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Inflation from nilpotent Kähler corrections

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Inflation and uplifting with nilpotent superfields Renata Kallosh and Andrei Linde JCAP11(2016)028 hysics with a chiral P S le ic t ar doi:10.1088/1475-7516/2016/11/028 strop A b -attractor behaviour, with inflationary predictions in excellent α and Marco Scalisi . Content from this work may be used 3 osmology and and osmology a C 1609.00364 physics of the early universe, string theory and cosmology, supersymmetry and We develop a new class of supergravity cosmological models where inflation is rnal of rnal Article funded byunder SCOAP the terms of the Creative Commons Attribution 3.0 License . ou An IOP and SISSA journal An IOP and Ernest Rutherford Physics Building,3600 McGill University University, Street, QC, Montr´eal H3ADeutsches 2T8 Elektronen-Synchrotron, Canada DESY, Notkestraße 85, 22607 Hamburg, Germany E-mail: , [email protected] [email protected] b a Any further distribution ofand this the work must title maintain of attribution the to work, the author(s) journal citation and DOI. Evan McDonough Inflation from nilpotent K¨ahler corrections Received September 17, 2016 Accepted November 4, 2016 Published November 11, 2016 Abstract. induced by terms in the potential K¨ahler which mix a nilpotent superfield J sector Φ. AsK¨ahlertransformation, the these new terms models are arecould non-(anti)holomorphic, arise intrinsically and for K¨ahlerpotential hence example driven.geometry. cannot due We be to Such show a removed terms that backreaction byof this of a mechanism inflation an is anti-D3 and very braneinternal dark general on geometry and energy, the allows of with string for the theorycorrections controllable a naturally bulk bulk SUSY unified lead field breaking description to isagreement at with hyperbolic, the the we vacuum. latest prove Planck When that data. the small perturbativeKeywords: K¨ahler cosmology ArXiv ePrint: JCAP11(2016)028 (1.1) D3’s is sponta- This fact has turned out to be . The latter is constrained by the 1 S , = 0 2 – 1 – S nilpotent superfield phase, in the context of supergravity and/or string theory, acceleration in terms of the superspace coordinates, one can easily check that the scalar part is replaced S = 1 flux background, famously used by ‘KKLT’ to construct de Sitter vacua in string -attractors from K¨ahlercorrections to hyperbolic geometry8 The string theory origins of constrained superfields, and connections to D-brane physics, The initial investigations of the nilpotent chiral multiplet in the context of global su- Some of the most interesting aspects have however emerged in the context of string After writing α 2.1 Bilinear2.2 nilpotent corrections4 Linear nilpotent corrections7 N 1 by a bilinear fermion.the This recent simply investigations reflects [6–8]). that, One in can this then case, recover supersymmetry the is original non-linearly Volkov-Akulov action realized9, [ (see10]. also D3’s and moduli stabilization via various contributions towere the then superpotential. further worked outplaced in in [37–44]. certain The geometries, fermions breakpackaged into arising supersymmetry constrained when spontaneously superfields. one (see An or e.g. example more [41]) is anti-branes, can provided by often the be recent works [43], where often involves the appearancecondition of [1–5] a which implies the absence of scalar degrees of freedom. neous rather than explicit, providing strong evidence for the compatibility of ‘uplifting’ via 4 Discussion 1 Introduction One of the keyour goals Universe, of unifying modern particlehas theoretical physics recently with physics been is cosmology atthat to significant obtaining both find advancements a early a towards pure and UV this late complete goal: times. description of it There has indeed been realized 11 3 Contents 1 Introduction1 2 Inflation from K¨ahlercorrections to flat2 geometry persymmetry [1–5] (see alsosymmetry [18, becomes19]), local havenearly [20–32]. been forty by years The now after recentment extended the in discovery to advent the of the of field. ‘de regime AdSde Coupling Sitter when Sitter supergravity a phase, [33], this supergravity’ nilpotent with has20–22], [ multiplet no marked to scalar a supergravity fields serious indeed involved. theory. gives develop- rise It to was a realizedan pure in [34, ]35 thattheory the [36], four is