Exponential potential for an inflaton with nonminimal kinetic coupling and its supergravity embedding
aa r X i v : . [ h e p - t h ] N ov Exponential potential for an inflaton with nonminimal kinetic coupling and itssupergravity embedding
Iannis Dalianis ∗ and Fotis Farakos † Physics Division, National Technical University of Athens,15780 Zografou Campus, Athens, Greece Institute for Theoretical Physics, Masaryk University,611 37 Brno, Czech Republic
In the light of the new observational results we discuss the status of the exponential potentialsdriving inflation. We depart form the minimal scenario and study an inflaton kinetically coupled tothe Einstein tensor. We find that in this case the exponential potentials are well compatible withobservations. Their predictions coincide with those of the chaotic type quadratic potential for aninflaton minimally coupled to gravity. We show that there exists a simple mapping between the twomodels. Moreover, a novel aspect of our model is that it features a natural exit from the inflationaryphase even in the absence of a minimum. We also turn to supergravity and motivate these sort ofpotentials and the non-minimal kinetic coupling as possible effective dilaton theories.
I. INTRODUCTION
Recently, the BICEP2 telescope has announced theobservation of a B-mode polarization of the Cosmic Mi-crowave Background (CMB) which for first time indicatesa non-zero value for the tensor-to-scalar ratio, r , at 7 σ C.L. [1]. This discovery, if confirmed by the future mea-surements, has striking cosmological implications. Thereported by BICEP2 values together with the Planckdata [2] provide a significant constraint on inflationarymodels. The value of r is directly related to the scale ofinflation and the type of the inflationary potentials. Thelarge field models, see e.g. [3–6] for reviews, are well sup-ported by the data. A particular example of large fieldmodels is the exponential potential which leads to powerlaw inflation [7–10]. Potentials of this sort can arise ina number of microscopic theories of matter and interac-tions and in particular in stringy set ups, see Ref. [11–13]for recent works, and Ref. [14, 15].However, cosmological inflation driven by exponentialpotentials and minimally coupled to gravity has been dis-favoured by the WMAP and Planck analysis, and, cur-rently poorly fits the BICEP2 data.In this Letter, in the light of the new results, we re-draw our attention to the exponential potentials and inparticular to an inflaton kinetically coupled to the Ein-stein tensor [16–23]. We find that the predicted valuesof a kinetically coupled inflaton with exponential poten-tial fit very well the observational data. Furthermore,the non-minimal kinetic coupling (or simply kinetic cou-pling) allows inflation to take place for a much widerrange of values for the potential parameter λ , see Eq. (1).This feature has an important consequence: inflation canterminate contrary to the standard case where inflationnever ends without the assistance of an additional mecha- ∗ Electronic address: [email protected] † Electronic address: [email protected] nism. Once the kinetic coupling effects become negligiblethe potential is too steep to drive inflation and the in-flaton’s energy density rapidly redshifts. If the createdradiation energy density becomes fast enough the domi-nant energy component in the post-inflationary universethen the standard thermal phase can be achieved.A very interesting aspect of the theory for an infla-ton with kinetic coupling and exponential potential isthat the inflationary predictions coincide to leading or-der in the slow-roll parameter with those of the chaotictype quadratic potentials. Although these two modelsappear to be unrelated we show that they are effectivelydescribed by the same dynamics during the inflationaryphase. Actually, any potential for an inflaton with ki-netic coupling can be transformed to a potential of an-other form for an inflaton minimally coupled to gravity.A similar observation concerning the Starobinsky the-ory was also made in the Ref. [24]. The exponentialpotential for an inflaton with kinetic coupling has obser-vational signatures that distinguish it from the quadraticpotentials: the post-inflationary evolution can be muchdifferent a fact that breaks the observational degeneracy.The observational support along with the fact thatexponential potentials emerge in generic stringy set-upsmotivate us to incorporate this higher derivative theoryinto a supergravity framework. Let us mention in pass-ing that there has been a renewed interest in embeddingchaotic inflation in supergravity triggered by the new ob-servational data [25–37]. Concerning our model, our re-sult is that inflation driven by the kinetically coupledinflaton can be successfully described in terms of super-gravity. We demonstrate that the kinetic coupling doesneither infer problematic instability issues nor the pres-ence of intermediate strong energy scales for the graviton-inflaton system.
II. MINIMALLY COUPLED INFLATON
Let us assume an inflaton minimally coupled to gravitywith an exponential potential V = V e − λφ/M P . (1)The scale factor grows like a ( t ) ∝ t /λ . In general theparameters λ and V should originate from some under-lying theory that implies their values. When the expo-nential potentials are applied for implementing the earlyuniverse inflation it appears that they fit quite poorly theobservational data. The slow-roll parameters ǫ V = M P (cid:18) V ′ V (cid:19) , η V = M P V ′′ V (2)yield the values ǫ V = λ / η V = 2 ǫ V = λ . Aninflationary phase occurs only if λ <
2. This theorypredicts the rather simple relation for the spectral index, n s and the tensor-to-scalar ratio, r , r = 8(1 − n s ) { for minimally coupled inflaton } (3)where 1 − n s = 6 ǫ V − η V = λ . The Planck collaboration[2] has excluded the exponential potential for the mea-sured spectral index value n s = 0 .
