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This content was downloaded from IP address 131.169.4.70 on 13/12/2017 at 23:04 JCAP10(2016)038 hysics P osmology le ark energy after the ic t ar h a single chiral superfield. 10.1088/1475-7516/2016/10/038 ently tunable due to a degen- niversity of Groningen, calar partner of the Goldstino examples that are favoured by doi: strop . A a,d ion to the author(s) [email protected] , . Content from this work may be used 3 osmology and and osmology and Diederik Roest Creative Commons Attribution 3.0 License C a,b,c 1608.03709 inflation, theory and cosmology, and c

We prove that all inflationary models, including those with d Sergio.ferrara@.ch rnal of rnal Article funded by SCOAP under the terms of the

ou An IOP and SISSA journal An IOP and Department, CERN, CH-1211 Geneva 23, Switzerland INFN — Laboratori NazionaliVia di Enrico Frascati, Fermi 40, 00044Department Frascati, of Italy Physics andMani Astronomy L. and Bhaumik Institute470 for Portola Theoretical Plaza, Physics, Los UCLA, Angeles,Van Swinderen CA Institute 90095-1547, for U.S.A. Nijenborgh Particle Physics 4, and 9747 Gravity, AG U Groningen, TheE-mail: Netherlands b c d a ArXiv ePrint: Received September 1, 2016 Accepted October 20, 2016 Published October 24, 2016 Abstract. Sergio Ferrara General sGoldstino inflation J end of inflation,Moreover, can the be amount embedded oferacy supersymmetry in in breaking minimal the is independ choicein wit for this the set-up. superpotential.the We The illustrate Planck inflaton data. our is general a procedure s with two Keywords: and the title of the work, journal citation and DOI. Any further distribution of this work must maintain attribut JCAP10(2016)038 ]. 1 2 3 5 6 8 8 ]. 12 that 8 , η 7 ] leaves 3 ], builds on 1 ameter ]. This paper ]. Despite this 3 2 [ for the flat scalar σ 2 oldstino field arising ¯ Φ) − ¯ Φ, the scalar potential’s ] and most recently [ nck results [ 07 at 2 . (Φ 0 11 ltiplet does not adress the 2 1 e embedding of inflation in ]. The subsequent develop- = Φ – n this manner. Models of dent tuning of the gravitino ts of the scalar potential. 6 fields: e.g. the masses of this 9 − < The construction of [ hler manifold instead [ , K s constructed [ s te measurement of the spectral 5 = ook a decade and a half before plete theory of quantum gravity. direction in field space as SUSY was realized a decade later that itino mass term and the positive een inflation and SUSY breaking. /A nd observationally succesful. The e.g. [ their non-trivial scalar potentials: t K A ials [ = r ]. In addition, constraints have been put 1 006 [ . 0 – 1 – ± 968 . = 0 s ]. Secondly, the construction employs an additional chiral 4 n -problem. With the conventional choice η ) gives order-one contributions to the second-slow roll par K that constitutes the sGoldstino, the scalar partner of the G S -attractors, providing an excellent agreement with the Pla α Supergravity poses strong constraints on the (s)Goldstino The versatility of the constructions with the sGoldstino mu A first step in this direction is provided by supergravity. Th First of all, it uses a shift-symmetric potential K¨ahler question whether itsGoldstino is inflation - necessary with inflationbreaking to taking - place extend have in been the the investigated same with field varying results, content see i latest CMB observations of Planckindex 2015 of provide a scalar very perturbations: accura 6 Example II: Goncharov-Linde inflation 7 Discussion 1 Introduction The framework of inflation is both theoretically appealing a 3 Superpotential flows 4 Stability issues 5 Example I: Starobinsky inflation Contents 1 Introduction 2 preliminaries K¨ahler manifold to avoid the A common issue in these investigations is the interplay betw provided two improvements over previous constructions. due