dimensional indeed a descriptionis supergravity of non-linearly theory anti-D3 realized. of branes a This in nilpotent demonstrated superfield, that wherein supersymmetry supersymmetry breaking by beneficial for cosmological applications:to one an can inflationary indeeddescription sector show of that generally inflation coupling simplifies and a the dark nilpotent energy overall field [11–17]. dynamics and allows for a unified JCAP11(2016)028 (1.2) = 4 super- d absence of a K¨ahlergeometry, has proven to be an optimal hyperbolic D3’s (see e.g. [48]). On the S and , flat ) ¯ S , S, D3 is placed on intersecting O7-planes. It = 1). -problem [49, ].50 The effect on inflation ¯ Φ η , P l (Φ M – 2 – δK correction to the K¨ahlerpotential. with a bulk modulus Φ, even in the = 0 , while keeping the superpotential independent of the S α δK [61–63] behaviour with cosmological predictions in great -attractor zero curvature α D3 on the bulk geometry, which would manifest itself in the for Φ. This is similar in spirit to ‘K¨ahleruplifting’ [59, ],60 where the term D3 on the bulk manifold and/or fluxes. In this letter, we take the opposite approach. We will show that inflation can be driven We will focus our investigation on the case of both In terms of physical applications, the nilpotent superfield However, in general there is no reason to expect the nilpotent field to be totally decou- We will prove that this procedure provides a unified description of early and late time , it is possible to obtain a comprehensive physical framework which describes the primordial 2 Inflation from corrections K¨ahler In to this flat section, geometry weK¨ahlerpotential would with like toinflaton show superfield. how inflation can arise simply from corrections to a by corrections to the potential K¨ahler of the form due to non-negligible53]. corrections Computing to these the correctionsnotoriously K¨ahlerpotential explicitly difficult in has task a and been concrete has considered string been compactification done in setting in e.g. only is [51– a a small number of cases, see e.g. [54–58]. agreement with the latestsummary observational of data the main [64–67]. findingswe We and will will perspectives work for conclude in future reduced in directions. Planck section4 mass Throughoutwith units the a paper, ( gravity theory as couplings betweenit the is nilpotent superfield the andof task the the of bulk model moduli. model undertential builders Typically, consideration, are to and argue suppressed in that and particular, such that do corrections the not corrections do lead to not to the affect K¨ahlerpo- an the dynamics which mix the nilpotent superfield the latter typically arisingwill from thoroughly string study theory the effects compactifications.providing all of Therefore, the the relevant in possible formulas and K¨ahlercorrections section2, to results.highlighting a we the In flat section3, differences K¨ahlergeometry, we will andthe discuss similarities the internal hyperbolic with manifold case is theautomatically hyperbolic, previous we lead flat will to case. prove that Interestingly, small when perturbative K¨ahlercorrections pled from the inflationary physics.backreaction of In the the context of string theory, this becomes a question of cosmology, where, at least qualitatively,of the the inflationary dynamics is due to the backreaction both a nilpotentinflationary superfield models and in an [45, thus ‘orthogonal46], appears nilpotent emerge that when superfield’, constrainedsimple an example superfields as of are used a ubiquitous to single in nilpotent construct string superfield theory, originallytool and studied in in not the [34, just construction35]. the of the one cosmological hand, scenarios it [11–17] allowshappens (see to also in easily the string uplift recent modelsother scenarios work of hand, with [47]). inflation it the On in generically supergravity, additioncontrol ameliorates analogous over of the to the stability what one phenomenology. properties Then, orS of by many the means of model an andexpansion inflaton of yields sector the better Φ Universe together and with a controllable nilpotent level one of dark energy and SUSY breaking. superpotential responsible for the uplift to dS is an JCAP11(2016)028 . S S K in (2.1) (2.2) (2.3) (2.5) (2.4) (2.6) linear or bilinear are forbidden since this . However, this direction | 0 S ¯ MS. S, W | + ¯ Φ) > , 0 | W (Φ M g | MS, parametrizes the contribution of the = + ¯ + . do not depend on ReΦ. On the other M 0 2 S 0 . W ¯ W 0 W Φ) , 3 M, = W = 0 (since the bilinear fermion, replacing the − (Φ = ¯ and = S,W g 2 S 2 S – 3 – W / + M K ) and a bulk geometry and/or fluxes (encoded in 3 S + ¯ S S = m D 2 S
