JHEP04(2016)027 , I U Springer April 5, 2016 : March 14, 2016 March 28, 2016 Ctor of Super- : : Published Received Accepted L Inﬂationary Evolution

Home , Renata Kallosh

JHEP04(2016)027 , i U Springer April 5, 2016 : March 14, 2016 March 28, 2016 ctor of super- : : Published Received Accepted l inflationary evolution. 10.1007/JHEP04(2016)027 r inflation. iii) The matter doi: at under certain conditions: t direction potential, Kähler ntial, providing a good fit to e potential Kähler and in the adratic in the matter fields , anced versions of the hyperbolic b Published for SISSA by belonging to a hidden sector, so that matter S , in models where the potential Kähler has a flat S during inflation. The simplest class of theories satis- [email protected] 2 , and Timm Wrase H a . 3 Andrei Linde 1602.07818 a The Authors. c Supergravity Models, Cosmology of Theories beyond the SM

We describe the coupling of matter fields to an inflationary se , [email protected] Wiedner Hauptstraße 8–10, A-1040 Vienna,E-mail: Austria SITP and Department of382 Physics, Via Stanford Pueblo, University, Stanford, CAInstitute 94305, for U.S.A. Theoretical Physics, TU Wien, [email protected] -attractor models with a flat inflaton direction pote Kähler b a inflaton direction. Such modelsα include, in particular, adv the observational data. If the superpotential is at least qu Coupling the inflationary sector to matter Open Access Article funded by SCOAP Abstract: Renata Kallosh, fying all required conditionsand is with provided by the models inflaton with Φ a and fla a stabilizer gravity, the inflaton Φ and a stabilizer superpotential. Keywords: ArXiv ePrint: fields have no direct coupling to the inflationary sector in th ii) There are nomasses tachyons squared in are the higher matter than sector 3 during and afte with restricted couplings toi) the The inflaton presence sector, of we the prove matter th fields does not affect a successfu JHEP04(2016)027 4 8 9 9 1 4 14 15 16 10 12 12 14 10 12 13 gical = 0 f ij ¯ fA = ′ ) g ij tion of these models A mmetry breaking ( 2 c ity inflation, which had gs: superpotential. This allows be equivalently represented tly achieved in theories with tructing realistic inflationary is includes, in particular, the on direction; is was related to ble for the implementation of g ij e inflaton field. In such theories, ]. More recent progress is related A 6 – ∝ 4 f ij ] based on the hyperbolic geometry of 12 ¯ fA – 7 + ′ ) g ij – 1 – A ( ]. 2 3 c – 1 -coupled matter -attractors [ S α (Φ) = 0 g ]. These models provide an excellent fit to the latest cosmolo 13 ]. Flatness of the inflaton potential in the original formula 16 – 14 It was found also that many successful versions of supergrav A.3 Models with tachyons and stable Minkowski vacuum with sy A.1 Tachyons in modelsA.2 with Tachyons in models with a non-flat potential Kähler 4.1 A simple4.2 model with a A single model matter with multiplet two matter multiplets 2.1 General case 2.2 Stabilization using2.3 moduli space curvature Cases with2.4 flat potential Kähler and Models restricted2.5 with couplin ‘Sgoldstino-less’ models with very first version of chaoticto inflation the in development supergravity of [ the a moduli class of space [ data [ flat potentials for seeminglyas