Reconstructing the inflationary f(R) from observations
Massimiliano Rinaldi, Guido Cognola, Luciano Vanzo, Sergio Zerbini
PPrepared for submission to JCAP
Reconstructing the inflationary f ( R ) from observations Massimiliano Rinaldi, a,b
Guido Cognola, a,b
Luciano Vanzo, a,b
SergioZerbini. a,b a Dipartimento di Fisica, Universit`a di Trentovia Sommarive 14, 38123 Trento, Italy. b TIFPA (INFN),via Sommarive 14, 38123 Trento, Italy.E-mail: [email protected], [email protected],[email protected], [email protected]
Abstract.
The BICEP2 collaboration has recently released data showing that the scalar-to-tensor ratio r is much larger than expected. The immediate consequence, in the context of f ( R ) gravity, is that the Starobinsky model of inflation is ruled out since it predicts a valueof r much smaller than what is observed. Of course, the BICEP2 data need verification,especially from Planck with which there is some tension, therefore any conclusion seemspremature. However, it is interesting to ask what would be the functional form of f ( R ) inthe case when the value of r is different from the one predicted by the Starobinsky model.In this paper, we show how to determine the form of f ( R ), once the slow-roll parameters areknown with some accuracy. The striking result is that, for given values of the scalar spectralindex n S and r , the effective Lagrangian has the form f ( R ) = R ζ , where ζ = 2 − ε and | ε | (cid:28) R theory, with a small deviation that, as we show, can be obtained by quantum corrections. Keywords:
Inflation, modified gravity, quantum field theory in curved space.
ArXiv ePrint: a r X i v : . [ g r- q c ] A ug ontents R and its one-loop quantum corrections 94 Discussion 11A Hubble flow functions and slow-roll parameters 11 In the light of several observational results obtained over the past decades, from COBEand WMAP to Planck, together with the recent BICEP2 observations [1], the importanceof single field inflationary models has been constantly sharpened, to the point that now ithas taken the form of a scientific paradigm. These models are mathematically equivalent tomodified gravity with Lagrangians L = (cid:112) − det( g ) f ( R ), provided one chooses as the relevantdynamical variable the metric field (see [2–6] for extensive reviews on the subject).Notwithstanding the lack of a general equivalence theorem, at least in the classicaldomain, they must also be considered “physically equivalent”, since the numerical values ofthe best tested observables is the same in both formulations, away from singularities [7, 8].By contrast, the limited knowledge on the quantum aspects of these models falls short to afull equivalence, although the quantum corrections discussed in Sec. 3 look like the effectiveColeman-Weinberg potential of scalar-tensor theories. As an example of the effects of fieldredefinitions, we recall that in flat space quantum field theory these lead to the same S -matrix. In any case, we believe that there are no reasons to suspect that the quantizedlinearized theory in either formulation are really different (for example, both have the samephysical degrees of freedom).Virtually all models of modified gravity are considered in the form f ( R ) = M p R +“something else”. The most popular is the Starobinsky model, where the additional termreads R / (6 M ) and M (cid:39) GeV [4, 9, 10]. In this paper we take the more radicalstep of omitting the linear Einstein-Hilbert term and keeping only the remaining stuff. Thefundamental reason is that the linear term tends to suppress precisely the tensor-to-scalarratio that we want to save from becoming too small in order to be consistent with the recentfindings of BICEP2 [1]. Although there is a lot of discussion about the reliability of thisexperiment, our findings are, at the end of the day, independent from BICEP2. In fact, aswe will show below, all we need is just a non-vanishing value of r . By assuming that thescalar spectral index n S and r are given, we show that, in the slow-roll regime, there is aunique and simple expression for the dominant term of the effective Lagrangian, which reads f ( R ) = R ζ where ζ = 2 − ε , with | ε | much smaller than one