¯ V S,W ¯ Φ ¯ Φ) S , − = (Φ Φ f K 1 2 = − = δK ReΦ. Note that higher order terms in is an unconstrained chiral multiplet, the works [70, 71] already considered bilinear 2 K S in eq. (2.5) to functions of the field Φ. This breaks the shift symmetry D3 brane (encoded in M in the scalar potential). , cannot get any vev). The constant phase described by eq. (2.2) is then the are arbitrary functions of their arguments, whose non-zero values can break S g K and e 0 is the flux induced superpotential [68] and is nilpotent, this approach has been put forward by [11–15]. The basic idea is to W and 0 S f W , such as Our starting point is the simplest realization of de Sitter phase in supergravity. This Now, let us just extend the internal K¨ahlergeometry with a chiral multiplet Φ which In order to produce inflationary dynamics, one must break the shift symmetry of In this letter, we intend to explore an alternative possibility to induce inflation: while In full generality, the only possible allowed corrections are either ¯ S In the context where 2 couplings. However, these were takenpotential. to be independent on ReΦ, thus not affecting the form of the inflationary Φ). Specifically, for thefor sake Φ of and simplicity, then we have choose a flat shift-symmetric K¨ahlerfunction can be encoded in the following set of K¨ahlerand superpotential [11–15, –22]: 20 D3. Then, the resulting scalar potential is a cosmological constant of the form where Note that one canpotential obtain in the supergravity latter andscalar eq. declaring part that (2.2) of by employing the usual formula for the scalar and the gravitino mass The latter setting stillThis provides is the obviously same a constant value flat (2.2), direction along as the both real axis ImΦ = 0. result of the delicate balance between the scale of spontaneous supersymmetry breaking of eventually will play the rolewould of describe the inflaton. a In a string theory interpretation, this framework hand, the orthogonal field ImΦ has a positive mass when turns out to bedependence not suitable for inflation, as it is too steep (due to the typical exponential Traditionally, this has been done byexample introducing a the Φ-dependence pioneering in work thewhere [69] superpotential (see and for the subsequentpromote developments [70, 71]). In the context and along the real axis of Φ and perturbs thekeeping original a flat direction Φ-independent creatingmix superpotential, an inflationary the we slope. two can sectors,the add break antibrane terms the and to original the bulk shift-symmetry the modulus. and K¨ahlerpotential encode which the interaction between where the shift symmetry for JCAP11(2016)028 (2.7) (2.8) (2.9) (2.11) (2.10) . ¯ S S ¯ Φ. Then, K ↔ Φ 3 = 0. The model g (Φ) in the superpotential f ¯ S. . -direction as the F-terms are S ¯ Φ S  ¯ Φ (i.e. ImΦ = 0) if the function Φ= ¯ Φ) | ¯ M, Φ. In addition, in order for ImΦ = 0 , , is symmetric under , ¯ Φ) = f  (Φ , Φ) f f ¯ g , W (Φ ¯ Φ S f ¯ ∂ (Φ Φ) or Φ 1 1 + ¯ 1 + Φ f  ∂ g 1 + Φ = 1 + ∂ 2 – 4 – ¯
Φ ,D  ≡ ¯ Φ Φ= = | − = 0 J (Φ) I ¯ Φ) Φ , F W K 2 1 Φ (Φ D − f Φ = ∂ K ¯ Φ. satisfies also an orthogonal nilpotency constraint [45, 46]). S is nilpotent (fermion interactions are subdominant during inflation) and the S This model still allows for an extremum along Φ = It is interesting to notice that the corrections (2.6) will affect the form of the K¨ahlermet- However, this turns out not to be an issue for the cosmological dynamics of the model In the following, we analyse the effects of the bilinear and linear nilpotent corrections Supersymmetry is spontaneously broken just in the One can simplify the following discussion by defining the function This is analogous to the reality property imposed on the holomorphic function 3 satisfies This class of couplings is wellmatter motivated from fields string theory generically as the appearsconjugate. for K¨ahlerpotential D-brane as a bilinear combination of the fields and their complex f A sufficient condition fortypical eq. corrections are (2.9) the to ones depending beto on be valid (Φ+ is a consistent that discuss truncation, this one later). must ensure positive mass of theequal orthogonal to direction (we for