theories unrelated with reasons, potentials Kähler may with in a flat fact direction. Th the inflaton potential has ato shift develop symmetry broken a only large byvarious class the versions of of chaotic flat inflation inflationary [ potentials suita 1 Introduction During the last years theremodels was in supergravity. a This significant potentials difficult progressKähler with task in a can cons flat be direction most corresponding efficien to th 5 Conclusions A Examples with tachyons 3 Models with orthogonal nilpotent4 multiplets Examples without tachyons Contents 1 Introduction 2 Coupling the inflationary sector to matter did not require flatness of the potential Kähler in the inflat JHEP04(2016)027 is i (1.2) (1.1) (1.3) U ], and in the 2 17 Λ , / = 0 or just 2 11 . ], where the ) ı ¯ 2 ¯ S ]. This helped 24 ¯ S U – i S 11 ( ≡ U 22 , − [ ) i 1 10 X ... x, θ , etc. may, in principle, ( 2 + ) + nary models with Kähler the inflaton Φ and a sta- tial. This separation can ijk ¯ S k S tachyons, we will find that he stabilizer belong to the re term ies, so we will start with a B , including all models men- U tter multiplets have all van- , r potential between the two efore they do not affect the S, on field [ j , ter multiplets. Under certain ature in the theory. However, ) uperfields general one could expect that eously describe inflation, dark theories to be discussed below ij odels of this type [ U hler potentials of these models ese fields and the matter fields vantages of the previously con- ¯ instabilities and a very compli- Φ; i ractions between all fields, ,S A , U Φ , (Φ i ijk and U infl B i ( n to particle physics and cosmology for a long 1 K 3! U ]. However, using the symmetries of the mat + 13 ) = j W ı ¯ U ¯ U i , ) + i – 2 – U ]. The formulation of such models is especially ij U ,S ( A 12 (Φ 1 2 mat infl K ) = W ) + ,S = ¯ S Φ , can be either a nilpotent superfield i W S, is very restricted. U S ( ¯ Φ; ]. ij , 21 A ]. We assume that the potential Kähler for the matter fields mat – (Φ 3 W , 18 infl 2 K = K denotes terms with higher powers in the . The stabilizer S ... In this paper we will reveal yet another advantage of inflatio More specifically, we will consider inflationary models with Various constrained superfields were studied in applicatio 1 , not only in the potential Kähler but also in the superpoten i be very useful, if oneour wants main to suppress results the will reheatingmore be temper general valid superpotential in which a can describe more direct general inte class of theor where The simplest option tohidden consider sector, is so that there theU is inflaton no field direct and interactions t between th canonical, and that theresectors: is no direct coupling in the Kähle simple in the context potential ofKähler is the flat theory in of the orthogonal inflaton nilpotent direction. s hyperbolic geometry it wasinto possible a to form cast with the theto original flat solve Kä direction the corresponding problem tofacilitated of the initial the inflat conditions development of for modelsenergy inflationary which m and could supersymmetry simultan breaking [ time, see for example [ the hyperbolic geometry of the moduli space [ Here depend on the inflaton sector.such a But dependence to in guarantee the