and decreasing for decreasing r .We also demonstrate that ζ is constant over a wide range of e-foldings that encompass theepoch of horizon exit of the relevant scales. We finally look for an explanation of the form The Palatini version of the theory does not fit very well with observational data. Here, M p = 8 πG is the reduced Planck mass. – 1 –f this term in the quantum corrections that are to be expected, according to the modernviewpoint of effective quantum field theory.The full structure of the effective Lagrangian is not in the class of attainable goals sothat, in brief, our main idea is to restrict considerations to the observable number of e-foldings during inflation and start with a single term R , on the ground that, at the scale ofinflation, conventional wisdom in quantum field theory would suggest a scale invariant world.Strictly speaking, the same requirement is also met by the quadratic Weyl action , and infact the attractiveness of conformally invariant actions as fundamental actions for quantumgravity has been repeatedly stressed in the past [11]. However, as we mentioned above, singlefield inflation is the favorite model and this leads uniquely to f ( R ) theories. We would like tonote that a similar approach was recently proposed in [12]. This model, dubbed “Agravity”,is much more ambitious than ours as it requires that the fundamental theory of nature doesnot contain any physical scale.The Einstein-Hilbert linear term hopefully emerges later in the cosmic evolution, asthe result of a scale invariance symmetry breaking induced by matter quantum corrections[11, 14]. Eventually, this is the necessary mechanism to stop inflation and ignite reheating.This scenario is different from the model of induced gravity proposed long ago by Sakharov[15, 16] in that the induced gravitational constant is renormalized and computable, so thatno tree level linear term is needed. In practice, this means that R is treated as the quantumside and R as the classical emergent side of one and the same gravitational field. This is notan entirely heretical idea, given the notorious difficulties in formulating a quantum theory ofthe Einstein action, with the notable exception of loop quantum gravity.In the next section we discuss how all these considerations reduce, in our scheme, toa single differential equation for a function α ( t ), given in terms of the slow-roll parameters,that fixes the exponent ζ . The equation is numerically solved and the solutions are discussedto find the effective Lagrangian. In this context, we also offer a brief comparison with othermodels. In Sec. 3 we propose an explanation of our results in terms of a quantum correctedde Sitter solution of R gravity. Finally, we conclude with some considerations in Sec. 4. Let us consider the general f ( R ) theory with Jordan-frame action S J = M p (cid:90) d x (cid:112) − det( g ) f ( R ) (2.1)in the vacuum and the cosmological J-frame metric ds = − dt + a d(cid:126)x , (2.2)where a ( t ) is the scale factor. The equations of motion are3 XH = 12 ( XR − f ) − H ˙ X, (2.3)¨ X = − X ˙ H + H ˙ X, (2.4) Inflation with a Weyl term has been studied in [13]. – 2 –here the dot refers to a derivative with respect to the (J-frame) cosmic time t , H = a − ˙ a is the Hubble function, and X ≡ df ( R ) dR , R = 6(2 H + ˙ H ) . (2.5)Our scope is to write the E-frame slow-roll parameters in terms of J-frames quantities onlyand for a generic f ( R ). One possible procedure is to write first the theory as an equivalentscalar-tensor theory in Einstein frame and then compute the slow-roll parameters as usual.The problem is, however, that it is not alway possible to follow this strategy, for example when f ( R ) contains terms like ln( R ), as it happens when loop corrections are taken in account.However, no matter the form of f ( R ), we know that the conformal transformationbetween Jordan and Einstein frames has the form ˜ g µν = Xg µν , where, from now on, tildedquantities refer to the Einstein frame. It has been proven that the spectrum of cosmologicalperturbations is invariant under conformal transformation (see e.g. [4]). As a consequence,also the observed spectra are invariant and