any value ofrespect Φ to and then thepositive for previously potential the developed thanks entire to cosmological nilpotent the evolution. cosmological supersymmetry Note breaking models along the11–17], [ both difference directions. which with yield a of the models [70, ]71 in order to assure consistent truncation along the real direction. along the extremum Φ = ric such as field is nilpotent and(anti)holomorphic, and eq. so (1.1) cannot holds.is be possible gauged In when away by addition, a the K¨ahlertransformation (whereas above this couplings are in general non- thus inducing non-zero off-diagonal terms and modifying the originally canonical as the field only scalars involved aremay the have some real relevant and consequencesthe imaginary for concluding components the section of post-inflationary of Φ. this evolution, paper. as The we off-diagonal commentseparately. in terms 2.1 Bilinear nilpotentLet corrections us focus on the effects of the sole bilinear corrections while keeping is then characterizedK¨ahlerpotential by such as the same Φ-independent superpotential given in eq. (2.5) and a JCAP11(2016)028 to 2 / 3 (2.13) (2.14) (2.15) (2.16) (2.12) m of order F and . Note that, in M ) implies a large 2 M from its canonical / 3 K m . The non-trivial structure and the gravitino mass, as (0) = 0), we have indeed 0 S (0) and , F M ) F 2 must be of the same of order of does not necessarily correspond f , ( , M . M O , as in eq. (2.4). At the vacuum, the (Φ) 0 (0) can still allow for a desirable low (0) and of the non-zero correction K¨ahler + (0) F F W F 2 1 corresponds instead to an f S F 2 2 2 M M M M |  + H. = − f 2 + 0 | ) 2 is necessarily large. ≥ 2 | 0 2 0 W – 5 – 0 3 W W M W W 3 S − 3 − D = − | ¯ S 2 2 / S is non-canonical, unlike the dS model defined by eq. (2.1) 3 Λ = M (Φ) = ¯ K S m V S or even higher, such as = ( K H V at the minimum and thus a considerable deviation of f The regime of small K¨ahlercorrections The cosmological constant and amplitude of the inflationary potential are thus deter- Note that, within this framework, a large value of To summarize, bilinear nilpotent corrections to a flat K¨ahlerpotential, such as the ones The combined effects of the SUSY breaking in is a constant. In this limit, the scalar potential can be indeed expanded as gravitino mass (e.g. orderstring TeV) theory and landscape). a Nevertheless, negligible the cosmological latter case constant (small (inform the eq. (2.5). spirit of the mined in terms of theto same be underlying parameters. very small The(albeit by cosmological in constant late-time is a cosmology, constrained while model-dependentcosmic the microwave way) background size by (CMB). of the the amplitude inflationary of potential the is curvature fixed perturbations in the to a very highSUSY gravitino mass, breaking which scale is is still indeed equal given to by where the K¨ahlermetric term and the modelsbe of of [11–17] the (in same these order). cases the A almost small vanishing fine-tuned CC valueK¨ahlercorrection of forces unity. In this case,the eq. Hubble (2.12) scale implies of that inflation the parameter This holds during the wholeM cosmological evolution until the minimum of the potential, since which makes once more explicitthis what regime, we the just CCin said is eq.(2.2). about given the Therefore, again magnitude the of as latter compensation between of eq. (2.8), cansolely due account to for spontaneous both SUSY inflation breaking of and the dark nilpotent energy. field Both acceleration phases are of the K¨ahlercorrectioncosmological still model, allows with to tunable have level of great the control CC over and the the scale phenomenology of of SUSY the breaking. which clearly allows for arbitraryminimum inflation of and the a potential residual cosmological (which constant is (CC). placed At at the Φ = 0, provided generates a scalar potential for Φ, along the real axis, given by JCAP11(2016)028 0 W , is , V , and (2.24) (2.20) (2.22) (2.23) (2.21) (2.18) (2.19) (2.17) 1 3 F m ∼ 2 H ¯ S, ¯ S, S · S . 2 , . ¯ 0 Φ Φ 2 Φ Φ W ϕ 2 2 3 . 2 (Φ) 2 2 , . 2 ¯ m Φ 0 F M − m m 2 M 2 Φ 2 1 + W + 1 2 2 2 M 2 . + (Φ) 24 − + 4Λ M 0 m Φ
M 2 ¯ F 0 2 S 2 2 2 + 4 0 2 W S W − + 2 0 m W M M 2 + 3 – 6 –
W 2 and = 8 4 comes from the condition that Λ be small. Thus ¯ Φ −
