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Sound Speed of Primordial Fluctuations in Supergravity Inflation Alexander Hetz and Gonzalo A. Palma Grupo de Cosmolog´ıa y Astrof´ısica Teórica, Departamento de F´ısica, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile. (Dated: October 11, 2016) We study the realization of slow-roll inflation in N = 1 supergravities where inflation is the result of the evolution of a single chiral field. When there is only one flat direction in field space, it is possible to derive a single-field effective field theory parametrized by the sound speed c at s which curvature perturbations propagate during inflation. The value of c is determined by the s rate of bend of the inflationary path resulting from the shape of the F-term potential. We show that c must respect an inequality that involves the curvature tensor of the Ka¨hler manifold un- s derlying supergravity, and the ratio M/H between the mass M of fluctuations ortogonal to the inflationary path, and the Hubble expansion rate H. This inequality provides a powerful link between observational constraints on primordial non-Gaussianity and information about the N = 1 6 supergravity responsible for inflation. In particular, the inequality does not allow for suppressed 1 0 values of cs (values smaller than cs ∼ 0.4) unless (a) the ratio M/H is of order 1 or smaller, and 2 (b) the fluctuations of mass M affect the propagation of curvature perturbations by inducing on them a nonlinear dispersion relation during horizon crossing. Therefore, if large non-Gaussianity is t c observed, supergravity models of inflation would be severely constrained. O 1 It is unlikely that observations will ever allow us to single field slow-roll inflation would be the only single 1 pindownthefundamentaltheoryunderlyinginflation[1– field theory such that c =1. s 5]. Instead, we might have to conform ourselves with Current constraints on primordial non-Gaussianity ] h gaining general insights into the origin of cosmological give c ≥ 0.024 (95% C.L.) after marginalizing over s t perturbations with the help of the effective field the- c˜ [14]. Future measurements will further constrain the - 3 p ory (EFT) approach to inflation [6] (see also Ref. [7]). coefficients of the EFT Lagrangian, narrowing down the e This consists of a powerful model-independent frame- class of fundamental theories that can host inflation to- h work to study the evolution of curvature perturbations getherwiththestandardmodelofparticlephysics. Thus, [ R during inflation, valid at energies around the Hub- it is important to understand how fundamental theories 3 ble expansion rate H. With the help of symmetry ar- map into the EFT expansion, especially in the case of v guments, it is possible to derive the most general class theories such as supergravity and string theory, charac- 7 5 of Lagrangians governing the dynamics of fluctuations terized for having a large number of scalar fields repre- 4 in a Friedman-Robertson-Walker (FRW) background, in senting moduli that must remain massive during infla- 5 termsofaGoldstonebosonfieldπ =−R/H,givenbyan tion. 0 operator expansion with nonlinearly related coefficients Giventhatsupergravitieshaveseveralscalarfields,itis . 