so are the spectral indices. We can exploit thisfeature to write an observed spectral index as a function of E-frame or J-frame quantities.In particular, we can avoid to write the f ( R ) theory as an explicit scalar-tensor theory andfulfill the aim of expressing the spectral indices as functions of H and X only.In E-frame the metric reads d ˜ s = − d ˜ t + ˜ a d(cid:126) ˜ x , (2.6)from which it follows that ˜ a = √ Xa, dtd ˜ t = 1 √ X , (2.7)and ˜ H = 1˜ a d ˜ ad ˜ t = dtd ˜ t ddt (cid:104) ln (cid:16) a √ X (cid:17)(cid:105) = 1 √ X (cid:32) ˙ X X + H (cid:33) . (2.8)By differentiating again with respect to ˜ t and by using the equation of motion (2.4) we find d ˜ Hd ˜ t = − X X , (2.9)and d ˜ Hd ˜ t = 1 √ X (cid:32) X X + 3 ˙ X ˙ HX − H ˙ X X (cid:33) . (2.10)By combining these expressions, we find that the first two Hubble flow functions, defined byEqs. (A.2) in the Appendix, can be written as˜ (cid:15) = 3 ˙ X ( ˙ X + 2 XH ) , (2.11)and ˜ (cid:15) = − X ( H ˙ X + X ˙ X ˙ H + 2 X H ˙ H − X ˙ XH )˙ X ( ˙ X + 2 HX ) . (2.12)– 3 –he function ˜ (cid:15) is quite complicated and we do not report it. It will be shown in a simplifiedform below. The expression (2.11) can always be inverted and written as˙ X = αHX, α = 2(˜ (cid:15) ± √ (cid:15) )3 − ˜ (cid:15) . (2.13)By definition, accelerated expansion in Einstein frame occurs as long as | ˜ (cid:15) | <
1, namelywhen 1 − √ < α < √
3. During inflation, it is known that ˜ (cid:15) is not constant thus α isa function of time. By keeping this in account, from the equation of motion (2.4) we obtainthe (exact) relation ˙ αH + ( α + 2) ˙ H = α (1 − α ) H . (2.14)In terms of the number of e-folding N (defined as dN = Hdt ), the expression above becomes( α + 2) H (cid:48) H + α ( α −
1) + α (cid:48) = 0 , (2.15)while (2.13) becomes simply X (cid:48) = αX (from now on, a prime will indicate a derivative withrespect to N ). Using these equations, it easy to express the first three Hubble flow functionsin terms of α and its N-derivatives only as˜ (cid:15) = 3 α ( α + 2) , (2.16)˜ (cid:15) = 8 α (cid:48) α ( α + 2) , (2.17)˜ (cid:15) = − α + 2) α (cid:48) α ( α + 2) + 2 α (cid:48)(cid:48) ( α + 2) α (cid:48) . (2.18)By using the equations of motion, the definitions (2.5), and the equation (2.15), we fi-nally find that, on the cosmological background (2.2), the f ( R ) theory is effectively equivalentto f ( R ) = f R ζ , ζ = 4 − α − α (cid:48) − α ) − α (cid:48) , (2.19)where f is an arbitrary constant. Of course, this cannot be taken as a fundamental actiongenerating field equations, except for when ζ is constant. However, it can be taken as anumerical value of the Lagrangian density characterizing the scale of inflation. We remindthat ζ = 2 corresponds to a scale invariant theory while ζ = 1 leads to the usual Einstein-Hilbert term.The expressions (2.19) are exact and we wish to determine α during inflation by meansof observational data. This is possible, of course, because the Hubble flow functions arerelated to the slow-roll parameters in the slow-roll approximation. Since we can determinethe slow-roll parameters, through the measure of ˜ n S and ˜ r in Einstein frame, we can alsodetermine the function ζ ( N ).Suppose first that α is constant. In this case, we find˜ (cid:15) = 3 α ( α + 2) , ˜ (cid:15) = 0 . (2.20)– 4 –n particular, the second equality implies that 2˜ (cid:15) V = ˜ η V and eq. (A.10) that˜ n s = 4 + 4 α − α ( α + 2) . (2.21)By using the Planck value ˜ n s = 0 . ± . α and, in turn, the tensor-to-scalar ratio by means of eq. (A.11), which yields 0 . < ˜ r < . r is marginally compatible with the recent findings of BICEP2 [1]. Althoughprobably too large, this result is useful as it helps us to check that the solutions to theequations of motion correctly reproduce inflation. In fact, if α is constant we can integratethe equations of motion exactly. The Hubble parameter turns out to be H ( t ) = α + 2 α ( α − t − t ) = ⇒ a ( t ) ∼ ( t − t ) α +2 α ( α − (2.22)for arbitrary t . We checked that the two possible central values of α obtained by solving eq.(2.21), correspond to one expanding and one contracting Universe, according to α = − .
150 = ⇒ a ( t ) ∼ t . , (2.23) α = +0 .