1, so as to be a small correction to a flat K¨ahler ¯ ¯ − Φ) = Φ), 0 ¯ 2 Φ − , , M 2 ImΦ (Φ) = W = M − Φ = 12 + (Φ (Φ F m As a concrete example, let us consider the classic model 1 is necessarily very large during inflation and hence the |  Φ f f f 2 1 = | 2 − 2 ImΦ and 2 ImΦ ) 2 1 − H m V m − M = /F = K K = (1 f It is important that we check the stabilization of ImΦ. The mass of ImΦ is 2ReΦ. The normalization of the inflationary potential depends only on √ = ϕ . The mass during inflation, expressed as a ratio to the Hubble constant We can also construct this model as a small perturbative correction away from a flat First consider the following K¨ahlerpotential, M If we imposepotential, the this condition again that gives the same quadratic potential (2.22). which corresponds to the choice with which corresponds to the choice of K¨ahlerpotential. Consider the following K¨ahlerpotential: given by This gives the scalar potential hence the only constraintthis on model allows forcase, low-scale the supersymmetry magnitude breaking of andmodel a is small a gravitino large mass. deviation In from this a canonical K¨ahlerpotential. Stability. given by, where we have used eq.(2.13) to relate Both the above termsand are ImΦ positive, is and effectively the stabilized first during term inflation, is dominant. regardless of The the above ratio precise is details large of The mass at late times, at the minimum of the potential, is given by or Example: quadratic inflation. of quadratic inflation. InWe the do following, so we in explicitly two construct ways, this which model have in low our and framework. high scale supersymmetry breaking respectively. JCAP11(2016)028 : g M . 2 / (2.31) (2.30) (2.29) (2.32) (2.27) (2.28) (2.25) (2.26) 3 m . as given by S ¯ S and is symmetric under S g , and is given by = 0). The most general g ), then this is a coupling . , g . ) ) f ¯ 2 to be a real and symmetric 0 , S 0 = ¯ 2 g W + , g g 2 , the task of finding the form of 2 ¯ Φ)] S Φ) g S , ( , 1, then imposes a condition on ¯ Φ) S. S , as follows O ¯ 0 , Φ)( . (Φ (Φ g , ¯ , g Φ) ¯ g -direction, with the F-terms equal to Φ + |  1 (Φ , 0 0 S Φ g (Φ f + g . | ) then the coupling is to Im 0 g (Φ W 0 and ¯ g 10 g 2 − |  M + 0 g − − + now requires solving a non-linear differential 1 g + ¯ m ¯ | S = MW is purely real ( 10 MW = 2 V M S S ¯ [ Φ provided the function , g g W – 7 –  ¯ + Φ) S + 1 , 2 2 ) + 2 2 D
0 2 ¯ 0 Φ) = (Φ M ¯ Φ , W g |  W ¯ when the CMB pivot scale exits the horizon during Φ. 3 g 3 − | (Φ = to unobservably large values, corresponding to high-scale − g 10 Φ − 0 − as in eq. (2.5). Similar to the previous case of section 2.1, 2 = 0 δK W 2 1 10 M W − (Φ) = W 1, this corresponds to high-scale supersymmetry breaking and ∼ Φ and = V = ( D ∼ V now receives a Φ-dependent correction. V K M is purely imaginary ( F S g ¯ Φ. In the following, we will then assume ↔ 1 forces 1 implies |  , while if g S | |  f | However, we can make some progress. In particular, in the regime Contrary to the models with a correction coupling to Supersymmetry is still broken purely along the The scalar potential of this model now involves derivatives of The normalization In the former case, the model is characterized by a K¨ahlerpotential such as which is similar to thebe expansion constructed (2.16). here via Therefore, the the same choice quadratic model eq. (2.22) can the potential can be expanded perturbatively in function of its arguments. Note that the term in If we make theof simplifying Φ assumption to that Re equation. This makes constructingthe models example (2.20), with an low-scale intractable supersymmetry problem. breaking, such as the exchange Φ along the inflationary trajectory Φ = which yields the desired inflationary potential As in (2.23), thement normalization that of the inflationarysupersymmetry potential breaking. in conjunction with the require- and the same Φ-independent we have a consistent truncation along Φ = Since (due to eq. (2.22) and the smallness2.2 of the CC) Linear also nilpotent toLet corrections a us very now large considereq. gravitino terms mass (2.6), in while the neglectingform K¨ahlerpotential which of the are this effects correction linear of is in the bilinear correction ( inflation (see e.g. [72]), along with the the condition JCAP11(2016)028 D3 was (3.4) (3.3) (3.5) (3.1) (3.2) α (note the hyperbolic -attractors’. α ¯ → ∞ Φ, the inflation- 0 and Φ → , ..., ¯ Φ ¯ S. + ¯ ∂ S, S  Φ S . This yields universal cosmological ∂ + . ϕ + 2 α ϕ 