1 parametrizing physics beyond the canonical single-field conceivablethattheinflationarypathexperiencedbends 0 paradigm. These coefficients, among which the sound in field space. In fact, it is well understood that, regard- 6 speedc playsaprominentrole,aredeterminedbytheul- less of how massive fields normal to the path are, these 1 s traviolet (UV) complete theory underlying inflation. To bendsaffectthedynamicsofπbyinducingc <1[15,16]. : s v quadratic order, the Lagrangian is given by [6] For instance, in two-field models one has [12, 16, 17] i X r L(π2) =MP2l|H˙|(cid:2)c−s2π˙2−a−2(∇π)2(cid:3), (1) cs =(cid:0)1+4Ω2/M2(cid:1)−1/2, (4) a where MPl is the reduced Planck mass and a is the scale whereΩisthelocalangularvelocitydescribingthebend, factor. cs also defines the strength of cubic operators andM isthemassoffieldsorthogonaltotheinflationary that may lead to primordial non-Gaussianity [8]. For path. InthisLetterwewilldeduceaninequalitythatany instance, if cs < 1 (with |c˙s| (cid:28) Hcs), then equilateral single field EFT derived from N = 1 supergravity must and orthogonal non-Gaussianity are produced, with fNL satisfy. It involves cs, H, and M, and it is given by parameters given by [9] (cid:0)c−2−1(cid:1)M2/H2 ≤O(1), (5) s fequil =−(c−2−1)(0.275+0.078c2+0.52c˜ ), (2) NL s s 3 fortho =(c−2−1)(0.0159−0.0167c2+0.11c˜ ), (3) where O(1) represents an order 1 number determined by NL s s 3 the geometry of the Kähler manifold underlying the su- wherec˜ isaparameterthatarisesatthirdorder[9]and pergravity of interest. As we shall see, to ensure that 3 it is determined by the UV-complete theory [10]. It has thehorizonexitisproperlyaccountedforbyEq.(1),one been conjectured [13] that every parameter of the EFT requires that (1−c2)2H2/M2c2 (cid:28) 1. If not, the dis- s s expansion, such as c˜ , is proportional to (1−c2). If so, persion relation describing the propagation of π during 3 s 2 horizoncrossingbecomesnonlinear,duetonewoperators in Eq. (1) with spatial gradients induced by the massive field normal to the path. This, together with Eq. (5) implies that cs cannot be much smaller than 1 (and su- V pergravity models cannot produce large primordial non- Gaussianity) unless the horizon crossing is described by an EFT with a nonlinear dispersion relation [18, 19]. In thiscase,thedynamicsoffluctuationswoulddepartcon- �2 siderably from the standard single-field EFT description of Eq. (1) used to connect models with observables. Before deducing the inequality, let us review some ba- sics about multifield inflation. We consider a general ac- (cid:82)√ tion S = −gL describing an arbitrary number of ~ tot ~ T real scalar fields φa(t,x), a = 1,...,N, minimally cou- N 1 � pled to gravity, with a Lagrangian M2 1 L= PlR+ γ gµν∂ φa∂ φb−V(φ), (6) 2 2 ab µ ν FIG. 1. An example of a path forced to have bends as it where R is the Ricci scalar constructed from the metric meanders down a two-field potential V. g , γ is a σ-model metric that depends on the fields, µν ab and V(φ) is the scalar potential. From now on, we work in units where MPl =1. In the case of N =1 supergrav- Thisequationrevealsthatwheneverthepathissubjected ity, γab and V are uniquely determined by the function to a bend, it is pushed towards the outer wall of the po- G = K +ln|W|2, where K is the Kähler potential, and tential. Ω plays a crucial role in the dynamics of fluctu- W thesuperpotential. Wewillcomebacktosupergravity ations, as it couples together curvature and isocurvature in a moment. To describe the background, we consider modes [21]. To be consistent with the view that there anFRWspacetimewithametricds2 =−dt2+a2(t)dx2, mustbeashiftsymmetryensuringaflatdirectioninthe where a(t) is the scale factor and x are comoving coor- potential, in this work, we assume that δ ≡ Ω˙/HΩ is Ω dinates. The expansion rate is H = a˙/a and it is given suppressed(steadybends),implyingthatc variesslowly: s by the Friedman equation 3H2 = φ˙20/2+V(φ), where |c˙s|(cid:28)Hcs. Onemayconsidersharpbends,inwhichcase φ˙20 ≡γabφ˙a0φ˙b0. The equations of motion for the homoge- features are produced in the spectra [16], and Eqs. (2) neous background fields φa0(t) derived from Eq. (6) are and (3) are not sufficient to constrain cs. D φ˙a+3Hφ˙a+γabV =0, (7) In order to characterize the evolution of H during in- t 0 0 b flation