Generalized Slow-roll Inflation in Non-minimally Coupled Theories
aa r X i v : . [ g r- q c ] A p r Generalized Slow-roll InflationinNon-minimally Coupled Theories
A. Sava¸s Arapo˘glu ∗ Faculty of Science and Letters, Department of Physics, 34469 Maslak, Istanbul, Turkey (Dated: July 14, 2018)
Abstract
The slow-roll field equations for the case of non-minimally coupled scalar fields are obtained intwo ways: first using the direct generalization of slow-roll conditions in the minimal coupling case tonon-minimal one; and, second, conformal transforming the slow-roll field equations in the Einsteinframe to the Jordan frame and then applying the generalized slow-roll conditions. We compare thedifference of two methods in calculation of the spectral index, n s , for a model example. The secondmethod seems to be more precise. ∗ Electronic address: [email protected] . INTRODUCTION Inflation is the most plausible scenario providing not only the successful explanation ofthe horizon, flatness, and monopole problems of the standard big bang cosmology [1–3],but also the primordial density fluctuations for the formation of the observed large-scalestructure of the universe (references [4–7] for reviews).In inflationary universe models, it is supposed that the nearly exponential expansion of theuniverse is driven by a scalar field (called inflaton) which is assumed to be minimally coupledto the gravity and slowly evolves in a nearly flat potential V ( φ ). In the so-called “slow-rollapproximation” [8] the most slowly changing terms in the field equations are neglectedwhich amounts to the approximation that the kinetic energy of the inflaton is considered tobe much smaller than the potential energy, that is, ˙ φ ≪ V ( φ ) and ¨ φ ≪ H ˙ φ . The existenceof inflationary attractors is necessary for slow-roll approximation to work. The slow-rollsingle-inflaton field models predict almost scale-invariant density perturbations consistentwith the observations of anisotropies in Cosmic Microwave Background (CMB).On the other hand, quantum field theory in curved spacetime necessitates a non-trivialcoupling between the scalar field and the spacetime curvature even if they are absent in theclassical theory. Actually there are many other indications that the inflaton couples to thecurvature of spacetime R (summarized in a nice way in [9]). Therefore, it is reasonable toconsider how the dynamics of the inflaton changes because of this non-minimal coupling.In general, one expects that the coupling is of the form ξφ R with a constant ξ , but thequantum corrections may change this situation and the behaviour of renormalization groupeffective coupling ξ becomes φ dependent also. The various asymptotics of the couplingconstant ξ in quantum field theory in curved spacetime were studied in [10]; recently in thisdirection the running of the non-minimal parameter ξ is analyzed within the non-perturbativesetting of the functional renormalization group (RG) [11] and the inflationary parameters inthe renormalization group improved φ theory at one-loop and two-loop levels are consideredin [12] and [13]. To cover all these effective models, then, one can consider a non-minimallycoupled inflaton field with a general coupling function of the form f ( φ ).We are now currently in an era stated commonly as the ‘precision cosmology’, imply-ing that the observational data sharpens and this allows one to compare the models moreprecisely. Inflationary models are examined and compared by the observations of Plancktogether with WMAP [14–16] and a joint analysis of BICEP2 [17] via the inflationary pa-rameters such as the spectral index n s , the tensor-to-scalar ratio r , the running of the spectralindex α = dn s /dlnk , and non-Gaussianity of the primordial perturbations. Indeed, discrim-inating the various inflationary models through the calculation of these parameters in bothminimally and non-minimally coupled theories is an active research area. Therefore, it isbeneficial to consider and compare the calculation of these parameters in non-minimallycoupled theories, and check whether there is any significant difference between minimal andnon-minimal cases, considering the recent bunch of papers appearing in the literature about2he subject.These parameters are obtained in the slow-roll approximation either considered directlyin the Jordan Frame (JF) through the “generalized slow-roll” approximation [18, 19], orby performing a conformal transformation to the