absence of structed inflationary models. ishing vacuum expectation valuesinflationary during dynamics. inflation, This is andinteractions rather ther between important, various because fields in couldcated lead multi-field to cosmological tachyonic evolution.one Meanwhile can in introduce the matter multiplets while preserving all ad potentials with a flattioned direction: above, can a be broadconditions consistently class to generalized of be by such specified adding models below, mat the scalar fields from the ma a very heavy field due to the presence of the sectional curvatu ¨he potentialKähler [ bilizer JHEP04(2016)027 , ] ], ). 3 in 24 S , 12 – 3 2 (1.6) – (1.4) (1.5) √ 22 10 ) = 0 as or has a ¯ Φ ¯ Φ = 0 can )(1+ i − ,S − to the Kähler Φ ,U ( S (Φ ¯ Φ and the Kähler ). In inflationary DZ 1.2 f ’s and that the Kähler ) = i -attractor models [ vestigated separately. See ] where also the references coupling of matter fields to α n eq. ( . ) might have some tachyonic cients, as explained in [ ] also satisfies this condition. tential. For definiteness, we rthogonal superfields [ lass of flat’ ‘Kähler models. 25 becomes much better, if the 3 l superfields al field Φ + R is a choice valid, for example, rmed in the past. For example The simplest potential Kähler , but the potential Kähler has a flat , 1.3 ns for the absence of tachyons. ollowing transformations , S, U α 2 ∈ 2 3 ¯ Φ = 0 is a stable minimum with in a potential Kähler as models ‘with a (Φ − − ). = W ) and ( , K 1.2 defines the bisectional curvature of the is either independent of Φ e confused with the manifold Kähler curvature. ¯ R Φ = 0. In such a case we find that all 1.2 ¯ Φ = 0 usually requires the superpotential a, a i.e. = 0 A ij . − − 2 A Φ + were studied in [ ¯ Φ=0 = 0 − 2 i – 3 – → Φ

U ]. Here W Φ 3 · K ¯ Φ is flat, , 1 Φ 2 [ ∂ W ] the curvature is 2 , ]. Magically, the hyperbolic = 12 3 ¯ ¯ – Φ) Φ , 10 W 2 − ↔ 2 [ ¯ Φ is stabilized at Φ Φ (Φ / and ¯ S 2 ¯ − Φ. The requirement that inflation takes place at Φ ] it was proposed to have a superpotential at least quadratic 2 ¯ Φ) 27 − K AS − + ], or, if necessary, by adding a sinflaton stabilization term for a discussion of these issues. 1 (Φ 23 K − , A.3 -attractor models [ = α 22 K The condition for inflation to take place at Φ Various studies of multi-sector supergravities were perfo A more general class of potentials Kähler described in [ We will find that in general our models in eqs. ( Our condition that the matter multiplets have vanishing vev We will often refer to these models which have a flat direction 2 ]. This condition may be relaxed in theories with nilpotent o be also implemented by either using the nilpotent orthogona respect to the field Φ In many of such models, the inflationary trajectory Φ proposed in [ to be a holomorphic function of the superfields with real coeffi tachyons disappear under the condition that ¨he manifoldKähler which may depend on the inflaton field. This is a subclass of our models considered in eq. ( certain restricted dependence onof the such inflaton type sector is fields. inflaton direction. potential of the form It requires the potential Kähler to be invariant under the f compatible with the current and future data, belong to this c inflaton field corresponds