it is useful to define the slow-roll parameters: where V =∂ V =∂V/∂φb. Furthermore, D is a covari- b b t ant time derivative whose action on a given vector Aa (cid:15)≡−H˙/H2 =φ˙2/2H2, η ≡(cid:15)˙/H(cid:15). (10) 0 is such that D Aa ≡ A˙a +Γa Abφ˙c, where Γa are the t bc 0 bc Slow-roll inflation corresponds to the regime where the Christoffel symbols derived from γ . The inflationary ab perturbations are defined as δφa(t,x)=φa(t,x)−φa(t). expansion rate H evolves very slowly for a long enough 0 To study them, it is useful to define vectors tangent and time, requiring that (cid:15) (cid:28) 1 and |η| (cid:28) 1. In this regime, normal to the path φa(t), respectively, given by onefinds3H2 (cid:39)V,and(cid:15)(cid:39)Vφ2/2V2. Ontheotherhand, 0 it is also customary to introduce a matrix proportional Ta ≡φ˙a0/φ˙0, Na ≡−DtTa/|DtT|. (8) to the Hessian of the potential V: Figure 1 illustrates an example where inflation is driven N ≡∇ V /V. (11) byatwo-fieldpotential. Fluctuationsalongthedirection ab a b TagiveuscurvatureperturbationsasR=−HTaδφa/φ˙0, This matrix contains information about the shape of the whereas those along Na correspond to isocurvature per- potential,anditissometimesusedtodefineslow-rollpa- turbations [20, 21]. Ta and Na allow us to define the rameters alternative to those of Eq. (10). These are [22] angular velocity Ω via D Ta ≡ −ΩNa [then Eq. (8) t tells us that Ω (cid:62) 0]. From Eq. (7) it follows that 1V Va (cid:15) ≡ a , η ≡min. eigenvalue{N}. (12) Va = VφTa + VNNa, where Vφ = TaVa and VN = V 2 V2 V NaV . As a result, projecting Eq. (7) along Ta one finds a However, there is a problem with these definitions: Both φ¨ +3Hφ˙ +V =0,whichresemblestheequationofmo- 0 0 φ (cid:15) and η may attain large values even with (cid:15),|η| (cid:28) 1. tionofasinglescalarfield. Ontheotherhand,projecting V V First, (cid:15) is not necessarily small because Ta and the Eq. (7) along Na one obtains, V direction of steepest descent of the potential (∝ V ) do a Ω=V /φ˙ . (9) not coincide in a bend. To see this, we may combine N 0 3 (cid:15) (cid:39) V2/2V2 with Eqs. (9) and (10) to find that (cid:15) is wherefiisanarbitraryunitcomplexvector. Itturnsout φ V (cid:112) given by that if fi = Gi/ GjG , then the right-hand side (rhs) j ofEq.(18)becomesindependentofsecondderivativesof (cid:15) =(cid:15)(cid:0)1+Ω2/9H2(cid:1). (13) V the superpotential W [28, 29], and reduces to [27] This shows that (cid:15)V may attain values of order 1 without ∇ V 2 1+γ (cid:20)2 (cid:21) necessarily violating slow roll. Second, η is not neces- i ¯fif¯≤− + −R(f) V V 3 γ 3 sarilysmallbecauseTa andtheeigenvectorassociatedto √ (cid:15) 4 (cid:15) ηV (the smallest eigenvalue of Nab) do not coincide in a + V + √ √ V , (19) bend(thatis,η (cid:54)=N TaTbwhenΩ(cid:54)=0). Toshowthis, 1+γ 3 1+γ V ab letuscomputeη inthecaseoftwo-fieldmodels. Inthis V whereγ ≡H2/m2 (m =eG/2 isthegravitinomass), case,wemaydefineaunitvectorha ≡cosαTa+sinαNa 3/2 3/2 and minimize the contraction hahb∇aVb/V by tuning α, and R(f) ≡ Ri¯k¯lfif¯fkf¯l is the Riemann tensor com- to obtain back ηV. To perform the contraction we need puted out of Ki¯projected along the complex vector fi. to be aware of the following identities [16]: We now have all the necessary elements to deduce the desired inequality: If the supergravity has only one chi- TaTb∇aVb =3H2(2(cid:15)−η/2)+Ω2, (14) ral field Φ, then the scalar field target space is spanned NaTb∇aVb =ΩH(3+δΩ+η−2(cid:15)), (15) by two real scalar fields, implying that ηV in the left- NaNb∇ V =M2+Ω2−(cid:15)H2R, (16) hand side (lhs) of Eq. (18) is precisely given by Eq. (17). a b Then, putting together Eqs. (17) and (18) we deduce where δΩ = Ω˙/HΩ, and R = RabcdTaNbTcNd is the the desired inequality. However, in order to explicitly Riemann tensor of the scalar field space projected along write down a simple version of it, it is useful to antici- Ta and Nb. M is the effective mass of the perturbation pate three relevant regimes: First, if there are no bends, along the direction Na, and it is the same quantity ap- η =(4(cid:15)−η)/2, and one recovers the bound deduced in V pearinginEq.