Einstein Frame (EF) and using the usualdefinitions of slow-roll parameters [8] in this frame; mostly the latter is preferred. Theexistence of attractor behaviour in inflation with non-minimal coupling is also demonstratedin [20] which is necessary for this approximation to work. The strong attractor inflationarybehavior in multifield inflationary models is also shown in [21]. Recently, Kallosh et al.[22] have shown that any inflationary model with a scalar-curvature non-minimal couplingasymptotes a universal attractor.In this paper, we obtained the slow-roll field equations in JF in two ways: in the firstmethod, we use directly the so-called ‘generalized’ slow-roll conditions in JF, [18, 19], andget the slow-roll field equations without any reference to EF; in the second method, we writethe slow-roll field equations in EF, as they are originally suggested, and get the correspond-ing ones in the JF via conformal transformations and generalized slow-roll approximationstogether. There is an interesting difference between the two methods: although the slow-rollFriedmann equations coincide, the scalar field equations do not match exactly which leadsto a difference in the calculation of, for example, the spectral index n s . It seems that the onederived from the EF slow-roll scalar field equation via conformal transformations provides amore precise value. We exemplify this result with a simple model.The plan of the paper is as follows: In section II, we present the set-up and the backgroundfield equations, both in the JF and the EF. In section III, we give two different ways of gettingthe slow-roll field equations in JF, and compare them in a simple model, showing explicitlytheir difference. Section IV is devoted to concluding remarks. II. SET-UP AND NOTATION
The action for the system of non-minimally coupled scalar field and gravity is S = Z d x √− g (cid:20) f ( φ ) R −
12 ( ∇ φ ) − V ( φ ) (cid:21) . (1)The field equations following from this action are f G µν + ( g µν (cid:3) f − ∇ µ ∇ ν f ) = 14 ∇ µ φ ∇ ν φ − g µν (cid:20)
12 ( ∇ φ ) + V ( φ ) (cid:21) , (2) (cid:3) φ − V ′ + f ′ f (cid:20) (cid:3) f + 12 ( ∇ φ ) + 2 V ( φ ) (cid:21) = 0 , (3)3here a prime indicates d/dφ . In flat Friedmann-Robertson-Walker spacetime, they become H = 16 f (cid:18)
12 ˙ φ + V ( φ ) (cid:19) − f ′ f H ˙ φ, (4) (cid:18) f ′ f + 1 (cid:19) ( ¨ φ + 3 H ˙ φ ) + f ′ f ( f ′′ + 1) ˙ φ + f ddφ (cid:18) Vf (cid:19) = 0 . (5)A conformal transformation of the formˆ g µν = Ω ( φ ) g µν , (6)Ω ( φ ) ≡ M f ( φ ) , (7)brings the action into the Einstein-Hilbert form with a canonical scalar field S = Z d x p − ˆ g (cid:20) M R −
12 ( ˆ ∇ ˆ φ ) − ˆ V ( ˆ φ ) (cid:21) , (8)where the new (canonical) scalar field ˆ φ is defined in terms of the JF scalar field φ throughthe relation d ˆ φdφ ≡ M s f + 3 f ′ f . (9)The scalar field potential in EF is also defined asˆ V ( ˆ φ ) ≡ M V ( φ )4 f = V ( φ )Ω . (10)Note that in Eq.(10), the right-hand side is written in terms of the JF scalar field φ . Towrite ˆ V in terms of ˆ φ one must use Eq.(9) to get ˆ φ ( φ ) and invert it to find φ ( ˆ φ ) which is inprinciple possible but in general difficult. The field equations following from Eq.(8) areˆ G µν = 1 M (cid:20) ˆ ∇ µ ˆ φ ˆ ∇ ν ˆ φ −
12 ˆ g µν ( ˆ ∇ ˆ φ ) − ˆ g µν ˆ V ( ˆ φ ) (cid:21) , (11)ˆ (cid:3) ˆ φ − ˆ V ′ ( ˆ φ ) = 0 , (12)where the covariant derivatives are with respect to the metric ˆ g µν and a prime indicates d/d ˆ φ . In flat Friedmann-Robertson-Walker spacetime, they becomeˆ H = 13 M d ˆ φd ˆ t ! + ˆ V , (13) d ˆ φd ˆ t + 3 ˆ H d ˆ φd ˆ t = − dd ˆ φ ˆ V . (14)4
II. SLOW-ROLL FIELD EQUATIONS IN JF