towill a assume flat that direction the of inflaton the direction po corresponds Kähler to the re matter fields, however, due tothe nilpotent the superfield nature was of required these to be models, of the the form matter directions during inflation. However, the situation to earlier work aremodels given. developed in This is [ different from our setting i potential and superpotential arefor quadratic the in superfields matter like fields squarks and leptons flat potential’. Kähler Note, however, thatFor this example should in not b potential in the inflaton direction Φ + also appendix In this case we areThe able case to of establish matter rather fields general with conditio non-vanishing vev’s has to be in 26 whereas the sinflaton Φ cases with JHEP04(2016)027 . (2.1) (2.2) (2.3) (2.4) (2.5) ]. S 2 26 | ) ) is at least i g n , mat ecouple ∀ W ,U e theories with Φ , + g S, ( ( . perpotential that is ı ¯ ... = 0 ) ¯ i U independent term and + mat . ionary trajectory. j + S g/f ,U e also only need the weaker mat and the following Kähler W . U i which are flat in the imag- W S (Φ (Φ) i g ! mat U ∂ 2 | f n W Φ adds a matter dependent term Sf mat k i elds with arbitrary dependence ∂ ∂ htly more involved, see [ Φ | (Φ) ∂ f = mat SW ) f ij i is not nilpotent, if one adds the sectional (Φ). W X (Φ) + g g/f S mat f ij g mat ) + SA i + on anti-holomorphic functions. Thanks W W = 2 that satisfies two criteria: 1) The Kähler , SA | n ¯ Φ ,U Φ ı ¯ i + ∂ ∂ ¯ U g U g (Φ mat i (Φ) + (Φ) and = g g ij U W g + ( (Φ) + mat A – 4 – ′ ]. In that case the formulation of the condition i + g ij g/f ) mat which only allows for an W : X g A ¯ 1 2 27 3 S,W | W and 2) the superpotential S g i 3 , mat + S ı ¯ ∂ 1 ¯ = 0. ¯ S W ) = − U = 0, i = (Φ) + S 2 2 [ K | 2 ,S U / (Φ) + + ( ¯ ij Φ) + 2 i S Sf ′ ∂ , g/f f mat (Φ mat g Sf ¯ | P Φ) = ¯ ij Φ) + (Φ ¯ W Φ W , The full model is then k A = Φ K i + ) 4 (Φ (Φ) + ∂ = K . i k g f (Φ) + | mat to the potential Kähler and takes Λ to be sufficiently small to d 1.3 g + U K = = 2

K ı ¯ Λ = depends on Φ and ¯ / U K W 2 i in the superpotential. i ) W U ¯ on holomorphic functions and U S i S S = 0 implies Φ ( P i ∂ − + = 0 we have U k i adds a matter dependent term to e ¯ Φ = 0 by construction. Also, as we will show in this paper, in th U = − g mat means on the inflaton field, without affecting stability of the inflat V ′ W ij (Φ). In general, we will study the case where the term in the su Now we couple this model to matter fields We find the following scalar potential Note that one may equivalently consider potentials Kähler These conditions seem to be satisfied for the MSSM and NMSSM. W The same conclusion can be reached for models in which f 4 3 A quadratic in the quadratic in the fields to condition that Here and superpotential nilpotent orthogonal superfields one canof introduce matter fi to the above requirements, we see that for inary direction, such as (Φ + 2 Coupling the inflationary2.1 sector to matter General case We start with a model of inflation defined via two fields Φ and equivalent to the reality condition mentioned above is slig a term linear in i.e. we specify that in ( curvature term where Φ Here potential is canonical We will consider nilpotent fields JHEP04(2016)027 inf )). V = 0 (2.9) (2.6) (2.7) (2.8) i 