(4). Giventhatweareassumingslowroll, Ref.[27](seealsoRef.[30]),informinguswhetheragiven we disregard δΩ +η−2(cid:15) in Eq. (15) before minimizing supergravity can produce inflation without bends. Sec- hahb∇aVb/V. In addition, we may assume that (cid:15)H2R is ond,if(c−2−1)M2/H2isoforderO((cid:15),η),thenΩ2 (cid:28)H2, s small compared to M2 +Ω2, which is generally true in and c is close to 1. Then, η is of order O((cid:15),η) and the s V supergravity models of inflation [23–26]. Then, we find inequality emerging from Eq. (18) does not give us in- (cid:32) teresting information beyond that of Ref. [27]. Third, if 1 M2 M2 η = (c−2−1) +2 +3[4(cid:15)−η] (c−2−1)M2/H2 (cid:29)|O((cid:15),η)|wemayneglecttheslow-roll V 12 s H2 H2 s parametersappearinginEqs.(17)and(19)toobtainour (cid:115)(cid:18)M2 3 (cid:19)2 M2(cid:33) main result: −2 H2 − 2[4(cid:15)−η] +9(c−s2−1)H2 , (17) 1M2 (cid:32)c−2−1 (cid:114) H2(cid:33) s +1− 1+9(c−2−1) s 6 H2 2 M2 where we used Eq. (4) to introduce c . Now, the crucial s (cid:20) (cid:21) point to appreciate here is that it is possible to have 1+γ 2 2 ≤ −R(f) − . (20) Ω (cid:29) M (and therefore c2 (cid:28) 1) without violating the γ 3 3 s conditions(cid:15),|η|(cid:28)1. Ifthisisthecase,themisalignment This inequality puts together the sound speed c , the islarge, andη acquiressizablevalues. Onthecontrary, s V ratio M/H, and additional information on the curvature if Ω=0, one recovers a suppressed value given by η = V of the Kähler manifold. (4(cid:15)−η)/2. Hence,(cid:15) andη arenotlegitimateslow-roll V V Let us now analyze the content of Eq. (20), which as- parameters unless the multifield path remains straight sumes that (cid:15),|η| (cid:28) 1 [31]. To start with, notice that (Ω=0). the smallest attainable value of the lhs of (20) is −3/4. Let us now deduce the inequality (5), valid for the This means that when the inflationary path is subjected scalar sector of N = 1 supergravity. Before consider- to strong bends, it is possible to have M2 > 0 simul- ing the case of a single chiral field, let us consider the taneously with a large and negative value of η . Next, general situation. Here, the target space corresponds to V the rhs of eq. (20) may acquire any desired value, de- a Kähler manifold spanned by complex scalar fields Φi. pending on the specific Kähler potential under consid- The F-tern potential is given by V = eG(Ki¯G G −3), i ¯ eration. Nevertheless, we expect this value to be of or- where G = ∂ G and K = ∂ ∂ G is the Kähler met- i i i¯ i ¯ der 1. First, the natural value of the curvature term is ric (derivatives are taken with respect to Φi and their R(f) (cid:39) O(1). For instance, canonical models are char- complex conjugate Φ¯ı). One may now define a Hessian acterizedbyR(f)=0whereassingle-chiral-fieldno-scale matrix ∇ V . Then, because η is the minimum eigen- i ¯ V models satisfy R(f) = 2/3 [32]. Second, there are two value of N , one necessarily has [27] ab classes of models of supergravity inflation depending on η ≤fif¯∇ V /V, (18) the value of γ = H2/m2 . On the one hand, we expect V i ¯ 3/2 4 the value of H during inflation to be of order 1014GeV �� or smaller, but not too much smaller (in order to have a �� tensor to scalar ratio r (cid:46) 0.1). On the other hand, we �� -�� expect the value of m at the end of inflation to be of � 3/2 order1TeVorlarger,butnottoomuchlarger(inorderto �� have a soft SUSY breaking scale close to the