In non-minimal inflationary models the method followed mostly in the literature is tomap the model in JF via conformal transformations to a model in EF, presumably due tothe fact that the field equations and the procedure to follow are simpler in EF. The slow-rollparameters in EF are defined as usual, ǫ ≡ M " ˆ V ′ ( ˆ φ )ˆ V ( ˆ φ ) , (15) η ≡ M " ˆ V ′′ ( ˆ φ )ˆ V ( ˆ φ ) , (16) ζ ≡ M " ˆ V ′ ( ˆ φ ) ˆ V ′′′ ( ˆ φ )ˆ V ( ˆ φ ) / , (17)where the prime denotes d/d ˆ φ . To proceed in EF one has to write ˆ V in terms of the EF scalarfield ˆ φ ; but since in general it is difficult to find ˆ φ in terms of φ in closed form, the generallypreferred strategy is to express each quantity of interest in terms of the JF quantities. Theslow-roll parameters, for example, are to be evaluated at ˆ φ hc which is the value of ˆ φ atwhich the scales of interest cross the horizon during the inflationary epoch. Although thecalculation of the value of the field at horizon-crossing is not an easy task in both frames,by assuming that the scales of interest cross the horizon after N e-folding before the end ofinflation, we can write e N ≡ ˆ a (ˆ t end )ˆ a (ˆ t hc ) = Ω end Ω hc a ( t end ) a ( t hc ) , (18)where φ hc appearing in Ω is the value of the JF scalar corresponding to ˆ φ hc . This allows usto consider the slow-roll parameters, mapped back to the JF, at correct time [23]. Thereforewe need slow-roll field equations and need to solve them to get a ( t ) and φ ( t ) in JF. We getthe slow-roll field equations in two ways in this section: first we apply the generalized slow-roll approximations [18–20] to get the approximate field equations assuming the existence ofinflationary attractors[20] in phase space. Second we write the slow-roll field equations inEF, and express them in terms of JF variables by applying the conformal transformationsEq.(6) together with the generalized slow-roll conditions. A. Slow-roll field equations in JF via generalized slow-roll conditions
The dynamics of inflationary models with a single minimally coupled inflaton is consideredin the “slow-roll approximation” [8] which amounts to the assumptions that the inflatonevolves slowly in comparison to the Hubble rate, and that the kinetic energy of the inflatonis smaller than its potential energy. These conditions are expressed in a compact way as5 ¨ φ | ≪ H | ˙ φ | ≪ H | φ | and ˙ φ ≪ | V ( φ ) | . The generalization of these conditions to scalar-tensor theories with a coupling function f ( φ ) which has a sufficiently fast convergent Taylorexpansion is | ¨ f | ≪ H | ˙ f | ≪ H | f | , (19)first pointed out by Torres, [19].Direct application of these conditions to the field equations in JF, Eqs.(4) and (5), leadsto the approximate field equations of the form H ≃ f V, (20)3 H ˙ φ ≃ V f ′ f − φV ′ . (21) B. Slow-roll field equations in JF via those of EF
The slow-roll field equations in EF are obtained from Eqs.(13) and (14) with the usualslow-roll approximations as ˆ H ≃ M ˆ V , (22)3 ˆ
H d ˆ φd ˆ t ≃ − dd ˆ φ ˆ V . (23)Applying the conformal transformations Eq.(6) in connection with the generalized slow-roll conditions Eq.(19) together, we get the slow-roll approximate field equations in JF. Theslow-roll approximated Friedmann equation Eq.(20) is exactly the same as the one obtainedin the previous section but the slow-roll approximated scalar field equation becomes3 H ˙ φ (cid:18) f ′ f (cid:19) ≃ V f ′ f − φV ′ (24)which is different from Eq.(21) derived in the previous section.The difference between the scalar field equations in JF implies that the results of calcu-lation of φ hc are different, and thus ˆ φ hc and slow-roll parameters are different in turn. C. Comparison through a simple example
In this section we compare the Eqs.(21) and (24) through the calculation of the spectralindex n s and check whether there is any meaningful difference between them in the light ofthe sharpening data of aforementioned observations.To calculate the spectral index n s , we must calculate slow-roll parameters in the EF andevaluate them at ˆ φ hc . But because of the possible technical difficulties in its calculation onecan use the slow-roll scalar field equation to get the functional form of φ ( t ) and then Eq.(18)6o find the corresponding value in JF, for example, for 60 e-foldings. In this way we comparethe possible difference between the slow-roll approximations we considered.The spectral index to the second order in terms of these parameters is given by [8, 24] n s = 1 − ǫ + 2 η + 13 (44 − c ) ǫ + (4 c − ǫη + 23 η + 16 (13 − c ) ζ (25)where γ ≃ .