to spontaneous SUSYmass breaking. and This SUSY allows breaking, for and an thus decouples indepen themultiplet two are ingredien set only bythe the inflaton SUSY multiplet breaking Φ orderthis free parameter. construction of allows such for constraints.ment arbitrary of Indeed, inflationary it potent the same models while replacing the flat with a hyperbolic K¨a multiplet success, it remains imperative to embed inflation in a UV-com on the remaining prediction of tensor perturbations: such theories turns out tothese be entail surprisingly an intricate, interplaydefinite due between to the supersymmetry negative (SUSY) definite breakingthe grav contributions. simplest It model t of quadratic inflation in supergravity wa overall factor exp( ruin the slow-roll conditions [ JCAP10(2016)038 - ], + α → ify 20 (2.1) (2.2) – ] and W K entire Φ ¯ Φ. 18 21 ∂ Φ), and , ¯ Φ). , )[ ( ≡ and ¯ Φ 12 ¯ ( W W W ials: K W W igher-order terms f Φ e D (Φ) (together with (Φ) = → ¯ +log( Φ) = Φ). These properties W , W , =0 along Φ= . K W ¯ = Φ (Φ Φ ( = s i.e. K K G ldstino inflation: all infla- W G , the inflationary predictions  ¯ rent physical phenomena. e imaginary component of Φ is tion. Moreover, we will demon- fied in terms of two quantities. ¯ ny other multiplets [ Φ) = constant in the final vacuum to W , , ¯ ¯ perfield framework, which can be Φ, i.e. Φ nction nstruction. The interplay between  (Φ rmation. The scalar potential reads ¯ Φ → G WD G Φ Φ D G ¯ ¯ Φ Φ Φ Φ as the superpotential contributions drop out. G K ¯ Φ Φ + ]). More recently, single-field realizations of – 2 – 3+ K ¯ 15 W − = 0, along the inflationary trajectory. This consis- = ] depending on the angle between the two directions  W ¯ Φ, specifying the trajectory along which inflation will and the covariant K¨ahler derivative ¯ Φ G K 3 14 ¯ Φ) specifies the geometry K¨ahler of the scalar manifold Φ e , 1 ¯ , Φ − ]. G − Φ ¯  Φ for arbitrary choices of and K¨ahler superpotentials sat- = 13 (Φ 17 K K V , K e ≡ 16 = = (Φ), which leave the scalar potential invariant. Indeed the ¯ Φ f Moreover, one can employ the transformation K¨ahler to simpl , while the holomorphic superpotential ] instead use the imaginary direction as inflaton and require h Φ V K ¯ 1 Φ 23 K Φ , K 22 ], we will make the following assumptions about these potent 6 with holomorphic ¯ f . The latter are covariant under the transformation K¨ahler tutes a shift symmetry of the inflaton field and implies that The potential K¨ahler vanishes, hence will be an even function of the imaginary component of Φ The superpotential will be a real holomorphic function of Φ, The potential K¨ahler will be invariant under Φ − In this letter we will prove a comparable versatility for sGo Following [ f The constructions of [ • • • W 1 − ) determines the scalar potential via Φ guarantee that the truncation to Φ = isfying the above criteria. K This implies that the invariant function K¨ahler satisfies take place, will besatisfied a along consistent the one: trajectory the Φ = field equation for th with the metric K¨ahler in terms of the metric The same issue was addressed in [ in terms of this variable. Note that the potential: K¨ahler of inflation and the sGoldstini (see also [ thus proves to be remarkably succesful in accomodating diffe 2 preliminaries K¨ahler The scalar dynamics ofThe general real potential K¨ahler supergravity models is speci is therefore manifestly invariant under this gauge transfo to achieve stability. tionary potentials can beinflation realized and in a SUSY single-superfieldnor breaking co on therefore the level poses ofstrate no SUSY that breaking, constrains one either on can duringdescribe or independently the after introduce dark infla a energy ofthought cosmological the of late as Universe. arising The after single-su a supersymmetric decoupling of a attractors were