96 signifies the presenceof a strong signal of gravitational waves r = 0 .
32, see Fig.1 of [2]. This value is too large even for the BICEP2 data[1]. The value r = 0 . + . − . gives n s = 0 . − . . and whena dust reduction is taken into account, r = 0 . + . − . , the n s value increases to n s = 0 . − . . . These imply thatthe exponential potential for a minimally coupled inflatoncannot fit well the observaional data. They are actuallytoo steep yielding a too large ǫ V .Apart from the indicated observational disproof thereare also some theoretical difficulties. The coefficient λ in the exponent has to be λ = O (0 . r = 0 .
16 requires λ = 0 .
14. This means that thefield value φ is suppressed effectively by a superplanckianmass M P /λ whereas, the case usually predicted by par-ticle models, as e.g. in the string theory where the stringand compactification scales appear, is that λ &
1. A re-lated issue is that a sufficient number of e-folds requiressuperplanckian excursions for the inflaton, ∆ φ = N λM P .In addition, it is well known that the exponential poten-tial cannot account for a complete inflationary theory,since the slow-roll never ends and an additional mecha-nism is required to stop it.In the following we will show that these shortcomingscan be addressed when the inflaton with an exponentialpotential has a kinetic coupling to the Einstein tensor. III. INFLATON WITH A KINETIC COUPLING
We shall consider a theory of a scalar field, that weidentify it with the inflaton, kinetically coupled to grav- ity. The Lagrangian of this theory reads L = √− g (cid:20) M P R − (cid:18) g µν − G µν ˜ M (cid:19) ∂ µ φ∂ ν φ − V ( φ ) (cid:21) (4)and V = V e − λφ/M P for exponential potentials. In a ho-mogeneous FLRW background the Friedmann equationand the equation of motion (EOM) for this kineticallycoupled φ field are [20] H = 13 M P " ˙ φ (cid:16) M − H (cid:17) + V ( φ ) , (5) ∂ t h a ˙ φ (cid:16) M − H (cid:17)i = − a V φ . (6)These equations yield the modified expressions for theslow-roll parameters ǫ = ǫ V H ˜ M − , η = η V H ˜ M − , (7)where the former is derived by the definition ǫ ≡ − ˙ H/H and, the later by differentiating the approximated EOMform where one finds that δ ≡ ¨ φ/H ˙ φ = 3 ǫ − η . The ǫ V and η V are the slow-roll parameters for the minimal case(2). When the new scale ˜ M is much smaller than theHubble scale, i.e. H ˜ M − ≫
1, we have the so called highfriction limit . In this limit the Eq. (5) and (6) take theform H = 13 M P (cid:20) ǫ V ǫ ˙ φ + V ( φ ) (cid:21) (8) ǫ V ǫ (cid:20) H ˙ φ + (3 ǫ − η ) H ˙ φ − ˙ ǫǫ ˙ φ (cid:21) = − V ′ (9)where we used that ¨ φ = (3 ǫ − η ) H ˙ φ . During slow-rollinflation it is ǫ, η ≪ ǫ = 2 Hǫ the EOMEq. (9) reads approximately3 H ˙ φ ≃ − ǫǫ V V ′ . (10)The Eq. (10) implies, after straightforward calculations,that the Friedmann equation (8) reads H = 13 M P (cid:20) ǫV V (cid:21) ≃ V M P (11)and, indeed, during the slow-roll regime the potentialdominates the energy density. The Eq. (11) and (10)are the master equations that describe the inflationarydynamics in the case of kinetic coupling.In the high friction limit the spectral tilt of the scalarpower spectrum has, in turn, a modified dependence onthe slow-roll parameters [21, 22] n s − ≡ d ln P R d ln k (cid:12)(cid:12)(cid:12)(cid:12) c s k = aH ≃ − ǫ + 2 η (12)due to the sensitivity of the P R to the modulated bythe first slow-roll parameter sound speed of the pertur-bations [22]. The high friction limit, albeit, leaves the P R amplitude and, in first order, the tensor amplitude P t unchanged. Therefore the tensor-to-scalar power ratioin terms of the slow-roll parameter ǫ has approximatelythe standard form. Summing up, in the case of kineticcoupling and for the exponential potential the n s and the r are related to leading order to ǫ as follows: r ≃ ǫ − n s ≃ ǫr ≃ − n s ) { for an inflaton with kinetic coupling } (13)Comparing with the expression (3) for the minimal casewe see that, here, the predicted value for the tensor-to-scalar power ratio is lower for a given n s . We illustratethe different behaviour of the r = r ( n s ) for the mini-mal and the kinetic coupling cases in the Fig. 1 wherethe BICEP2 and the Planck (for running spectral index)contours are in the background. The exponential po-tential in the minimal case predicts zero running and isexcluded by the Planck data.The expression for the number of e-folds before theend of inflation, N ∗ , at which the pivot scale k ∗ exits theHubble radius, is also modified. It reads N ∗ ≡ ˆ t ∗ t e Hdt = 1 M P ˆ φ ∗ φ e (1 + 3 H ˜ M − ) VV, φ dφ = φ e − φ ∗ λM P + V λ M P ˜ M (cid:16) e − λφ ∗ /M P − e − λφ e /M P (cid:17) . (14) A. Inflation terminates