electroweak � �� -�� symmetry breaking scale). Hence, in models where m 3/2 �� did not evolve significantly during inflation, one expects �� γ (cid:29) 1. To be conservative, in the present Letter we as- �� sume γ (cid:38)1, implying that the rhs of Eq. (20) is of order 1. Therearestringinspiredmodels,suchaslargevolume �� scenarios [33], where γ (cid:28) 1. However, in these models � � � � � �� one finds 2/3−R(f)∼γ, implying that the rhs eq. (20) �/� continues to be of order 1 [34]. It is important to consider in our analysis the range of FIG. 2. The parameter space M/H vs c . The inequality is s parametersforwhichthetwo-fieldsystemadmitsanEFT satisfiedattheregionsboundedbythesolidlines,oppositeto oftheform(1). Thiswilldependonthevaluesofthera- thelabelsindicatingbenchmarkvaluesfortherhsofEq.(20). tios M/H and Ω/M (or equivalently c ). We plot these s rangesinFig.2: First,thedynamicsoffluctuationsisde- scribedbyasingledegreeoffreedomaslongastheGold- the qualitative content of this result will remain valid. stone boson’s energy ω satisfies ω2 (cid:28)M2/c2 [17]. Then, In those cases the rhs of Eq. (20) will not change, and s requiring that horizon crossing ω (cid:39) H happens within we only need to worry about the lhs. There, cs will this regime, we obtain the bound H2 (cid:28) M2/c2. This have a similar dependence on the masses of heavy fields s bounddefinestheregionattherightofthe“MultiField” to that of Eq. (4) [35]. Moreover, the form of η will V areainFig.2,which,forreference,islimitedbythecurve change due to the new matrix entries determined by the H2 = 0.5M2/c2. Within this region there are two addi- masses of the extra fields. However, it turns out that s tionalsubregimes: Ifω2 (cid:28)M2c2s/(1−c2s)2 thedispersion these contributions do not modify ηV importantly if the relation is of the form ω(k) = c k and the theory is de- massesoftheextrafieldsareofthesameorderthanthat s scribed by Eq. (1). For energies ω2 (cid:39) M2c2/(1−c2)2 along Na. In particular, in many models of supergravity s s or larger, the spatial gradients in the kinetic term of there is only one chiral field Φ that evolves during in- the massive field (normal to the path) dominate over flation, while the rest remain stabilized, having the role its mass (k2 (cid:29) M2), inducing new operators in Eq. (1) of breaking SUSY [36]. In those types of models our in- that make the dispersion relation ω(k) forthe Goldstone equality remains exactly valid. This is because here the boson π nonlinear [17–19]. In this case, the evolution only evolving fields are the two real scalar fields entering of π is determined by an EFT that departs significantly Φ,givingusbackthesameexpressionforηV ofEq.(17), from(1),implyingdifferentrelationsbetweenobservables deducedundertheassumptionthattheinflationarypath and the parameters of the theory. Thus, we deduce that was embedded in a two-field manifold. the EFT (1) is valid as long as [17]: In conclusion, we have derived a direct link between observables and N = 1 supergravity inflation. By con- (1−c2)2H2/M2c2 (cid:28)1. (21) s s straining non-Gaussianity, we are able to infer nontrivial The region satisfying this bound corresponds to the area information about the supergravity responsible for in- above the dashed curve satisfying (1−c2s)2H2/M2c2s = flation. If inflation is described by the EFT (1), then cs 0.5, separating the domains “Single Field” and “Nonlin- cannotbesuppressed,andnon-Gaussianitymustbesuch ear dispersion relation” in Fig. 2. In addition, the in- thatfNeqL,orth (cid:46)O(1)(cs (cid:38)0.4). Onthecontrary,ifobser- equality (20) is satisfied at the regions bounded by the vations ever confirm sizable non-Gaussianity feq,orth (cid:38) NL solid lines, opposite to the labels indicating benchmark