put forward [ K K supergravity theory can be written in terms of fu the K¨ahler JCAP10(2016)038 e )) ϕ ) at 0 (3.1) (3.2) (2.3) metry ϕ (Φ( ( W W ) result in 0 = 0. Then = ϕ . The flow is -space (using ( ϕ 1 ϕ W W , respectively. T he flow becomes hor- ). For the examples of , ϕ  als, and fully fixes the rameter ambiguity in the Φ( ¯ T | ional physical requirement amples that will be relevant = 0 signals that one has to T ation that the superpotential | + rst-order differential equation tions: given a value e can take it at t can be hard to find a closed 2 the same Minkowski minimum ates Φ and T . ), to continue in ntial. This is illustrated this for W tial equation. While this always ϕ  ation (be it of the chaotic, hilltop ) . 3.2 = α e-dimensional and hence can always l. This can be seen as the analogon ) , the superpotential (3 non-negative scalar potential in this log ϕ / 2 V ( ϕ 2 α in the flow. Note that this implies that V 3 √ 2 ) and illustrated in figure any − − ). Viable inflationary potentials are much + e W ϕ 3.2 = 2 2 = / 2 W hy 3 3 – 3 – p r , T , K ± 2 2 √ = ¯ Φ) ϕ/ would result in a discontinuous second derivative of the − dϕ dW ϕ (Φ ) allows one to introduce SUSY breaking independently. The to be non-negative everywhere with a minimum that can be Φ= 1 2 ϕ ( is non-negative. The different initial conditions V − V ] describing domain wall solutions in AdS dual to RG-flows, se . = V 1 24 fl K will be a small correction to V . = 0) = 0. Note that this corresponds to the choice of a supersym W ) in figure ϕ ( ϕ ( ]. W increases at least as fast as exp( V 26 ). We will assume ϕ , ϕ ( 25 Its asymptotic behaviour, however, follows from the observ A special feature occurs in the case of a Minkowski minimum: t We would like to stress that one can embed W also [ take the other branch of solutions, with opposite sign in ( either Minkowski or De Sitter. Without loss of generality on the above equation always has a one-parameter family of solu preserving Minkowski minimum; all otherwith choices broken would have supersymmetry. The horizontal flow at therefore satisfies some point, one can follow the flow defined by ( counterpart to this simplicity is a non-autonomous differen has a solution andexpression allows for for a simple numerical treatment, i Note that frame.K¨ahler this can Iton should be the theory be and achieved stressed onlyare amounts for that to a this arbitrary convenient gauge is fixing. K¨ahler not potenti Ex an addit different superpotential yielding theof same scalar ‘fake potentia supergravities’ [ In terms of the canonically normalized inflaton field Given a specific scalarfor potential, this is a non-autonomous fi way in a single-superfield,or allowing for plateau a type) descriptionscalarpotential of as for infl a well given as dark energy. Moreover, the one-pa at large always well-defined since izontal at describing a flat and hyperbolic manifold in terms of coordin flatter, and hence ¨he geometriesK¨ahler above one finds the same signs on both sides of 3 Superpotential flows Along the inflationary trajectory, the scalarbe manifold brought is to on canonical form by means of a field redefinition Φ = a specific superpotential). We will refer to this as the SUSY superpote JCAP10(2016)038 a ], ), 3 ϕ 29 2 (3.3) (3.4) / 3 ial. p 2 , . 1 ) and sinh( → ±∞ → ±∞ ϕ c superpotentials de- rpotentials translates e converted into real 2 ent paradox with [ / ) while it is non-zero at the 3 ign flip of the SUSY solu- . , 3.4 ubble scale by far. In this p ϕ tions on the scalar nor the  = 0. In contrast, generic, 3 2 1 − - 1 - 2 - 3 ... ϕ totic regions and cross through 2 potential cannot deform a super- )+ arge , x ence from the non-SUSY solutions. ) ϕ , x ( blue. ) - 1 asymptotics ( ... ne dotted and the non-SUSY ones dashed) V ϕ ... + 6 √ ϕ + 2 − / ϕ e 3 2 / - 2 3 − √ ± 1 √ – 4 – e (Φ) is a real holomorphic function. This ensures  ± ( e 0 ϕ ( W 2 is consistent and unambiguous. Vice versa, solutions 0 / W )) for a given scalar potential with a Minkowski minimum. 