3 = 0 and it is again explicitly symmetric ¯ ¯ Φ Φ | ¯ / Φ 2 α log Φ Φ S | 3 2 p α 2 Φ + Φ + 3 Φ + −   – 8 – r log ¯ ± Φ and analogue of the flat and shift-symmetric K¨ahlerpo- log ¯ exp α Φ = = α 3 1 ∂ 3 V − ϕ Φ − ∂ + = curved = 0 ¯ Φ V Φ K K K = V Φ). controls the value of curvature of the internal field-space, given by / 1 α ↔ and a chiral one Φ, where the latter spans a flat internal manifold (i.e. S . α 3 (see [73, 74] for an analysis of its properties in relation with the physical implica- / 2 − -attractors from K¨ahlercorrections to hyperbolic geometry When the field Φ moves away from this boundary, in the direction Φ = Some working examples of this phenomenon were already found in [76–80]. However, One can make the inversion and rescaling symmetries of this K¨ahlerpotential explicit α = K ary implications are veryexponential peculiar deviation as from they a dS generically phase lead such to as a scalar potential which is an thus inducing a boundarysymmetry under in Φ moduli space, placed at both Φ the general framework with a varying K¨ahlercurvature in terms of the parameter The K¨ahlerpotential (3.2) then implies non-trivial kinetic terms of the field Φ, such as predictions in excellent agreement with the latest observational data [64–67]. developed by [61–63] and theFurther corresponding studies family of have models clarifiedK¨ahlergeometry has of that been the dubbed the ‘ sole attractorand inflaton nature sector, with is independently a simply ofbe certain the a realized SUSY peculiar special by breaking feature resistance means directions of to of the a the single-superfield other setup fields [85, involved86 ] [81–84]. (see also It87, [ can88], indeed in the case of by performing a K¨ahlertransformation (which leaves the physics invariant) and obtain [75] when expanded at large values of the canonical field R 3 In the previous sectiontent we have superfield considered the casezero of K¨ahlercurvature). K¨ahlercorrections which However, typical mixpactifications a K¨ahlerpotentials arising nilpo- have from often string ageometry theory logarithmic com- dependence ontions). the moduli When and the describe latterbrane a and is a expressed bulk in field terms may of be half-plane described variables, by the the presence following of K¨ahlerpotential: an where the parameter The latter can be regarded as the tential (2.5). It indeed vanishes at Φ = with respect to a shift of the canonically normalized field JCAP11(2016)028 (3.6) (3.7) (3.8) 1 and , via a ). This S , such as |  S , for linear 0 f | → ∞ MW D3 brane provides = 2 0 (or Φ 1 V → Φ, if we expand around , / n | Φ , | n ) g ¯ S + =1 ∞ n , S 2 0 P W ¯ Φ)( = 3 , . In these two cases, the scalar potential S − (Φ 2 g , g M + n | ¯ – 9 – S = Φ | S 0 n ). In the K¨ahlerframe defined by eq. (3.2), it be- V f ) thus generating non-trivial cosmological dynamics. ¯ Φ) ϕ , ,S =1 ∞ n (Φ (Φ f P W produces again a pure de Sitter phase such as the one given = = by means of eq. (3.3). This implies that any pertubative simply breaks the original scale-symmetry in Φ of the system = f W ϕ δK W W some coefficients, is naturally small during inflation (i.e. n , for bilinear nilpotent terms (see eq. (2.16)), and g 2 M . The resulting situation strikingly resembles the flat one and we find simply and = 1, for bilinear and linear corrections respectively). In the inflationary regime, n 1 MS f V ), thus easily allowing for plateau inflation as given by eq. (3.5). + ϕ 0 ( |  direction as given by eq. (2.10), in the case of bilinear nilpotent corrections, and as g the primordial acceleration whichwith then the gets bulk a geometry. dependence on Φ, due to the interaction implies that any polynomialinfinity), e.g. K¨ahlercorrection in such as Φ (or in 1 Inflation happens around the boundary of moduli space at Φ with Unlike the flat case, the geometricto variable Φ the has non-trivial canonical kinetic terms, field being related In the caseeq. of (3.5) small with K¨ahlercorrections, one realizes an exponential fall-off such as | one can then consider the expansions (2.16) and2.31). ( K¨ahlercorrection in Φ will naturally turnV into exponential terms in the scalar potential The pure nilpotentflationary acceleration energy phase, rather equalK¨ahlermixing than terms to induce the (3.8), the CC inflationary thus slope. (see serves Qualitatively, e.g. as an eq. the (2.22)), Hubble in- whereas the perturbative and nilpotent terms (see eq. (2.31)). W One can then proceed in analogy to the previous section2 by including mixing terms There are, however, some differences with respect