O(1)thenthesupergravityresponsibleforinflationmust values for the rhs of Eq. (20). We may further simplify havehadnontrivialpropertiespertainingtotheevolution the form of our inequality: Because of Eq. (21) and the of perturbations during horizon crossing: (a) the ratio fact that the rhs of Eq. (20) is of order 1, we may take M/H was of order one or smaller, and (b) the propa- 9(c−2−1)(cid:28)M2/H2 andTaylorexpandthesquareroot gation of curvature perturbations during horizon cross- s in Eq. (20) to recover Eq. (5). As a result, for super- ing was characterized by a nonlinear dispersion relation gravityinflationtobedescribedbyEq.(1), c cannotbe as a result of their interaction with the field normal to s much smaller than unity. Indeed, in Fig. 2 we see that the trajectory. In such a case, we would need to under- c cannot be smaller than c (cid:39) 0.4 when the rhs of the stand the generation of primordial perturbations within s s inequality is 1. a regime that so far has not been studied in the context In the case of supergravities with many chiral fields, of supergravity inflation. Of course, our bound c (cid:38) 0.4 s 5 must be taken with some caution, as the rhs of Eq. (20) 2015 results. XVII. Constraints on primordial non- may acquire large or small values depending on the spe- Gaussianity,” arXiv:1502.01592 [astro-ph.CO]. cific model under consideration. Notwithstanding, our [15] A. J. Tolley and M. Wyman, “The Gelaton Scenario: Equilateral non-Gaussianity from multi-field dynamics,” analysisgivesvaluableinformationabouttheconnection Phys. Rev. D 81, 043502 (2010) [arXiv:0910.1853 [hep- betweensupergravityandobservables,thatcouldbefur- th]]. ther complemented with constraints on c˜ (which also 3 [16] A. Achucarro, J. O. Gong, S. Hardeman, G. A. Palma depends on the Kähler geometry). and S. P. Patil, “Features of heavy physics in the CMB power spectrum,” JCAP 1101, 030 (2011) We are grateful to Ana Achućarro and Daniel Green [arXiv:1010.3693 [hep-ph]]. for useful comments and discussions that helped to [17] A. Achucarro, V. Atal, S. Cespedes, J. O. Gong, improve the first version of this manuscript. This work G. A. Palma and S. P. Patil, “Heavy fields, reduced was supported by the Fondecyt Project No. 1130777 speeds of sound and decoupling during inflation,” Phys. (G.A.P.), the Conicyt Anillo project ACT1122 (G.A.P), Rev. D 86, 121301 (2012) [arXiv:1205.0710 [hep-th]]. [18] D. Baumann and D. Green, “Equilateral Non- and the Conicyt Fellowship CONICYT-PCHA/Mag´ıster Gaussianity and New Physics on the Horizon,” JCAP Nacional/2014-66185 (A.H.). 1109, 014 (2011) [arXiv:1102.5343 [hep-th]]. [19] R. Gwyn, G. A. Palma, M. Sakellariadou and S. Syp- sas,“Effectivefieldtheoryofweaklycoupledinflationary models,”JCAP1304,004(2013)[arXiv:1210.3020[hep- [1] A.H.Guth,“TheInflationaryUniverse: APossibleSolu- th]]. tion to the Horizon and Flatness Problems,” Phys. Rev. [20] C. Gordon, D. Wands, B. A. Bassett and R. Maartens, D 23, 347 (1981). “Adiabatic and entropy perturbations from inflation,” [2] A. D. Linde, “A New Inflationary Universe Scenario: A Phys. Rev. D 63, 023506 (2001) [astro-ph/0009131]. Possible Solution of the Horizon, Flatness, Homogene- [21] S. Groot Nibbelink and B. J. W. van Tent, “Scalar ity,IsotropyandPrimordialMonopoleProblems,”Phys. perturbations during multiple field slow-roll inflation,” Lett. B 108, 389 (1982). Class. Quant. Grav. 19, 613 (2002) [hep-ph/0107272]. [3] A.AlbrechtandP.J.Steinhardt,“CosmologyforGrand [22] C. P. Burgess, J. M. Cline, H. Stoica and F. Quevedo, Unified Theories with Radiatively Induced Symmetry “InflationinrealisticD-branemodels,”JHEP0409,033 Breaking,” Phys. Rev. Lett. 48, 1220 (1982). (2004) [hep-th/0403119]. [4] A.A.Starobinsky,“ANewTypeofIsotropicCosmolog- [23] J. Ellis, D. V. Nanopoulos and