3 = 0 at the minimum - 3 ϕ ± W ϕ √ ( e = = 0 ] geometry K¨ahler to describe plateau inflation, while the W 3 W 17 W W ϕ, W , ± 6= 0. Prototypical examples are cosh( 16 = 2 ) in ( ϕ . Such superpotentials were put forward in combination with W ) allow one to reconstruct the full holomorphic superpotent 3.2 V ) 1 3.2 ϕ 6 √ : 0 − − ) is the assumption that ϕ - 1 ( non SUSY : ] and hyperbolic [ = 0 has an AdS instead of Minkowski vacuum. W . Left: the flow ( ϕ 28 - 2 , We would also like to mention that our models have holomorphi The special role of the SUSY superpotential resolves an appar The aformentioned difference between SUSY and non-SUSY supe 27 A direct corollorary is that any superpotential that has the - 3 = 0 at a different point 2 2 1 0 3 to the flow equation ( that the truncation to a real variable Figure 1 in terms of exp( flat [ which shows that infinitesimal transformationssymmetric Minkowski of minimum the into super ation, non-SUSY its one. superpotential Due indeed to has the a s non-continuous differ pending on afunctions complex scalar field Φ. The reason that these can b minimum present context is potential. fullyK¨ahler Moreover, general: a similar we expansion have holds not at made l any assump Right: three different superpotentials in orange (the SUSY o for the same flow with the resulting scalar potential inthe solid gravitino mass andregime, the the scale superpotential of can SUSY be breaking expanded exceed as the H respectively. non-SUSY superpotentials have opposite signsW in both asymp of which the SUSY one touches into the asymptotic signs - 1 - 2 - 3 JCAP10(2016)038 V (4.3) (4.2) (4.1) in an ,  ; however, supergravity hat, in the ′ ) ¯ 2 Φ V R G ¯ increasing with Φ decreasing with decreasing with + , viours where the 2 2 2 D R ¯ Φ V + in terms of the real Φ ¯ 2 Φ V G G ] that is valid for any − )( e . The same applies to the Φ 32 . , V + G h the and K¨ahler the super- ]. ¯ written as Φ ϕ Φ Φ 31 34 ϕ and hence the Hubble scale is ass of the orthogonal direction, G D G real line. However, these models , w-roll inflation: ons proportional to Φ f [ W d to ¯ Φ WV V + G 2 33 V [ 2 Φ + G R + 2 ( Starobinsky model from h a D-term potential, the issue of stability does 2 ϕ 2 2 ¯ Φ ϕ ) 1 one can include higher-order stabilizer W V ¯ Φ G 4 Φ Φ ≤ V G + . In turn, ¯ 2 Φ. An important aspect of this concerns the α ′′ R +( 2 + V ) is the curvature of the manifold, K¨ahler which ϕ ) supergravity extensions with an F-term action – 5 – ) it follows that the average of the scalar masses ¯ Φ  R R W Φ ( ¯ Φ R 2.2 f G ¯ Φ G + 2 . The first two terms of this expression above therefore Φ G 2 2 Φ G log( ¯ W Φ W G ¯ Φ Φ ¯ Φ ∂ Φ G Φ = 2 ]. ∂ G 2 1 the model is stable, with the sinflaton mass ¯ Φ 30 1 the model is unstable, with the sinflaton mass Φ 2+ m 2+ G = 1 the model is unstable, with the sinflaton mass −  α > − α< α  G = 0, the above formula relates the spin-0 and spin-1/2 masses G e = e ) vanishes. However, for Φ ¯ 3 or ) for the hyperbolic case and vanishes for the flat case. Note t Φ = 3 or 3 or = / G Φ α / / 4.3 2 2 2 2 ¯ R Φ (3 m − ¯ Φ Φ − − / 3 Φ V 2 = 1 is exactly the dividing line between both asymptotic beha = ¯ Φ ] in order to achieve asymptotic stability. , and therefore also during inflation. In the present set-up, G − this reduces to Φ α Φ R > R< R 17 = V G ϕ , 8 R = the gravitino mass. This includes the flat case. K¨ahler the gravitino mass. For For For in terms of the chiral scalar curvature multiplet the Hubble scale. An example is From the general scalar potential ( 2 If the inflaton is embedded in a massive vector multiplets wit • • • m 3 variable asymptotically much smaller