to the flat case, which are worth In all the works cited above, the inflationary attractor dynamics arises due to a Φ- S • • • • = of the form Conversely, a Φ-independent by eq. (2.2). flat geometry). The case where the bulk field Φ is coupled to a nilpotent sector highlighting: comes manifest that such a (corresponding to a shift-symmetry in takes the form (2.12) andto (2.29), the respectively. flat The case stability as conditions well. results to be identical dependent superpotential K¨ahlerpotential equal to (3.1) or3.2), ( has been investigated by [16, 17]. in the K¨ahlerpotential (3.2),W while keeping athe superpotential same just formulas dependent inthe on terms of thegiven geometric by variable eq. Φ. (2.28), Therefore, in SUSY the is case broken of just terms in linear in JCAP11(2016)028 0 0), W (3.9) ' = (3.10) (3.13) (3.11) (3.12) 2 / 3 m = 0, that is . n , c Φ=1 ϕ . Note that, for some choices of | 2 α =1 f n ∞ F , 3 2 X n , Φ n ∂ q n 2 + . The contribution of the K¨ahlercor- − 2 Φ  0 e 2 ϕ n M . n W α c 2 c 3 3 M =1 q ∞ = 0 − X n =1 − ∞ determines indeed the inflationary plateau as 2 n X n e 0 + M − 2 0 n c W – 10 – 1 W = =1  ∞ ). 3 X n 2 0 g / -attractor model by considering − V 3 and obtained by means of eq.2.12). ( The minimum of = 0) = α 2 (Φ) = 1 + = m and ϕ ϕ F M ( V f = 0). is order Hubble (or higher), the gravitino mass V = g = 0 (i.e. Φ = 1), provided V M ϕ Λ = by means of eq. (2.11). Therefore, during inflation (at Φ f , eq. (3.9) has provided quadratic inflation in the flat case (see eq. ()). 2.21 n c We conclude this section with some concrete examples by focusing just on and the gravitino mass M On the other hand, similar to the case of flat K¨ahlergeometry, small corrections corre- One can easily obtain a simple One can then control the residual cosmological constant of this model, at the vacuum Finally, also in the hyperbolic case, the model allows for a residual cosmological constant. This framework allows for remarkable phenomenological flexibility and one can repro- 1 which corresponds to very small K¨ahlercorrections ' in terms of thethe canonical potential inflaton can be set at which is still related to spond to very high SUSYbetween breaking scale, which is order Hubble or higher (the compensation of the potential, by tuning the several contributions, which add to Examples. bilinear nilpotent corrections ( F the coefficients However, once we assume hyperbolicscalar K¨ahlergeometry for potential the reads bulk field Φ, the corresponding Although the magnitude of This is given bythe the potential, finite which contributions canfirst of derivatives be the of correction placed K¨ahler the terms at functions at Φ the = minimum 1 of (provided we impose some conditions on the duce several other‘E-model’ known [89], models defined of by the inflation. potential Another example is given by the so-called rections to the SUSYpotential breaking (see scale eq.(2.13) might indeed for become bilinear important corrections). at the minimum of the can be still tuned to phenomenologically desiderable values (e.g. order TeV). given by eq. (3.8)).mass Nevertheless, as this one results can again still to obtain be a decoupled desirable from low the value of the gravitino JCAP11(2016)028 in 2 (4.1) (3.14) /M 2 0 W = 3 0 F , 0 S. F + (Φ) n 2 B Φ) -attractors. α − (Φ) + (1 A 2 – 11 – 0 V M ) = acquires a dependence on the inflaton, such as 0) and the consequent cosmological dynamics is an in the geometric field Φ will contribute to the scalar ,S W ' B (Φ (Φ) = , will be immediately spoiled by any generic polynomial W F S and Φ m A = = 1 returns the original Starobinsky model of inflation [90]. This is for the entire cosmological evolution, thus providing alone the neces- can be tuned in order to change the residual CC ( W 0 α S F 4 ) with exponential terms, thus preserving the typical attractor behaviour). (3.5 = 1 and ϕ ( n V The regime of small K¨ahlercorrections is definitively important as one would expect While we have explicitly studied the case of a Φ-independent superpotential, where A general feature of these models is that SUSY is broken purely in the direction of the This simplifies the description of the fermionic sector, whose non-linear terms disappear from the supe- 4 gravity action, as already pointed out in the last reference of [11–15]. which for realized via the choice where the constant K¨ahlercorrection in Φ. One can prove that, in theexpansion case of of hyperbolic