177 = ⇒ a ( t ) ∼ t − . . (2.24)This information is important to remove the degeneracy that will shop up below, when wewill take a time-dependent α .As a further check of our findings, we transform the f ( R ) expression found above intothe corresponding tensor-scalar theory. The associate scalar potential has the general form V ( ˜ φ ) = XR − f ( R )2 κ X , (2.25)while the (Einstein frame) scalar field ˜ φ is identified through the relation X = exp (cid:32) κ (cid:114)
23 ˜ φ (cid:33) . (2.26)By using the results above, we find that the potential can be written as V = V exp (cid:20) − κ (cid:114) p ˜ φ (cid:21) (2.27)for some constant V and for (see eq. (2.20)) p = ( α + 2) α = 1˜ (cid:15) V . (2.28)The potential is the one proposed by Lucchin-Matarrese in [19] and the fact that p is theinverse of the slow-roll parameter confirms that the solution is of power-law type also inEinstein frame and has the form ˜ a ∼ ˜ t p .We now look at the more realistic case of a time-dependent α during inflation. With thehelp of Eqs. (A.10) and (A.11), we can write the scalar spectral index and the tensor-to-scalarratio respectively as˜ n S − C (3 α + 2) α (cid:48) α ( α + 2) − α + (3 C + 5) α ) α (cid:48) α ( α + 2) − Cα (cid:48)(cid:48) α ( α + 2) − α ( α + α + 1)( α + 2) , (2.29)– 5 – igure 1 . Plot of α ( N ) (red curve) with the initial condition α (120) = 1 − √
3. The green dottedcurve is the asymptotic value α (cid:39) − . α = 1 − √ and ˜ r = 48 α ( α + 2) − (cid:0) π − C + 36 C α − − π α (cid:1) α (cid:48) ( α + 2) ++ 192 α (cid:0) C + 8 Cα − π α + 8 Cα + 30 α (cid:1) α (cid:48) ( α + 2) + 32 α (cid:0) C − π (cid:1) α (cid:48)(cid:48) ( α + 2) . (2.30)If we consider ˜ r and ˜ n S nearly constant during inflation (and determined by experiments) wecan numerically solve the system of differential equations (2.29) and (2.30). The combinationof the two equations above yields a first order differential equation, which is quadratic in α (cid:48) therefore there are two solutions. Thanks to the results obtained above in the approximationof constant α , we know, however, which one corresponds to an expanding Universe, that isthe negative one. The constant of integration is fixed assuming that inflation ends at a given N end , such that α ( N end ) = 1 − √
3, see Eq. (2.13). The results are plotted in Figs. (1) and(2) in the case when we take the Planck central value ˜ n S = 0 . r = 0 . n S , ˜ r ) yields a solution with an almost constant α (hence ζ ) except for a sharptransition (to a lower ζ ) towards the end of inflation. The result is accurate since the measuredvalues of ˜ n S and ˜ r are related to the spectra of fluctuations exiting the Hubble horizon about60 e-folds before the end of inflation at N = N exit . The corresponding scales all exit thehorizon in a range of about 10 e-folds centered around N exit [21]. In this interval, we see that α is extremely close to a constant. By changing the value of ˜ n S to keep in account Planckerror bars modifies the asymptotic values of α of a negligible amount.One might suspect that our result holds only if the BICEP2 measure of ˜ r is correct.However, if we change the value of ˜ r we find exaclty the same qualitative behaviour but with Note that Eq. (2.29) does not coincide with Eq. (2.21) when α = const. The reason is that the higher orderexpression for n S includes the term ˜ (cid:15) that does not, however, contain derivatives of α but is still consideredbeing of higher order than ˜ (cid:15) and ˜ (cid:15) , see eq. (A.10). – 6 – igure 2 . Plot of the exponent ζ ( N ) (red curve) of R ζ . The green and the blue lines are theasymptotic values ζ (cid:39) . ζ (cid:39) . Figure 3 . Plot of α ( N ) for various values of ˜ r and with the same initial condition as in Fig. (1). different asymptotic values for α and ζ . In particular, we see that if ˜ r decreases then α ( N exit )tends to 0 − (see Fig. (3)) while ζ grows towards ζ = 2 (see Fig. (4)). This is consistent withthe fact that in de Sitter space (that would be one of the possible cosmological solutions of R theory) one has ˜ (cid:15) V = 0 and, for single-filed inflation, ˜ r = 16˜ (cid:15) V = 0. This is the secondimportant result of this paper: the exponent ζ is uniquely fixed by the values of ˜ n S and ˜ r .We note that our model is phenomenologically