than leading term in ( stability of the truncation tothe the sinflaton? inflaton field: what is the m Away from a critical point, the latter two terms are correcti 4 Stability issues We have thus identifiedpotential a to large generate degeneracy a in specificwill the inflationary have choice model different along of properties the bot away from Φ = which is a generalization of the mass supertrace relations o set the mass of the orthogonal sinflaton direction during slo during slow-roll inflation these will be suppressed compare mass of the inflaton field, which is set by is given by Note that terms [ and any not arise for any manifold. K¨ahler A particular example is the where SUSY limit with in the new minimal formulation [ Anti-de Sitter background. In the non-SUSY case, it can be re is given by JCAP10(2016)038 ] ) ]. 17 35 4.1 th a , (5.1) (5.2) (5.3) (4.4) ]. 16 40 e potential theory, it can be 4 2 = 60 e-folds with a R al mass formula ( + N connection), the mass R unction given by a Pade . Requiring the resulting tial to generate a very light btaining an exact expression ), where the profile function terpolates between different curvature. ase, however, we can obtain l inflation, the check of stabil- rofile that also has a regular equal in this vacuum) are set 003 at l of the form . 2.3 ential , ve comparable masses and hence → ∞ r limit of the present model. d needs to be checked for specific . ll-off. As a result, the Starobinsky ]. It was demonstrated in [ mples. The first example concerns = 0 )) , ov-Akulov model, as shown by [ 2 ot generate a De Sitter plateau [ r T 39 T  = 0 yields two constraints on the ¯ Φ ( – ϕ V f 3 , 2 37 Φ ϕ / / 2 V 3 V ¯ bT cT Φ √ T 0 and Φ 97 and − . + = 0. In order to obtain the full Starobinsky e − K 1+ a 2 → T − / = 0 + 3 – 6 – 1 s − T  )= n T 2 = 0 is the dividing line between (in-)stability in the T ( VR ( 0 H R f ]. Originally formulated as an with potential K¨ahler ( = W 2 36 T = 3 = m . V W ∞ = T . This implies that a massive sGoldstino can only be stable wi ¯ Φ ¯ W Φ W = 1 in all subsequent plots. At large positive field values, th ¯ Φ Φ H K = V As a sideremark, in the rigid limit (where there is no K¨ahler Apart from the model-independent analysis during slow-rol As discussed before, although its existence can be proven, o ) has a regular Taylor expansion around A superpotential that is dominated by a single monomial cann 4 T ( large class of inflationary models with other origins [ scalar potential to have a Minkowski vacuum at three parameters of this Ansatz. The remaining parameter in rephrased in terms of a canonical scalar field with scalar pot asymptotes to a plateau with anmodel exponentially suppressed shares fa its inflationary predictions We will set in terms of hyperbolic coordinates compatible with both asymptotic limits implies that the massesby of the both spin-1/2 scalar mass components terms (which and are are independentformula of takes the the K¨ahler particularly simple form In contrast to their set-up,can where be both made scalar components veryinflaton ha massive, field. we are Therefore employing it a is flat impossible poten K¨ahler to take5 a non-linea Example I: StarobinskyWe will inflation illustrate the general procedurethe in Starobinsky two model concrete of exa inflation [ f ity over the entire inflationarycases. range is Once model-dependent one an ends up in a SUSY Minkowski minimum, the origin that such plateau models can be described by a superpotentia Taylor expansion around potential, it turns out that one must choose an inflationary p where for the underlyinga superpotential very can good beapproximant, approximation very by hard. making an In Ansatz this for c the profile f rigid limit. An example is the linear realization of the Volk non-vanishing and positive curvature; JCAP10(2016)038 5 on (5.6) (5.5) (5.4) ϕ hler ge- 4 he way up . . lation to 3  !! etween the values 3 φ We therefore have 8 3 2Φ √  r 2 nates:

! φ 4sinh 2 3 tion up to a field-dependent neral discussion implies that e the inflationary predictions 1 − r e) and its approximation in terms 13 sinh the imaginary component of Φ otential and a hyperbolic (dashed) ement

 hat there is a superpotential that − 3 ). This demonstrates that one can ular inflationary model. Moreover, 2 2Φ ! √ etween the two (green). Right: the mass φ  5 20 15 10 2 3 - 5 . α r 3 + 74cosh

). For concreteness, we will take a specific √ / ! - 1 φ 5.1 2Φ 8 3 − 1+6cosh 5 . 1). This particular superpotential was actually – 7 – ). e 1 , r 22 sinh 5 =

  5.1 − − 3 , T Φ √ 4  + 27 43 cosh ) = (7

tanh 3 6)  √ 3 a, b, c Φ √ φ/ (  4 = 1 will have an instability when embedded in a hyperbolic K¨a 2 1 α with St = sinh V 1 ) 109 cosh 2; indeed the scalar potential becomes infinite at this line. 0 ) is a smooth function that never vanishes and interpolates b φ ( φ W W π/ ). This can be remedied by going to a flat metric. K¨ahler The re ( d 109 in an almost monotonic manner. From the moment of CMB all t 3 )= d / φ = 2.3 √ . Left: the scalar potential of Starobinsky inflation (orang ( = 1 to obtain the corresponding superpotential in flat coordi 1 4 3 2 5 d ± V α Turning to the stability of the orthogonal direction, the ge The resulting scalar potential is equal to Starobinsky infla for the imaginary inflaton partner in the case of a Pade superp 2 - 1 to restrict to a horizontal strip of the flat geometry. Note that the superpotentialequals encounters a singularity when Figure 2 of a Pade superpotentialm (blue). We also indicate the ratio b plotted as the dotted orange line in figure plateau models with the real line implies that one must use the coordinate replac factor obtain arbitrarily accurate approximationsthe to general argument this put partic forwardexactly in reproduces this letter the guarantees scalar t potential ( element with coefficients ( and flat (dotted) potential. K¨ahler approximations of the Starobinsky potential ( ometry ( with to the end ofwill inflation, be it virtually only identical changes to a those few of percent, Starobinsky and (figure henc 1 and 179 Note that JCAP10(2016)038 ki ds 3). (6.3) . √ (6.2) (6.4) (6.1) (6.5) 3 entire (2 9. / π/ ± = 1 α ] with the specific 16 . USY. We will intro- ) with separate param- 3Φ) ǫ. √ 6.4 . tion will be stable on the e negative Hubble scale or omain between =  ever, the one-parameter de- e of this combination deter- ) dition, we have discussed an escribe SUSY breaking and a ϕ 0 , o incorporate SUSY breaking. W -attractors with cosh( al fall-off than the Starobinsky 2 ϕ e can introduce SUSY breaking α W / δ ll these aspects follow from the e arbitrary scalar potentials with ∂ 3 odels: they both follow from a similar ng for a nice illustration of our gen- gravity model of inflation by Gon- 3Φ) otential ( ential is p to be zero or positive. The first case = 0, at which √ ( , 2 2 ϕ T T 3Φ) + δ 3 δ, − √ 1 + 1 − tanh = 2 α ǫ − 0 √ 3Φ) tanh( 4 tanh( W W 2 √ ) ǫ – 8 – ϕ 3 2 2 ]. / + )=1+6 , sinh( 3 17 2 T 2 3 ( δ p ), this class is given by the general Ansatz of [ 1 ¯ f ( Φ, the resulting scalar potential reads 3 √ 2 3Φ) 5.5 − = √ 2 ǫ W ]. It has a flat geometry K¨ahler and a superpotential that rea . The different parameter choices are illustrated in figure = tanh sinh( ǫ 28 = 2 , q 2 V 0 ). Note that the sinflaton masses are equal in the SUSY Minkows 27 V 2 = W 0 ¯ Φ for the flat model. One can check that the same holds along the W W It therefore belongs to the same class of models as 5 As follows from the general discussion, the imaginary direc The original GL model has a Minkowski minimum with unbroken S There appears to be an interesting relation between the two m 5 which was considered for different