( u ) exp (cid:18) − (cid:90) u h (cid:48) ( w ) w − h ( w ) dw (cid:19) du (cid:21) , where B is an integration constant. In our case of (A10),(A14) and (A10) take the form: R = 6( β − β + 3)( β − A − β F − β + BF − (1+ β ) / , (A14) f ( R ) = 6( β − β + 3)( β − A − β F − β + B β + 1 β − F (1 − β ) / . This just coincides with (26) with A − β = 9 / ( β − from (33) (for M = 1) and B = 2 γ ( β − β − [1] J. Martin, H. Motohashi, and T. Suyama, Phys. Rev. D87 , 023514 (2013), arXiv:1211.0083 [astro-ph.CO].[2] H. Motohashi, A. A. Starobinsky, and J. Yokoyama,JCAP , 018 (2015), arXiv:1411.5021 [astro-ph.CO].[3] H. Motohashi and A. A. Starobinsky, Europhys. Lett. , 39001 (2017), arXiv:1702.05847 [astro-ph.CO].[4] N. Tsamis and R. P. Woodard, Phys.Rev.
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