equivalent to the Starobinsky one when thetensor-to-scalar ration has the value predicted by the latter, namely ˜ r = 0 . r tends to lower the exponent ζ – 7 – igure 4 . Plot of ζ ( N ) for various values of ˜ r and with the same initial condition as in Fig. (1). Itis evident that, as ˜ r → + , ζ → − . monotonically. This observation helps to compare our results with the recent work [23],where a model R + R β is investigated yielding results comparable to ours. At 60 e-foldsbefore the end of inflation and with a best fit value of β (cid:39) . r = 0 . r = 0 . ζ = 1 . ζ ( N, ˜ r ) versus ˜ r for fixed N . It is to be noted that in [23] theEinstein-frame potential is approximated to match exactly our model, because the Einstein-frame scalar field is assumed to take values much larger than the Planck mass. The smallnumerical discrepancy is mostly due to the fact that, in our calculations, we used up thethe third Hubble flow functions, which corresponds to the order ˜ (cid:15) V (see the details in theappendix), while the authors of [23] considered only the order ˜ (cid:15) V . The other notable resultin [23] is the relation of the model with SUGRA models, whereas our assumption is to traceits origin to the (necessarily present) quantum corrections affecting scale invariant gravity,in a viewpoint similar to that of Salvio and Strumia [12].We can also compare our findings with the well-known Higgs inflationary models [24–28]. In the former versions, the lower bound on the Higgs mass , necessary to the existenceof inflation, together with the form of the effective potential, imply a value of ˜ r much smallerthan the one found by BICEP2. However, as recently observed in [29, 30], near the criticalvalue of the Higgs mass the radiative corrections change the effective potential in such away that ˜ r is sensibly larger. Even if the Standard Model (SM), implemented with Einsteingravity, is non-renormalizable, this is a conceptually important observation as it saves thetheory from being ruled if the experimental value of ˜ r is higher than the one predicted by the“bare” Higgs inflationary model (or, equivalently, by the Starobinsky model). In the presentcase, we stress that our model is in principle independent from the SM parameters because thesource of inflation is, in fact, just an extra degree of freedom encoded in the function f ( R ).In principle, this model can be always transformed into an equivalent single-field action. This is 2 σ away the value measured by LHC. – 8 –hen, the non-renormalizability of this theory is reflected in the non-renormalizability ofthe corresponding inflaton potential, which is required to be a general function since thevery large values of the field during inflation are not suppressed by the Planck mass. Onthe other hand, the inflationary slow-roll parameters are computed from the essentially freetheory which governs the small fluctuations around the background. Loop corrections canbe computed in the “in-in” formalism and can be meaningful even for non renormalizabletheories (see, e.g., Ref. [31]). R and its one-loop quantum corrections In the previous sections, we have shown how the simple effective Lagrangian R ζ , with ζ ≤ f ( R ) gravity models at one-loop level in a de Sitter back-ground have been investigated. A similar program for the case of pure Einstein gravity wasinitiated in Refs. [40–42] (see also [43, 44]). Furthermore, such approach also suggests apossible way of understanding the cosmological constant issue [42]. Hence, the study of one-loop generalized modified gravity is a natural step to be undertaken for the completion ofsuch program, with the aim to better understand the role and the origin of quadratic cor-rections in the curvature. An alternative approach, which is in some sense alternative, hasbeen proposed by Reuter and collaborators [45], see also the review papers [46] and [47, 48],where quantum gravity effects in astrophysics and cosmology are studied. The quantizationof the Einstein-Hilbert theory plus quadratic curvature terms has been discussed in manypapers, beginning with the detailed study in flat space presented in the seminal paper [54].A preliminary discussion of a quadratic model based on one-loop on-shell results has beenpresented in [55]. Recently, a deformation of Starobinsky models has