reasons in [ inflationary line (figure Again this has amodel. plateau, however, with a different exponenti The resulting scalar potential still has a minimum at 7 Discussion In this letter we havea outlined Minkowski a or general De frameworkgeneracy to Sitter realiz in vacuum as the sGoldstino superpotentialPerhaps inflation. the corresponds clearest Moro illustration to is the provided by freedom the t superp Pade approximant choice Along the middle of this strip, Φ = eters for SUSY breakingaccurate and approximation of the Starobinsky cosmological inflation. constant. In ad minimum for the hyperbolica and positive flat gravitino cases,general mass and discussion during asymptote on to inflation, the th stability respectively. issues. A 6 Example II:An Goncharov-Linde inflationary inflation model with a simplereral superpotential, considerations, allowi is actuallycharov provided and by Linde the (GL) [ first super plateau at large Φ = Ansatz. In terms of curved coordinates ( where one has to restrict the imaginary component of Φ to the d Non-negativity of the scalar potential requires 2 duce two independent deformationspositive of cosmological this constant. model in The order generalized to superpot d leads to a Minkowskimines vacuum, while the in cosmological the constant.by second means Independently case of the of the valu this, parameter on JCAP10(2016)038 2 , , ] , is much (2010) 011 ϕ ]. 1 W 11 , , SPIRE JCAP IN , ] [ False vacuum inflation ing, set by 3.0 2.5 2.0 1.5 1.0 0.5 arXiv:1510.09217 [ ionary direction, which is to Sagnotti for discussions -attractors ]. le and the Planck scale this α Improved constraints on t by INFN-CSN4-GSS. ]. for the generalized GL-model with arge implications for inflation. . Wands, ]. ion therefore requires a different SPIRE IN Similarly, our general construction - 1 ] [ results. XX. Constraints on inflation SPIRE SPIRE IN astro-ph/9401011 (2016) 031302 [ IN ] [ . One might worry that this exceeds the ]. ] [ 2015 V 116 Natural chaotic inflation in supergravity 10 in Planck units. Therefore this allows for - 2 ∼ SPIRE ) – 9 – IN Planck 2 and Keck Array cosmic microwave background data 3) in blue, orange and green, respectively. ] [ / 2 (1994) 6410 /H 2 Pl arXiv:1502.02114 p Superconformal inflationary [ General inflaton potentials in supergravity , M D 49 (1 hep-ph/0004243 arXiv:1011.5945 [ , [ Phys. Rev. Lett. 1 0) , , 3log( collaborations, P.A.R. Ade et al., ]. / (1 2 , New models of chaotic inflation in supergravity 0) p (2016) A20 , SPIRE arXiv:1311.0472 Phys. Rev. & [ , IN (2000) 3572 GHz band 594 ϕ ] [ 2 1 (2011) 043507 )=(0 95 - 1 - 2 85 ǫ,δ Keck Array D 83 (2013) 198 and collaboration, P.A.R. Ade et al., ]. 11 - 1 . The superpotentials (left) and scalar potentials (right) SPIRE arXiv:1008.3375 IN A general feature of our models is that the scale of SUSY break Planck [ Phys. Rev. JHEP [ Phys. Rev. Lett. Astron. Astrophys. BICEP2 with inclusion of cosmology and foregrounds from BICEP with Einstein gravity [1] [7] R. Kallosh, A. Linde and D. Roest, [6] R. Kallosh, A. Linde and T. Rube, [2] [4] E.J. Copeland, A.R. Liddle, D.H. Lyth, E.D. Stewart[5] and R. D Kallosh and A. Linde, [3] M. Kawasaki, M. Yamaguchi and T. Yanagida, - 2 the observationally preferred modelsemploys of plateau a inflation. frame K¨ahler however with fully a broken by shiftmechanism the symmetry to superpotential. protect along This itself the construct from inflat quantum corrections with l Acknowledgments We would like to thankand Renata for Kallosh, collaboration Andrei on Linde related and work. Augus SF is supportedReferences in par larger than the inflationary Hubble scale, set by Figure 3 Planck scale. However, duedoes to not happen the until hierarchy between the Hubb parameter choices ( JCAP10(2016)038 ] , ] , ]. 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