2.5 U ¯ Φ ationary V ) . . . g mat ) ) k W k k Φ Φ Φ ) + ∂ ∂ ∂ ) = 0 (see eq. ( = 0. Similarly, the g ) ( g i mat g mat 2 g ). mat inf U | g mat W g W ) mat V j W , g W Φ 2.5 + mat g ∂ )) mat W )) g ∂ i mat + g + W ∂ + ( 2 W j es for g | g mat W ′ 3 g g mat ∂ n sector and the other block j k ) + g W ∂ mat Φ − W l + ( g + ( g | ) g mat ′ mat W ¯  + ′ U = 0 to g∂ + ) i i ) i δ W g W ∂ g f i + mat ( + j 3 ( g U j ∂ mat ). ′ g + j ∂ mat W 3 g ( − U | U W g )) mat W 2.1 ) ¯ ) Φ + − ) g mat + =0 due to eq. ( + Φ W ) ) g f k + ( mat i + ( g f W jl mat mat g ′ Φ K mat V ′ )( j g ∂ U ı ¯ g f mat W g ∂ ¯ mat g mat Φ are unchanged and are the same for ∂ W W W = 0. Analogously, we can see that + ¯ )( U )( W f )( + )) W mat ). W i 2 + k + g j mat j | + k j – 5 – − g U ∂ g Φ g ∂ mat W g ∂ Φ ( f g | g i Φ W 2.5 mat ∂ j )( ∂ i V 3 )( )( ∂ )( V W i ))( ∂ ( i U W ))( = 0 reduces for ∂ g − ∂ mat g g ( f mat mat g l ¯ f mat mat + mat l ¯ + + j 2 g V mat g | ¯ ′ W mat ¯ U U W W i g W ) = 0 is actually a critical point at which the mass matrix W Φ l U W i i f i i ( W i | ∂ ∂ W i U g ı ∂ ∂ ¯ + mat ∂ ∂ ∂ ¯  l + ¯ (  g ¯ ( mat + U U + + ¯ k + W U ¯ P ′ U l ¯ V = 0 is a critical point: g j e ′ l ¯ g W g ) mat ¯  + + ( ¯ ) i U ¯ ∂ i ( U + ı ¯ k l ∂ + ı l ( ¯ ∂ = 0 implies = ı ¯ W e ¯ U U ( g U ¯ U mat g U i il ¯ g l mat U l mat g g ∂ mat mat inf g Φ + mat U ¯ W P Φ ( + P W W ∂ + V ( ( Φ i + W i W + W i l i k ∂ k ∂ + ij V g ∂ mat X K ∂ e ij g ( e ¯  mat ( ∂ i ∂ ¯ Φ ( ¯ V Φ δ ( W + + + ( + + + W Φ i Φ Φ i j ∂ j = ∂ ∂ V X K ı ¯ K X ı ¯ ¯ So we can conclude that the matter sector does not affect the infl V U ¯ + + ( + = 0. This follows from eq. ( U ¯ + ( + +  i ∂ = i = U ∂ V V i i = 0. at ∂ ∂ i Φ V Let us first show that We can see analogously that Now let us look at mixed derivatives The interesting fact is that Now we check how the inflaton sector affects the matter sector: ∂ U and is block diagonal withcorresponding one to block the corresponding matter to sector. the inflato second derivatives with respect to Φ and/or which is just the inflaton potential resulting from ( So we have shown that We see that the above expression vanishes for We see that every term will be multiplied by a term that vanish and the scalar potential in this case reduces to sector at all. for JHEP04(2016)027 ) . g mat g )¯ W (2.16) (2.13) (2.12) (2.14) (2.11) (2.10) ne can + g ) mat . , g ( W g mat ! ij k g ∂ W mat f ij Φ ( ∂ )) does not appear W + ) ¯ − fA ij g ) 1.3 ∂ . g k + )( mat 3 ıj ¯ ) ′ Φ ) ) − W 2 g g | mat ) g ij g∂ g r directions g mat A + | A W ¯  · + ( i f j g (see eq. ( W mat he diagonal entries of the 2 ′ δ g ∂ )) i jl c ı ¯ ¯ g W A U ∂ ( = 0 or more generally, when + ( ¯ ( U g + ′ mat 5 )( ) + , + f ij ) k + g ij ij g f W ¯ g , )). Φ g g ¯ mat mat A g fA ∂ )( mat + 1 − ) cA g mat ) (2.15) c 2.3 W W ) + g W k il f ility. jl ¯ (  mat k ′ ≡ W g f ı l ¯ mat ∂ Φ ) ∂ Φ ¯ ( ij W + ( A ij f U g ij )( W ∂ ′ g∂ f ij l g∂ A ij g + ∂ ¯ X ( fA + ∂ g ( + ¯ mat  2 + via k i ′ ′ c ′ ) e ) + ( + l ¯ g ) g c W mat g k g ¯  ( ¯ ′ + ( U g ı ( ¯ il ) ¯ l Φ + ′ A ¯ ¯ g W Φ – 6 – ∂ mat ) A k U ∂ l ij g · ( = 0 where we are left with Φ = 0 (see eq. ( ( ) l Φ ∂ 2 2 A W g i i | l ∂ g ( c P K mat )( g A ¯ 2 g U X Φ U mat | + + ( c + ¯ W k Φ k = g g e W are independent and it is not possible to diagonalize mat e ¯  ij = 0, where we are left with 2 g ı k + ¯ K for )( ij ∂ c e ¯ i W + A ( ij f ∂ g ij )+ 1 = U ij + A g ¯ c A mat = 0 has thus the following form V V f/g ∂ 1 ¯ mat Φ ı ¯ 1 l ¯ + ( i c V c Φ