Inflation takes place for ǫ <
1. From the expres-sions (7) in the high friction limit we take that ǫ ≃ ǫ V / (3 H ˜ M − ) = λ / (6 H ˜ M − ). Hence, a first commentis that inflation can occur for much larger values for theparameter λ than in the standard case, i.e. λ ≫ O (0 . H ≃ V ( φ ) /M P , we find that inflation is realized onlyfor sufficient large values for the potential V ( φ ) >
12 ( λ ˜ M M P ) . (15)The above condition reads in terms of the inflaton field φ < M P λ ln V λ ˜ M M P ! . (16)Thereby, in the case of the kinetic coupling an inflationwith an exponential potential does have an end contraryto the minimal case. Indeed, we see from the expressions Minimally coupled Inflaton Inflaton with kinetic coupling
ExponentialPotentials
FIG. 1: Theoretical predictions for n s and r for an inflatonwith standard exponential potential (dashed line) and an in-flaton kinetically coupled to gravity (solid line) compared tothe BICEP2 data [1]. The red contours constrain models withtensors and, additionally, with running of the spectral index dn s /d ln k ≃ − . blue contour , since both models predict eitherstrictly zero (dashed line) or close to zero, O (10 − ), (solidline) running. for the slow-roll parameters (7) that when inflation ends, ǫ = 1, the Hubble scale has a value H e H e = ( λ −
2) ˜ M . (17)Unless λ > √ H e ≫ ˜ M which implies that λ ≫ H . ˜ M , the φ -inflaton field evolves in the standard way. It has beenshown [9] that for λ > / a ∝ t / . This corresponds to stiff mat-ter behaviour, p = ρ , with energy density scaling like ρ ∝ a − . Therefore given that the field φ , after the endof inflation, has transferred sufficient part of its energydensity into entropy it will remain a fast redshifting andsubdominant component of the universe energy density.In other words the absence of a minimum, which appearsto be a negative feature of the exponential potential, may not be problematic in kinetically coupled to gravity in-flaton scenarios. Nevertheless, we do not claim that thismodel should not receive further completion. An addi-tional mechanism may exist that generates a minimum,however, there is no need here the position of this mini-mum to be necessarily linked with the end of inflation. B. The predicted values for the inflationaryobservables
The tensor-to-scalar ratio . The scalar spectral indexvalue measured by Planck is n s = 0 .
96. Inflation drivenby an inflaton kinetically coupled to gravity and with anexponential potential predicts (13) r = 0 .
16 (18)when we plug in n s = 0 .
96. This value is in wellagreement with the BICEP2 data.
The number of e-folds . The potential energy dur-ing inflation is V ( φ ∗ ) = V e − λφ ∗ /M P = 3 . × − r ∗ M P which gives the following value for the exponent: λφ ∗ = (cid:20) − ln (cid:16) r ∗ . (cid:17) + ln (cid:18) V M P (cid:19)(cid:21) M P . (19)For r ∼ .
16 and V = M P the exponent takes the value λφ ∗ ≃ M P . Taking into account that in the high fric-tion limit it is λ ≫
1, the relations (16), (19) and thevalue of the ∆ φ = φ e − φ ∗ from Eq. (28) we find that thenumber of e-folds are approximately given by the secondterm of (14) i.e. N ∗ ≃ V ( φ ∗ ) − V ( φ e ) λ M P ˜ M (20)During slow-roll inflation in the limit 3 H ≫ ˜ M wecan write H ≃ [ ˙ φ (1 + 3 ǫ V /ǫ ) / V ] / M P ≃ V / M P .Thereby, from the first slow-roll parameter value at thepivot scale ǫ ∗ = ( λ ˜ M ) / H ∗ ≃ ( λ ˜ M M P ) / V ( φ ∗ ) andthe end of inflation ǫ e ≃ ( λ ˜ M M P ) / V ( φ e ) = 1 we findthat V ( φ ∗ ) ≃ V ( φ e )(2 N ∗ + 1) (21)and ǫ ∗ ≃ V ( φ e ) V ( φ ∗ ) = 12 N ∗ + 1 . (22)Hence, using the tensor-to-scalar ratio r ∗ = 16 ǫ ∗ we takethe following relations at the pivot scale: r ∗ ≃ λ ˜ M M P ) V ( φ ∗ ) ≃ N ∗ + 1 (23)and 1 − n s = 42 N ∗ + 1 . (24)If we ask N ∗ ≤
60 we take that r ∗ ≥ . r ∗ = 0 .