also been discussed alsoin [56].In our case, the starting point is complete different, since the Einstein-Hilbert term willbe not considered, and all degrees of freedom are the ones related to R . Let us first revisitthe classical solutions associated with the Lagrangian (cid:112) − det( g ) R (see also [57]). Thecondition for the existence of de Sitter solution, 2 f ( R ) = RX ( R ), is automatically satisfiedby the equations of motion (2.3) and (2.4). Thus, we have the de Sitter solution ds = − dt + exp (cid:18) tt (cid:19) d (cid:126)x , (3.1)with arbitrary t and nonvanishing constant curvature R = t . This solution will be ourbackground solution for the calculation of quantum corrections. On the other hand, R = 0is also a solution, which may yield Minkowski space but also a radiation-dominated Universe– 9 –ith scale factor a ( t ) = a t / . Finally, there exist other solutions, discussed in [57] and [58],that satisfy the non-linear differential equation2 H ¨ H − ˙ H + 6 H ˙ H = 0 , (3.2)but that will not be investigated here, because we are interested in the one-loop quantum cor-rections to the classical de Sitter solution. In passing, we also observe that the Schwarzschildmetric is a spherically symmetric static solution of R .For the quantum corrections, we consider the classical Euclidean gravitational action I E [ g ] = − (cid:90) d x (cid:112) det( g ) f ( R ) = − b (cid:90) d x (cid:112) det( g ) R , (3.3)where b is a dimensionless arbitrary constant. At the classical level, its role is irrelevant, but,as we will see, it becomes crucial at the quantum level. We also note that the R Lagrangianhas the important property that the one-loop off-shell effective action coincides with theon-shell one, since the off-shell factor 2 f − RX identically vanishes. Furthermore, for thismodel the “scalaron” mass is also vanishing, and it is possible to show, by making use ofthe techniques presented in [36, 37, 39], that the R action is one-loop renormalizable, thecounterterm being generated by the “bare” dimensionless parameter b . As a consequence,the renormalized b ( µ ) is running and, at one-loop level, it reads b ( µ ) = γ ln (cid:18) µ µ (cid:19) + b , γ = − π . (3.4)Similar quantum corrections have been investigated in [49].With regard to this result, a clarification seems to be necessary. In fact, it is well knownthat, when working in an arbitrary background, the complete renormalizability requires alsothe R µν R µν term, which, in our de Sitter background, is proportional to R . However,what we have in mind here a particular case of the general scale-invariant action with L ∼ αR + βC µνσρ C µνσρ . This is asymptotically free, it has zero energy for any asymptoticallyflat initial data when α, β ≥ β = 0 withoutdestroying these properties is an open question that we hope to answer soon.Returning to the main point, the discussion above leads to the one-loop effective La-grangian L = − b R (cid:18) γb ln R µ (cid:19) . (3.5)If the free parameter b is chosen such that γb (cid:28)
1, one finds that L ∼ − b R (cid:18) R µ (cid:19) γb , (3.6)which provides a possible explanation of the phenomenological modified gravitational La-grangian found in the previous section. From Eq. (2.19), we see that we can write ζ = 2 − α (cid:48) + 3 αα (cid:48) + 2 α − , (3.7)– 10 –nd we have shown that observational data constrain the second term to be much smallerthan unity. We then propose that this deviation from ζ = 2 is generated by the one-loopcorrection, namely that − α (cid:48) + 3 αα (cid:48) + 2 α − γb . (3.8)In fact, γ < b ( µ ) is a decreasing running coupling constant, compatible withthe asymptotic freedom of quadratic gravity [50–53]. For example, for ˜ r = 0 . b = 0 .