the defined K¨ahlergeometry bypotential functions eq. (3.2), any Taylor The situation ishigher different amount in of the fine-tuningof case in inflation. of order As to flatinflation, preserve a K¨ahlergeometry, defined the clear as by original example one superpotential-driven of generically model this needs circumstance, a the famous model [69] of quadratic exponential deviation from aof an dS anti-D3 plateau brane, at responsiblebulk the for geometry the Hubble induces inflationary the scale. acceleration, typical whose behaviour The interaction of physical with picture the is that these terms arisingparticularly as interesting subleading as dynamicalsmall perturbative effects. (as K¨ahlercorrections in inflation The the happens case inflaton at of Φ Φ hyperbolic are geometry naturally is the inflationary dynamicsK¨ahlercorrections is considered here purely are K¨ahlerdriven,by incorporated one the into superpotential. may a In model wonder this of what case, inflation happens that if is driven the sary acceleration for inflationrections and (which the cannot residual be gauged CC.ingredient away Interestingly, by in the a order non-trivial K¨ahlertransformation) become to K¨ahlercor- thethe have fundamental controllable vacuum level of of the supersymmetry potential breaking (see and the dark example energy defined at by eq. (3.9)). nilpotent superfield the case of Minkowski vacuum). 4 Discussion In this work weby have developed terms models in ofeven the inflation in K¨ahlerpotential the in which absence supergravity of where mixwould a have inflation the superpotential in is for inflaton mind driven the isstudied field inflationary given sector. the with by effects The an a physical of anti-D3either situation nilpotent brane these one flat interacting superfield, additional or with terms hyperbolic, aSitter when bulk and space. the geometry. found internal The We that have geometry outcome thisinflation, of is generically dark a the allows energy scenario bulk for and which field inflation supersymmetry allows that is breaking. for exits flexible to phenomenology de in terms of JCAP11(2016)028 ]. (2009) (1972) 09 16 (1973) 109. SPIRE IN JHEP , (1979) 2300 B 46 (1989) 569 ]. ]. JETP Lett. D 19 , Nonlinear realization of (1978) 451[ SPIRE B 220 ]. IN 41 Phys. Lett. SPIRE ][ , IN ]. SPIRE ][ ]. IN Phys. 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SPIRE SPIRE IN IN Finally, we note that there are many possible generalizations of the models presented Another interesting avenue of research would be to investigate the consequences of [ [ integration realizations of supersymmetry in general supersymmetry algebra from supersymmetric constraint 066[ models 438 [ supergravity [2] E.A. Ivanov and A.A. Kapustnikov, [3] U. Lindstr¨omand M. Roˇcek, [4] R. Casalbuoni, S. De Curtis, D. Dominici, F. Feruglio and R.[5] Gatto, Z. Komargodski and N. Seiberg, [7] S. Ferrara, R. Kallosh, A. Van Proeyen and T. Wrase, [1] M. Roˇcek, [6] R. Kallosh, A. Karlsson and D. Murli, [8] R. Kallosh, A. Karlsson, B. Mosk and D.[9] Murli, D.V. Volkov and V.P. Akulov, [10] D. Volkov and V. Akulov, [11] I. Antoniadis, E. Dudas, S. Ferrara and A. Sagnotti, here, similar to thesidered series in of [11–15]. developmentspossible of It for superpotential-driven would K¨ahlerpotential models driven be models of interesting of inflation inflation. to con- understand theAcknowledgments extent to which the same is We thank Andreiwork. Linde and Further, TimmRobert we Wrase Brandenberger, gratefully Wilfried for acknowledge Buchm¨uller,Jimand very Cline, stimulating Alexander Keshav helpful discussions Dasgupta, Westphal. comments Lucien onResearch EM on Heurtier Council related is a of topics supported Canada draftsupport with (NSERC) by of by via the the a this TheBoncompagni PGS collaborative Natural Ludovisi, D research Sciences n´eeBildt’. fellowship. center and MS SFB Engineering acknowledges 676 financial and byReferences ‘The Foundation Blanceflor which may have important consequences forof (p)reheating, primordial or curvature leave an perturbations. imprint in the spectrum the modified K¨ahlerpotentials studiedSuch here K¨ahlerpotentials for will the leadbilinear, fermions, to e.g. both derivative at interactions early of and the late inflaton times. with the fermion JCAP11(2016)028 ]. , 4 , 12 05 Eur. (2015) , JHEP JHEP , SPIRE Teor. , , IN Phys. (2015) 025 ]. 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