16 we take N ∗ ≃ The parameter λ and the mass scale ˜ M . We canrecast the expression (23) in such a way that the product λ ˜ M is given in terms of measured quantities H ∗ , r ∗ λ ˜ M = r H ∗ r / ∗ . (25)Also, given the measured amplitude for the scalarpower spectrum, P R , the Hubble parameter reads H ∗ = 1 . × − r / ∗ M P and the (25) is rewritten as λ ˜ M = 6 . × − r ∗ M P . Therefore the free mass scaleparameter ˜ M is directly related to the parameter λ ofthe inflationary potential. For example if λ ∼
10 thesuppression scale of the non-minimal derivative couplingis ˜ M ∼ − M P . The running of the spectral index . In the high fric-tion limit we find that dn s d ln k = − dǫd ln k = − λ ˜ M M P V ( φ ∗ ) ! ≃ − N ∗ + 1) . (26)Although there is a running of the spectral index here,contrary to the minimal case, the n s running is not thatsignificant so as to reconcile the different data accordingto the BICEP2 suggestion [1]. The variation of the field value during inflation .According to (16) the value of the field at the end ofinflation is φ e = ( M P /λ ) ln(2 V /λ ˜ M M P ). Plugging inthe value λ ˜ M = 6 . × − r ∗ M P we find φ e ≃ M P λ (cid:20) . − (cid:16) r ∗ . (cid:17) + ln (cid:18) V M P (cid:19)(cid:21) . (27)Hence ∆ φ = φ e − φ ∗ ≃ h . − ln (cid:16) r ∗ . (cid:17)i M P λ (28)where φ ∗ is given by the expression (19). Although thevariation of the φ field is possibly well subplanckian weemphasize that the φ field is not canonical. Actually,only during an exact de-Sitter phase one can recast thekinetic term for the scalar field (4) into a canonical form,where ˜ φ = √ H ˜ M − φ is the canonically normalizedfield and H = constant. Inflation is a quasi de-Sitterphase where ˙ H = − ǫH ≪
1. Hence, a crude estimate∆ ˜ φ ∼ √ H ˜ M − ∆ φ ∼ . √ r − / ∗ ( H/H ∗ ) M P , wherethe expression (25) has been used, indicates that thecanonical field variation is superplanckian and the Lythbound [38] is not violated, see also [39–41].We note that, despite the new nonliner interactionof gravity to the inflaton field, no non-Gaussian fluctua-tions larger than those in general relativity are produced.It has been actually shown [22] that in the single fieldcase the non-Gaussianities of curvature perturbationsare suppressed by slow-roll as in [42].Let us now briefly comment on the above results.The condition λ ≫
1, implied by the observational data,is welcome for several reasons. Firstly, we have a sub-planckian suppression scale M P /λ , which in particularmodels can be identified e.g. with the string scale or thecompactification scale. Secondly, the (non-canonical)field φ experiences subplanckian excursions which maybe prerequisite in some more complete models thatincorporate this effective theory. Moreover, in thepost-inflationary phase when H ≤ ˜ M the energy densityof the inflaton redshifts faster than radiation withoutspoiling the hot Big-Bang scenario (thermal phase, BBNfor λ &
20 [43, 44]). Here, we have assumed that asufficient reheating has taken place. On the other hand,if λ < M , introduces new non-renormalizable inter-actions. However, in a single field inflation with kineticcoupling the strong coupling scale of the inflaton-gravitonsystem can be identified with the Planck scale [22, 23].This fact implies that the standard calculations for theinflationary phase can be safely carried out and the re-sults presented in this section are reliable. IV. EFFECTIVE SUPERGRAVITYDESCRIPTION
In this part, we want to embed our results into super-gravity. Exponential potentials are of particular interestin supergravity since they are connected to the super-string dilaton. In fact previous work has pointed outsome interesting aspects of the model. First, the non-minimal derivative coupling has only been constructedin the framework of the new-minimal supergravity [45–48], a supergravity which is believed to be related to theheterotic string (for a discussion see for example [49]). Inparticular, the specific coupling can be found among thecouplings of the dilaton in the effective heterotic stringaction [50]. Moreover, as has been pointed out by earlierwork, it is not possible to explicitly introduce a superpo-tential for this field in a new-minimal supergravity frame-work, since it is forbidden by the R-symmetry [51, 52].Thus, as has been shown recently, the only consistentself-coupling of our superfield is via the gauge-kinematicfunction, and eventually this generates a potential [53].Interestingly enough, this is how the dilaton superfield isexpected to couple to the gauge superfields.It is then evident that the model studied in the previ-ous sections, which