027 while, for ˜ r = 0 . b = 0 . b increasesfor decreasing ˜ r . This result implies the intriguing possibility of measuring almost directlyquantum gravitational effects by means of cosmological observations. We have shown that the assumption of slow-roll evolution during inflation uniquely fixes theform of f ( R ), yielding a non-integer power of R , namely R ζ , with ζ slightly smaller than 2.This expression does not reduce to the Starobinsky model in any limit. We have also shownthat a possible explanation for this result comes from one-loop quantum corrections to theclassical Lagrangian R that can account for the deviation from the scale-invariant case R .Our results relies only upon the value for the scalar perturbation index measured by Planckand on the hypothesis that the scalar-to-tensor ratio is nonvanishing (including the valuefound by BICEP2).The picture that emerges is of an inflationary Universe governed by a scale-invariant ex-tension of general relativity together with quantum corrections. In opposition to the Starobin-sky model, our proposal does not have a built-in mechanism that can terminate inflation.We are confident, however, that inflation can be stopped in some other way and we hope toreport soon new results in this direction. Acknowledgments
We thank R. Percacci for valuable private communications, G.P. Vacca for comments andsuggestions, and all the participants to the First Flag Meeting “The Quantum and Gravity”(Bologna, May 28-30, 2014) for stimulating discussions.
A Hubble flow functions and slow-roll parameters
We recall the definitions of Hubble flow functions [18] (cid:15) = H i H , (cid:15) i +1 = ˙ (cid:15) i H(cid:15) i , (A.1)where H i is the initial value of the Hubble function. In particular, we are interested in (cid:15) = − ˙ HH = − H (cid:48) H , (A.2) (cid:15) = ¨ HH ˙ H + 2 (cid:15) = H (cid:48)(cid:48) H (cid:48) − H (cid:48) H ,(cid:15) = d ln (cid:15) ( N ) dN , – 11 –here the prime stands for the derivative with respect to N , the e-fold number, determinedby dN = Hdt . Note that these are exact expressions.In single-field inflation, with a generic potential V ( φ ), φ being the inflaton field, therelevant equations of motion are (here 8 πG = 1)3 H = 12 ˙ φ + V, ¨ φ + 3 H ˙ φ + V φ = 0 , (A.3)where V φ = dVdφ . For suitable potentials, there exists a slow-roll regime such that the kineticterm and the second time derivative of the inflaton can be neglected, so that3 H (cid:39) V , H ˙ φ (cid:39) − V φ . (A.4)In this regime, we find that (cid:15) (cid:39) (cid:18) V φ V (cid:19) , (A.5) (cid:15) (cid:39) (cid:32) V φ V − V φφ V (cid:33) , (A.6) (cid:15) (cid:15) (cid:39) (cid:34) V φφφ V φ V − V φφ V (cid:18) V φ V (cid:19) + 2 (cid:18) V φ V (cid:19) (cid:35) . (A.7)If we define the slow-roll parameters (valid in the slow-roll regime only) as (cid:15) V = 12 (cid:18) V φ V (cid:19) , η V = V φφ V , (A.8)we can, at the lowest order, write the relations (cid:15) (cid:39) (cid:15) V , (cid:15) (cid:39) (cid:15) V − η V . (A.9)We recall that all exponential potentials of the form V ∼ exp( kφ ) yield η V = 2 (cid:15) V for allconstants k .For the calculations shown in the main text, we are interested in the expressions thatlink directly the slow-roll parameters to the Hubble flow functions. One can show that [20] n S (cid:39) − (cid:15) − (cid:15) − (cid:15) − (2 C + 3) (cid:15) (cid:15) − C(cid:15) (cid:15) , (A.10) r (cid:39) (cid:15) (cid:20) C(cid:15) + (cid:18) C − π (cid:19) (cid:15) (cid:15) + (cid:18) C − π (cid:19) (cid:15) + (cid:18) C − π (cid:19) (cid:15) (cid:15) (cid:21) , (A.11)where C = γ E + ln 2 − (cid:39) − . γ E is the Euler constant. References [1] P. A. R. Ade et al. [BICEP2 Collaboration], Phys. Rev. Lett. (2014) 241101.[2] S. Nojiri and S. D. Odintsov, Phys. Rept. (2011) 59; eConf C (2006) 06 [Int. J.Geom. Meth. Mod. Phys. (2007) 115] ; Int. J. Geom. Meth. Mod. Phys. (2014) 1460006. – 12 –
3] S. Capozziello and M. De Laurentis,Phys. Rept. (2011) 167.[4] A. De Felice and S. Tsujikawa, Living Rev. Rel. (2010) 3.[5] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. , 451 (2010).[6] V. Faraoni and S. Capozziello, Beyond Einstein gravity : A Survey of gravitational theories forcosmology and astrophysics , Fundamental Theories of Physics, Vol. 170 (Springer, Dordrecht,2010).[7] V. Faraoni, E. Gunzig and P. Nardone, Fund. Cosmic Phys. , 121 (1999).[8] N. Deruelle and M. Sasaki, Springer Proc. Phys. (2011) 247.[9] A. A. Starobinsky, JETP Lett. (1979) 682 [Pisma Zh. Eksp. Teor. Fiz. (1979) 719].[10] A. A. Starobinsky, Phys. Lett. B (1980) 99.[11] D. G. Boulware, G. T. Horowitz and A. Strominger, Phys. Rev. Lett. , 1726 (1983).[12] A. Salvio and A. Strumia, JHEP (2014) 080.[13] N. Deruelle, M. Sasaki, Y. Sendouda and A. Youssef, JCAP (2011) 040.[14] S. L. Adler, Rev. Mod. Phys. (1982) 729, [Erratum-ibid. (1983) 837].