K. A. Olive, “No-Scale ical Models Without Singularity,” Phys. Lett. B 91, 99 SupergravityRealizationoftheStarobinskyModelofIn- (1980). flation,”Phys.Rev.Lett.111,111301(2013)[Phys.Rev. [5] V. F. Mukhanov and G. V. Chibisov, “Quantum Fluc- Lett. 111, no. 12, 129902 (2013)] [arXiv:1305.1247 [hep- tuation and Nonsingular Universe. (In Russian),” JETP th]]. Lett.33,532(1981)[PismaZh.Eksp.Teor.Fiz.33,549 [24] J.Ellis,D.V.NanopoulosandK.A.Olive,“Starobinsky- (1981)]. like Inflationary Models as Avatars of No-Scale Super- [6] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan gravity,”JCAP1310,009(2013)[arXiv:1307.3537[hep- and L. Senatore, “The Effective Field Theory of Infla- th]]. tion,”JHEP0803,014(2008)[arXiv:0709.0293[hep-th]]. [25] J. J. M. Carrasco, R. Kallosh and A. Linde, “Cosmo- [7] S.Weinberg,“EffectiveFieldTheoryforInflation,”Phys. logical Attractors and Initial Conditions for Inflation,” Rev. D 77, 123541 (2008) [arXiv:0804.4291 [hep-th]]. Phys.Rev.D92,no.6,063519(2015)[arXiv:1506.00936 [8] X. Chen, M. x. Huang, S. Kachru and G. Shiu, “Ob- [hep-th]]. servational signatures and non-Gaussianities of general [26] J. J. M. Carrasco, R. Kallosh and A. Linde, “α- single field inflation,” JCAP 0701, 002 (2007) [hep- Attractors: Planck, LHC and Dark Energy,” JHEP th/0605045]. 1510, 147 (2015) [arXiv:1506.01708 [hep-th]]. [9] L. Senatore, K. M. Smith and M. Zaldarriaga, “Non- [27] L. Covi, M. Gomez-Reino, C. Gross, J. Louis, GaussianitiesinSingleFieldInflationandtheirOptimal G. A. Palma and C. A. Scrucca, “Constraints on mod- Limits from the WMAP 5-year Data,” JCAP 1001, 028 ular inflation in supergravity and string theory,” JHEP (2010) [arXiv:0905.3746 [astro-ph.CO]]. 0808, 055 (2008) [arXiv:0805.3290 [hep-th]]. [10] For example, in DBI models [11] one has that c˜ = [28] M. Gomez-Reino and C. A. Scrucca, “Locally stable 3 3(1−c2)/2 [9] whereas in single field EFT’s obtained non-supersymmetric Minkowski vacua in supergravity,” s by integrating out a heavy field in canonical multi-field JHEP 0605, 015 (2006) [hep-th/0602246]. models one finds c˜ =3(1−c2)/4 [12]. [29] L. Covi, M. Gomez-Reino, C. Gross, J. Louis, 3 s [11] M. Alishahiha, E. Silverstein and D. Tong, “DBI in the G. A. Palma and C. A. Scrucca, “de Sitter vacua in sky,” Phys. Rev. D 70, 123505 (2004) [hep-th/0404084]. no-scale supergravities and Calabi-Yau string models,” [12] A. Achucarro, J. O. Gong, S. Hardeman, G. A. Palma JHEP 0806, 057 (2008) [arXiv:0804.1073 [hep-th]]. and S. P. Patil, “Effective theories of single field infla- [30] A. Borghese, D. Roest and I. Zavala, “A Geometric tion when heavy fields matter,” JHEP 1205, 066 (2012) bound on F-term inflation,” JHEP 1209, 021 (2012) [arXiv:1201.6342 [hep-th]]. [arXiv:1203.2909 [hep-th]]. [13] D. Baumann, D. Green, H. Lee and R. A. Porto, “Signs [31] One could keep (cid:15), η and δΩ appearing in Eq. (14)-(17) of Analyticity in Single-Field Inflation,” Phys. Rev. D and(19) to obtain a more general version of Eq. (20). 93, no. 2, 023523 (2016) [arXiv:1502.07304 [hep-th]]. However,theconclusionsofouranalysiswouldnotvary. [14] P. A. R. Ade et al. [Planck Collaboration], “Planck [32] E.Cremmer,S.Ferrara,C.KounnasandD.V.Nanopou- 6 los,“NaturallyVanishingCosmologicalConstantinN=1 [34] However, these models require at least two chiral fields. Supergravity,” Phys. Lett. B 133, 61 (1983). [35] S. Céspedes and G. A. Palma, “Cosmic inflation in [33] J. P. Conlon, R. Kallosh, A. D. Linde and F. Quevedo, a landscape of heavy-fields,” JCAP 1310, 051 (2013) “Volume Modulus Inflation and the Gravitino Mass [arXiv:1303.4703 [hep-th]]. Problem,” JCAP 0809, 011 (2008) [arXiv:0806.0809 [36] D.BaumannandD.Green,“SignaturesofSupersymme- [hep-th]]. try from the Early Universe,” Phys. Rev. D 85, 103520 (2012) [arXiv:1109.0292 [hep-th]].