fits well the observational data, de-serves a thorough investigation in the framework of new-minimal supergravity. Inflationary models in supergrav-ity with chiral superfields and higher derivative couplingscan be also found in [54, 55]. The new-minimal supergravity [45] is the supersym-metric theory of the gravitational multiplet e am , ψ αm , A m , B mn . (29)The first two fields are the vierbein and its superpartnerthe gravitino, a spin- Rarita-Schwinger field. The lasttwo fields are auxiliaries. The real auxiliary vector A m gauges the U (1) R chiral symmetry. The auxiliary B mn is a real two-form appearing only through its dual fieldstrength H m , which satisfiesˆ D a H a = 0 (30)for the supercovariant derivative ˆ D a .A special feature of new-minimal supergravity is thatis contains a gauged R-symmetry, U (1) R . This R-symmetry places restrictions on the supersymmetric the-ories one can write down. To illustrate this let us takea short detour through the global theory. The globalR-symmetry acts on the theta parameters asR θ α = e − iφ θ α → R dθ α = e − iφ dθ α (31)and on a chiral superfield asR Φ( x, θ ) = e inφ Φ( x, e − iφ θ ) (32)where n is the R-charge. A renormalizable supersymmet-ric theory reads L = ˆ d θ Φ ¯Φ + ˆ d θW (Φ) + c.c. (33)for W (Φ) = f Φ + m Φ + λ Φ . (34)It is then easy to realize from (31) and (32) that in atheory that respects the R-symmetry, only one of theterms in the superpotential (34) would be allowed. Forexample W (Φ) = m Φ → n = 1 . (35)In general the superpotential carries R-charge n = 2. Innew-minimal supergravity this R-symmetry becomes lo-cal and is gauged by the auxiliary field A m , which trans-forms as δ R A m = ∂ m γ. (36)We are now ready to incorporate the aforementionedmodels of inflation into supergravity. The inflaton fieldwill be accommodated into a chiral superfield Φ. Ashas been found in earlier work [51], in order to con-sistently couple the chiral superfield to curvature via anon-minimal derivative coupling, it has to carry a van-ishing R-charge, but from the previous discussion thistells us that a superpotential for this superfield is forbid-den, since a superpotential has to bear R-charge n = 2.On the other hand, if there exist other chiral superfieldswith a non-vanishing R-charge coupled to supergravity,this will again lead to ghost instability due to the factthat the auxiliary field A m will seize to be a Lagrangemultiplier (giving the equations of motion for H a ) andwill have quadratic terms. Thus one may not introduceany R-charged chiral superfield in this model, and thusno superpotential. It has been found that there exists anindirect way to introduce a potential for our superfield viaa gauge kinematic function, thus extending D-term infla-tion [56, 57] to higher derivative D-term inflation [53].In order for our Lagrangian to be manifestly supersym-metric, we derive it in a superspace framework [48]. Ourmodel in superspace is L = − M P ˆ d θEV R + ˆ d θ E K ( ¯Φ , Φ)+ iM − ∗ ˆ d θ E [ ¯Φ E a ∇ a Φ] + c.c. + 14 ˆ d θ E f (Φ) W ( V ) + c.c. + 2 ξ ˆ d θEV , (37)where V R is the gauge superfield of the R-symmetry andit also carries the Ricci scalar on its highest component, V is a U (1) gauge superfield and W α ( V ) = −
14 ¯ ∇ ∇ α V (38)is the standard field strength chiral superfield in new-minimal supergravity.The K¨ahler potential K (Φ , ¯Φ) in (37) will be consid-ered to be canonical, nevertheless other forms are pos-sible. In fact during inflation the form of the K¨ahlerpotential is not relevant. We write z and F mn respec-tively: z = Φ | = φ + iβ (39)the dynamical scalar component of the chiral superfield Φand, F mn the field strength of the dynamical U (1) gaugefield C m = 14 ¯ σ ˙ ααm [ ∇ α , ¯ ∇ ˙ α ] V | . (40)Then the component form of the superspace Lagrangian(37), after integrating out the auxiliary sector reads [53] e − L = M P R + z (cid:3) ¯ z + ˜ M − G ab ∂ a ¯ z ∂ b z − ξ Re f ( z ) −