[15] A. D. Sakharov, Sov. Phys. Dokl. (1968) 1040 [Dokl. Akad. Nauk Ser. Fiz. (1967) 70][Sov. Phys. Usp. (1991) 394] [Gen. Rel. Grav. (2000) 365].[16] Y. B. Zeldovich, JETP Lett. , 316 (1967) [Pisma Zh. Eksp. Teor. Fiz. , 883 (1967)].[17] P. A. R. Ade et al. [Planck Collaboration], “Planck 2013 results. XXII. Constraints oninflation,” arXiv:1303.5082 [astro-ph.CO].[18] J. Martin, C. Ringeval and V. Vennin, Phys. Dark Univ. (2014).[19] F. Lucchin and S. Matarrese, Phys. Rev. D (1985) 1316.[20] J. Martin and C. Ringeval, JCAP (2006) 009.[21] A. R. Liddle and S. M. Leach, Phys. Rev. D (2003) 103503.[22] I. L. Buchbinder, S. D. Odintsov and I. L. Shapiro, Effective action in quantum gravity , IOPPublishing, Bristol, 1992.[23] G. K. Chakravarty and S. Mohanty, arXiv:1405.1321 [hep-ph].[24] F. L. Bezrukov and M. Shaposhnikov, Phys. Lett. B (2008) 703.[25] A. O. Barvinsky, A. Y. Kamenshchik and A. A. Starobinsky, JCAP (2008) 021.[26] A. De Simone, M. P. Hertzberg and F. Wilczek, Phys. Lett. B (2009) 1.[27] A. O. Barvinsky, A. Y. Kamenshchik, C. Kiefer, A. A. Starobinsky and C. Steinwachs, JCAP , 003 (2009).[28] A. O. Barvinsky, A. Y. Kamenshchik, C. Kiefer, A. A. Starobinsky and C. F. Steinwachs, Eur.Phys. J. C (2012) 2219.[29] Y. Hamada, H. Kawai, K. -y. Oda and S. C. Park, Phys. Rev. Lett. (2014) 241301.[30] F. Bezrukov and M. Shaposhnikov, Phys. Lett. B, (2014) 249.[31] S. Weinberg, Phys. Rev. D (2005) 043514.[32] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani, and S. Zerbini, Phys. Rev. D , 046009 (2008).[33] K. Bamba, R. Myrzakulov, S. D. Odintsov, and L. Sebastiani, “Trace-anomaly driven inflationin modified gravity and the BICEP2 result,” arXiv:1403.6649 [hep-th]. – 13 –
34] S. Choudhury and A. Mazumdar, “Sub-Planckian inflation and large tensor to scalar ratio with r ≥ . , 386 (2014); D. Chialva andA. Mazumdar, “Super-Planckian excursions of the inflaton and quantum corrections,”arXiv:1405.0513 [hep-th]; Y. -F. Cai, J. -O. Gong and S. Pi, “Conformal description of inflationand primordial B-modes,” arXiv:1404.2560 [hep-th].[35] R. Costa and H. Nastase, JHEP , 145 (2014).[36] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov and S. Zerbini, JCAP , 010 (2005).[37] G. Cognola and S. Zerbini, J. Phys. A , 6245 (2006).[38] G. Cognola, E. Elizalde, S. Nojiri and S. D. Odintsov, Open Astron. J. , 20 (2010).[39] K. Bamba, G. Cognola, S. D. Odintsov and S. Zerbini, “One-loop Modified Gravity in de SitterUniverse, Quantum Corrected Inflation, and its Confrontation with the Planck Result,”arXiv:1404.4311 [gr-qc].[40] G.W. Gibbons and M.J. Perry, Nucl. Phys. B146 (1978) 90.[41] S.M. Christensen and M.J. Duff, Nucl. Phys.
B170 (1980) 480.[42] E.S. Fradkin and A.A. Tseytlin, Nucl. Phys.
B234 (1984) 472.[43] S.D. Odintsov, Europhys. Lett. (1989) 287; Theor. Math. Phys. (1990) 66; T.R. Taylorand G. Veneziano, Nucl. Phys. B345 (1990) 210.[44] D. V. Vassilevich, Int. J. Mod. Phys. A , 1637 (1993).[45] O. Lauscher and M. Reuter, Phys. Rev. D , 025013 (2001).[46] M. Niedermaier and M. Reuter, Living Rev. Rel. , 5 (2006).[47] M. Reuter and H. Weyer, JCAP , 001 (2004);M. Reuter and H. Weyer, Phys. Rev. D , 124028 (2004);E. Bentivegna, A. Bonanno and M. Reuter, JCAP , 001 (2004);M. Reuter and F. Saueressig, JCAP , 012 (2005);A. Bonanno and M. Reuter, Int. J. Mod. Phys. D , 107 (2004);A. Bonanno and M. Reuter, Phys. Rev. D , 043508 (2002).[48] A. Codello and R. Percacci, Phys. Rev. Lett. , 221301 (2006);A. Codello, R. Percacci and C. Rahmede, Int. J. Mod. Phys. A , 143 (2008);P. F. Machado and R. Percacci, Phys. Rev. D , 024020 (2009);A. Codello, R. Percacci and C. Rahmede, Annals Phys. , 414 (2009);A. Bonanno, A. Contillo and R. Percacci, Class. Quant. Grav. , 145026 (2011);[49] I. Ben-Dayan, S. Jing, M. Torabian, A. Westphal and L. Zarate, “ R log R quantum correctionsand the inflationary observables,” arXiv:1404.7349 [hep-th].[50] E. S. Fradkin and G. A. Vilkovisky, Phys. Lett. B , 262 (1978).[51] E. Tomboulis, Phys. Lett. B , 77 (1980).[52] I. G. Avramidi and A. O. Barvinsky, Phys. Lett. B , 269 (1985).[53] I. G. Avramidi, “Lectures Notes in Physics” M 64 , 953 (1977).[55] G. Cognola and S. Zerbini, J. Phys. A: Math. Theor. – 14 –
57] G. Menegoz and R. Percacci, ”Cosmological solutions in fourth-order gravity theories”,unpublished report (2010).[58] T. Clifton, Class. Quant. Grav. , 5073 (2007)., 5073 (2007).