577 is the Euler’s constant and c ≡ ln γ ) ≃ . φ hc any difference in slow-roll field equations will make a difference in the spectral indexobviously.We consider a specific model, the model of induced-gravity inflation[25–31], to comparethe calculations, S = Z d x √− g (cid:20) ξφ R −
12 ( ∇ φ ) − V ( φ ) (cid:21) (26)together with the Ginzburg-Landau potential V ( φ ) = λ φ − v ) . (27)In this model, ξ is the coupling strength of scalar field with curvature, and the non-minimalcoupling makes the Planck mass a dynamical quantity through the relation M Pl = √ πξv .For this model, the new inflation initial conditions lead to the result for the spectral indexto first order n s ≃ − ξ (28)if we use Eqs.(20) and (21), and give the result n s ≃ − ξ ξ , (29)if we use Eqs. (20) and (24). Although the difference between the two results differs bysecond order in the (small) parameter ξ , it is thought to be crucial because the inflationarytheoretical predictions have now reached to level of second order in the slow-roll parameters.The existence of a difference between the two approaches is interesting in that Morris [32]shows that JF field equations, if expressed in terms of EF variables, agree with the EF fieldequations directly obtained from the EF action, provided that some consistency conditionsare satisfied, and that these conditions are always met. This result implies that the twoframes are, at least, mathematically equivalent which in turn implies that one can work inone frame, if there is any advantage of simplicity over the other, and then can go to theother frame. Further Kaiser showed that the spectral indices are the same in JF and EF[33]. The route that we follow here is in the reverse order: we obtained the approximate JFequations of motion from those of EF expressed in terms of JF variables and we compareit with the approximate equations of motion obtained directly in the JF. The difference inthe results does not seem to be because of the mathematical in-equivalence of the framesbut stems from the fact that the slow-rolling approximation is a very critical issue and must7e applied carefully for the non-minimal coupling case. From our point of view the methodthat first writing the slow-roll field equations in EF and then expressing them in terms ofJF variables together with the generalized slow-roll parameters seems to be safer and moreprecise.The change in the scalar field equation can be expected on the ground that the conformaltransformations themselves are dependent on the JF scalar field φ and that the ‘generalized’approximation directly in the JF cannot give exactly the same scalar equation obtained viaconformal transformations from that of the EF. IV. CONCLUSION
In this work we have considered a general scalar-tensor theory of a single scalar fieldnon-minimally coupled to the curvature with a general coupling function f ( φ ). The aim ofconsidering such a system is application to single-field inflationary models in the slow-rollapproximation. The non-minimally coupled models in the inflationary context is an activeresearch area and, with the proliferation of recent high-precision observations, is consideredto be as important and viable as the standard minimally coupled models. This is obvious inthat there are many papers appearing frequently in the literature.Therefore we have thought that it is better once again (earlier in [18], [19], [20]) to considerthe slow-roll field equations in such models by getting them in two different but related ways,comparing them and systematising the result. A very careful analysis leads to a (presumably,though small) difference in the JF slow-roll scalar field equation of motion which may bemeaningful in the light of current precise data. This difference may cause important changesin the parameter space of non-minimal models studied currently. The model example thatwe choose is in the form that is studied in the literature frequently.The method that first getting the slow-roll field equations in the EF and then by applyingconformal transformations in conjunction with the generalized slow-roll approximation seemsto be safer and more precise than getting the JF approximate equations directly in the JF byapplying the such generalized conditions. One can expect that the conformal transformationof the slow-roll field equations in EF had to give the analogous equations in the JF; but havingpreformed the conformal transformations to JF, some additional terms are introduced whichare able to be eliminated by the generalized conditions. Thus we suggest to use Eqs.(20)and (24) as the more precise and correct set of slow-roll field equations in JF.Another issue that must be clarified is what could be the problem with the use of thegeneralized slow-roll approximation directly in the JF, i.e. Eqs.(20) and (21). To answer thisquestion one must remember that the slow-roll approximation is originally defined in the EFand it has a well-motivated physical content; direct generalization of this approach to theJF can possibly miss some underlying physical principles even if it seems mathematicallyproper. (An example of such a point in the literature is to disregard the necessity of attractor8ehaviour for such an approximation to work in the non-minimal context.) [1] A. H. Guth, “Inflationary universe: A possible solution to the horizon and flatness problems,” Physical Review D , vol. 23, pp. 347–356, Jan. 1981.[2] A. D. Linde, “A new inflationary universe scenario: A possible solution of the horizon, flat-ness, homogeneity, isotropy and primordial monopole problems,”
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