14 Re f ( z ) F mn F mn + 14 Im f ( z ) F mn ∗ F mn , (41)where ξ is the Fayet-Iliopoulos (FI) parameter of massdimension two. Note that this Lagrangian (41) does not contain ghost states or instabilities. The theory we have discussed in the previous sectionsis reproduced by setting f ( z ) = ξ V e λz/M P (42)then we directly get an exponential type potential V = 12 ξ Re f ( z ) = V λβ/M P ) e − λφ/M P . (43)The form of the function 1 / cos x implies that the β -dependent part of the potential will be stabilized to val-ues h / cos( λβ/M P ) i = 1 and the φ will be the inflatingfield. The mass of the imaginary field . It is important the ef-fective theory to be a single field one. Otherwise newenergy scales in the intermediate appear [22]. We wantthe imaginary field to be integrated out. Its mass in the β -direction minimum is m β ≡ ∂ V∂β (cid:12)(cid:12)(cid:12)(cid:12) β =0 = V λ M P e − λφ/M P ≃ λ H . (44)Therefore if λ ≫ β field is decou-pled and the effective theory is a single field one. In a sin-gle field theory with non-minimal derivative coupling thestrong coupling scale of the graviton-inflaton system isdetermined by the graviton only interactions and can beidentified approximately with the Planck mass [22, 23].Hence, the single field results of the section III remainvalid here as well where the inflaton is the real part ofthe complex scalar z = φ + iβ .In addition we can assume that the C m field where, F mn = ∂ m C n − ∂ n C m , is coupled to charged matter in U (1) gauge invariant manner. Then, rewriting the La-grangian in terms of the cannonically normalized fieldthe physical coupling will scale as ∝ / Ref(z). Assumingthat at the end of inflation the gauge coupling is less thanone then there is no any strong coupling problem duringinflation according to the form of the potential (43).Let us finally note that the FI term in (37) is not theonly way to introduce a potential. In fact since the su-perfield Φ appears in the gauge kinematic function, onemay instead of the U (1) sector of (37), introduce a hiddenstrongly coupled sector L sc = 14 ˆ d θ E g e − ˜ λ Φ /M P W ( V ′ ) + c.c. (45)This sector, if we allow for a non-vanishing scalar con-densate of the gauge fermions [58–60] h λ ′ α λ ′ α i = Λ , (46)will lead to a non-perturbative superpotential W NP = Λ g e − ˜ λ Φ /M P . (47)The (47) will introduce a potential of the form V = Λ ˜ λ g M P e − ˜ λ ( z +¯ z ) /M P . (48)This potential may have interesting cosmological im-plications. Here we mention that such kind of non-perturbative effects may be used for stabilizing the infla-ton runaway direction [61]. However, as we emphasizedin the text, the stabilization may not be necessary and itis disconnected to the end of inflation. V. COMPARISON TO THE QUADRATICCHAOTIC MODELSA. Similarities
The predictions of the inflationary model characterizedby an exponential potential for an inflaton with a kineticcoupling are astonishingly similar to the predictions ofthe quadratic chaotic potential V ϕ ( φ ) = m ϕ / ϕ minimally coupled to gravity. The spectral index, thetensor-to-scalar ratio and the running of the spectral in-dex read in the case of a ϕ potential1 − n s,ϕ ≃ N ∗ + 1 ,r ∗ ,ϕ ≃ N ∗ + 1 ,dn s,ϕ d ln k ≃ − N ∗ + 1) (49)which are exactly the same with those of the section III.This coincidence of the predictions can be also seen fromthe form of the ǫ ϕ and η ϕ slow-roll parameters ǫ ϕ ≡ M P V ′ ϕ V ϕ ! = 2 M P ϕ = m M P V ϕ , (50) η ϕ ≡ M P V ′′ ϕ V ϕ = ǫ ϕ . (51)The (50) and (51) are of the same form with the ǫ and η slow-roll parameters for exponential potentials for aninflaton with kinetic coupling. Indeed it is ǫ ≃ λ ˜ M M P V ( φ ) ←→ ǫ ϕ (52)for λ ˜ M = 2 m . (53)The number of e-folds N ∗ have also a similar dependence.Hence, the relations (13), between the r and n s of thesection III, hold for the quadratic potential as well. Weadd that both theories do not generate important non-gaussianities. These facts imply that there is an under-lying relation between these two models that we are in-vestigating in the following. B. Correspondence between the dynamics of aninflaton with kinetic coupling and a minimallycoupled inflaton
When the inflaton field φ has non-minimal derivativecoupling to the Einstein tensor during slow-roll regimeand in the high friction limit H ≫ ˜ M its dynamics aregoverned by the system of the equations (11) and (10): H ≃ V ( φ )3 M P , H ˙ φ ≃ − ǫǫ V V ′ ( φ ) . (54)According to the previous subsection there is a clear hintof a correspondence between the non-minimally coupledinflaton with exponential potential and the minimallycoupled inflaton with quadratic potential. There shouldbe a generic transformation of the form ϕ = g ( φ ) , V m ( ϕ ) = V [ g − ( ϕ )] (55)such that the above system of equations (54) is recastinto H ≃ V m ( ϕ )3 M P , H ˙ ϕ ≃ − V ′ m ( ϕ ) , (56)where V m ( ϕ ) a potential for the field ϕ minimally coupledto gravity. After straightforward calculations the EOMof (56) is written in terms of the φ field as3 H ˙ φ ≃ − V ′ ( φ )[ g ′ ( φ )] , (57)where the prime denotes derivative with respect to theargument field. This equation is equivalent to the EOMof the system (54) if [ g ′ ( φ )] = ǫ V /ǫ or g ′ ( φ ) = V / M P ˜ M . (58)Therefore the new field ϕ reads in terms of the field φϕ = ˆ V / M P ˜ M dφ . (59)This formula can be also found in [62]. Substituting V = V e − λφ/M P we take ϕ = − V / λ ˜ M e − λφ/ M P (60)and, as well, the inverse function g − ( ϕ ) = φ = − (2 M P /λ ) × ln( − λ ˜ M ϕ/ V / ). It follows that the po-tential V m for the minimally coupled ϕ field reads V m ( ϕ ) = V [ g − ( ϕ )] = 12 λ ˜ M ϕ . (61)Hence, the exponential potential is mapped to thequadratic potential: V m ( ϕ ) ≡ V ϕ ( ϕ ) = 12 m ϕ , (62)where m = λ ˜ M / ϕ field governed by thestandard slow-roll regime system of equations: H ≃ V ϕ ( ϕ )3 M P , H ˙ ϕ ≃ − V ′ ϕ ( ϕ ) . (63)We also see that, e.g., for 50 e-fold of inflation for theminimally coupled field ϕ [5] one also takes 50 e-folds ofinflation for the non-minmally coupled field φ , see for-muli (28) and (60). The excursion for the field ϕ is su-perplanckian.Summing up, the system of equation (54) for an in-flaton with exponential potential and kinetic couplingis recast into the system (63) after the transformation(59). The system (63) describes the slow-roll inflationdriven by the minimally coupled field ϕ characterized by”chaotic-type” quadratic potential. Therefore, the pre-dictions of these two theories naturally coincide and anydifferences are expected to be at the level of the slow-rollparameters ǫ and η . C. Differences and potential observationalsignatures
The essential point is the search for observational sig-natures that break the degeneracy between the two mod-els. The period between the end of inflation and theinitiation of the radiation dominated era plays here acrucial role since each model predicts a different cosmicpost-inflationary evolution. The exponential inflationarypotential fits well the data only if it is too steep for con-ventional inflation to occur. It is the non-minimal deriva-tive coupling acting in the very high energies that enablesthe implementation of the slow-roll regime. At the end ofinflation the Hubble parameter has the value H e ∼ λ ˜ M .Given that λ > √ w = − / w = 1 due to the steepness ofthe potential and the absence of a minimum. This is thedecisive difference between the quadratic chaotic modelsand the one presented here. Entropy production.
In order the standard hot BigBang scenario to be realized some of the inflaton energyneeds to be converted to radiation. This may happen viathree different ways. The first is the gravitational particleproduction [63] where the entropy is produced due to thevariation of the scale factor a ( t ) with time, i.e. due to thetime varying gravitational field. The density of particlesproduced at the end of inflation is found to be ρ R ∼ . g p H e (64)where g p is the number of different particle species cre-ated from the vacuum. The common expectation is that g p ∼ − H e reads in terms of observable quanti-ties H e (2 N ∗ + 1) / ≃ H ∗ ≃ − r / ∗ M P where the Eq.(21) was used. Hence the radiation is a small fractionof the total energy, ρ R /ρ φ ∼ − g p r ∗ / N ∗ . Once the kinetic regime ( w = 1) commences the radiation energydensity will increase relatively to the dominant φ fieldenergy density as ρ R ρ φ ∝ a ( t ) . (65)The radiation domination era will take over the stiff mat-ter era before the BBN, however, there is a stronger con-straint coming from the backreaction of gravity waves, aswe mention in the following.A second way that entropy production can be accom-plished is via instant preheating [64]. This requires acoupling of the form δ L = h φ χ (66)and further Yukawa couplings of the χ particles to othermatter fields. After the production of the χ particlestheir effective mass hφ grows together the φ field value.The energy density of the χ particles and of the prod-ucts of their decay can soon become important. Hence,this coupling may result in a much faster and sufficientreheating of the universe.Finally, a third way to implement the post-inflationarythermal phase is to proceed to a completion of the theoryvia terms that can generate a minimum to the potential.When φ ∼ φ min the φ field oscillates about the minimumand eventually decays (assuming no overshooting). Astudy of the decay process when non-minimal derivativecouplings are present has been performed in [65]. Never-theless, the position of the minimum can be at field values φ min ≫ φ e and the above discussion remains relevant forthis case too. Gravitational waves.
The existence of a post-inflationary phase stiffer than radiation influences therelative amplitude and the tilt of the stochastic gravi-tational waves [66]. The energy density of relic gravitywaves scales like ρ g ∝ a − when the background energydensity is characterized by w > / Gauge fields.
The inflaton interaction with the gaugefields, see the Lagrangian (41), is a particular featureof the supergravity inflationary models with kinetic cou-pling. The time dependence of the gauge kinetic functionbreaks the conformal invariance of the gauge field sectorand leads to amplification of the quantum fluctuationsof the gauge bosons during the nearly de-Sitter phase.Furthermore, the coupling of the inflaton to a gauge fieldalso contributes to the reheating of the universe [68].In summary, the exponential potential for an inflatonwith kinetic coupling predicts the same values for the cos-mological perturbation parameters. However, the post-inflationary phase may be much different and modify thepredictions. The end of inflation is followed by a stiffmatter kinetic regime and, hence, the radiation domina-tion phase may start much later. This fact alters theexpected number of e-folds which can be larger [69]. Thekinetic regime if long enough may affect the tilt of thegravitational waves giving more power to small scales.These effects are much milder if a preheating takes placeor if there is minimum in the low energy effective poten-tial. Finally, the supergravity realization of this infla-tionary scenario may lead to interesting gauge field gen-eration effects. These issues are important and deservea separate study in order to enable a definite observa-tional distinction between the quadratic chaotic inflationand the exponential potential for an inflaton with kineticcoupling.
VI. CORRESPONDENCE BETWEENPOTENTIALS FOR INFLATON WITH KINETICAND MINIMAL COUPLING
In the finale, let us depart from the exponential poten-tial example and comment on the general case of mono-mial potentials V ( φ ) = m − n +4 φ n . (67)According to the transformation (59), ϕ = ´ g ′ ( φ ) dφ , thefield ϕ reads ϕ = 2 n + 2 φ n/ m n/ − M P ˜ M , (68)and it appears to be minimally coupled to gravity duringthe inflationary phase. Its evolution is governed by thepotential V m ( ϕ ) = V [ g − ( ϕ )] where V m ( ϕ ) = m − n +4 (cid:18) n + 22 m n/ − M P ˜ M ϕ (cid:19) n/ ( n +2) . (69)During slow-roll there is direct correspondence betweenthe potential V ( φ ) for the kinetically coupled inflatonand the V m ( ϕ ) for the minimally coupled inflaton: V ∝ φ n ←→ V m ∝ ϕ nn +2 . (70)Let us look into specific examples, starting from the quar-tic Higgs-like potential. We find that V ( φ ) = λ q φ ←→ V m ( ϕ ) = λ / q (3 M P ˜ M ) / ϕ / , (71) i.e. the quartic potential for an inflaton with kinetic cou-pling [20] is equivalent to ϕ / monomial potential for aninflaton with minimal coupling [34]. Also, the quadraticpotential V ∝ φ with kinetic coupling corresponds toa linear potential V m ∝ ϕ , and, the linear V ∝ φ withkinetic coupling to the V m ∝ ϕ / ; see also Ref. [70]for relevant monomial potentials in stringy and [34] insupergravity set ups.We comment that for the case n = − V ( φ ) = m φ − for positive field values is in-stead depicted to an exponential potential: V ( φ ) = m φ ←→ V m ( ϕ ) = V m (0) e − MP ˜ Mm ϕ . (72)We also observe that for any positive power n of themonomial potential V ∝ φ n the corresponding minimalcase potential cannot have a power greater than two.In other words the quadratic potential, V m ∝ ϕ is thesteepest monomial potential that the V ∝ φ n , n > n ≫
1. The exponential potential can be seen asthe limiting case. Indeed, we find that V ( φ ) ∝ e − φ ←→ V m ( ϕ ) ∝ ϕ (73)which has been the potential investigated in the previoussections. VII. CONCLUSIONS
In this Letter we discussed the exponential potentialsas candidates for describing the early universe inflation-ary phase. We found that when the inflaton scalar field iskinetically coupled to the Einstein tensor the exponentialpotential predictions fit very well the data. The modelpredicts tensor-to-scalr ratio r = 0 .
16 for n s = 0 .
96 and r ∗ & .
135 for N ∗ .
60 e-folds. We found no significantrunning of the spectral index. These results coincide withthe predictions of the quadratic V ∝ ϕ potential. Theunderlying reason for this coincidence is that there existsa simple mapping of the one model to the other. Fur-thermore, inflation with exponential potential naturallyterminates even in the absence of a minimum. This maylead to specific observational signatures that distinguishthis model from the quadratic inflationary potential. Wealso found that this model can be successfully describedin a supergravity framework. This allows microscopictheories to accommodate and theoretically motivate thisinflationary candidate.We would like to thank C. Germani and E. N. Saridakisfor discussion and A. Kehagias for insightful comments.This work is supported by the Grant agency of the Czechrepublic under the grant P201/12/G028.0 [1] P. A. R. Ade et al. 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