W ¯ ¯ ′ ¯ ¯   U W U i U i i ) k ∂ K δ δ and ij e − ¯  + ∂ ′ g ¯ + mat V U = = ) + i ¯ = ij g f g ∂ W mat + V ) ). So we will now discuss more speciﬁc cases. In particular, o ( ! 2 ¯  . In this case we deﬁne A ¯  ij | W ) that controls the couplings that are cubic in the ∂ ∂ g ij f/g ij g ¯ i V V U f | ( g mat ,( mat ijl ∂ A A k ,S 2.13 ıj ij ¯ + ij g ∂ e W W ¯ Φ ( (Φ ∝ A V ij Φ ij + j l ∂ V ∂ V ∂ ∂ ijk f ij ∂ ¯ ¯   ( X K ı ı i ¯ ¯ V B ∂ ∂ ¯ ¯ U + + + ( fA k e 1

= + ′ = = = ) V 2 j g ij V Let us now check the derivatives along two holomorphic matte In general ∂ Note that j i A M 5 ∂ ( ∂ i 2 ∂ and we used that where we defined at all in the mass matrix and therefore does not affect the stab diagonalize the mass matrix above, whenever The matter mass matrix at c and This again simplifies substantially for So we find amass positive matrix. contribution from the inflaton sector to t This simplifies substantially for the mass matrix ( JHEP04(2016)027 = U al- be- ′ ) aton g ij (2.18) (2.22) (2.21) (2.19) (2.17) (2.20) = 0 or 2.5 A where ( f 2 T c )), we need U Σ 2.3 , U , Σ is a diagonal g . = ¯ . = 0 A g · ′ he rows and columns ) ! g = 0 A and a flat po- Kähler g ij ′ ! A ıj ¯ ) being the eigenvalues of ) A ∗ g , g ij ( 6= 0, we cannot calculate g 0         2 i U 2 g ij . rix. We find the following A λ A c . nt ( · . 0 = 0 2 U cA . ors of tee the absence of tachyons. (Φ) g c 2 f ij 2 2 ¯ f ) A ): in order to have | , vanishes when either λ ( A ⇒ c cλ g ij ! f ij ¯  2 i A 2 2 ⇒ ıj ¯ ) ± | 0 0 0 0 ) ¯ λ ], i.e. we write g fA i cλ ¯  g (Φ) that we discuss further below in ∝ ¯ ⇒ g ı ¯ at the quadratic level in the superpo- λ A g ij 1 g 31 ¯ A ( 2 1 i · A · i f ij ¯ c λ g cλ in the inflationary sector, i.e. if we want cA U (Φ) λ g in the superpotential (see ( g k A ¯ ¯ 1 fA A = 0 in more detail (see subsection 2 1 S ( j = 0 0 0 0 0 ¯ ( λ ¯ cλ U + ¯

 k f ij + e i i 6= 0 ′ Φ k ) 2         ) – 7 – | U e g ¯ ¯  fA ) g g ij ¯ g ı ¯ g∂ | A (Φ) + → ¯ k A · A (Φ) = + ( (Φ) can be arbitrary ¯ arbitrary c g 2 ′ ! g ij 2.17 ′ g 2 c ) | + e A A Σ g ( g ij | · Σ V k A c , f , e ( are real and non-negative with = 2 ! , g c n in eq. ( + 2 i Σ to the matter fields = 0 T µ 2 · ΣΣ 0 V ¯ c U S f ij 6= 0 and its conjugate as the off-diagonal terms. To find the eigenv Σ M † (Φ) = 0 1 0 ¯ g ij

, . . . , λ fA U g ij 1 (Φ) = 0, we need either that the leading matter coupling to the inﬂ = = λ cA A

′ g ij 2 ) Φ )) or in models with vanishing A g ij ∂ M Φ A ∂ 1.5 )( matrix we do a Takagi factorization [ . k ). Whenever such couplings are present and 2 Φ 2.4 2.3 M g∂ + ′ . Now we can perform a unitary transformation and rearrange t g g ( ¯ The other off-diagonal term in the mass matrix, = 0. If we want to use the nilpotent field A ¯ Φ · = 0 Φ 6= 0, then we require g f ij 2 to bring the matrix into the following block diagonal form tential in ( K A Now it is trivialeigenvalues to of determine the matrix the eigenvalues of the mass mat A f tential (see eq. ( i.e. there is no coupling of is a unitary matrix whose columns are orthonormal eigenvect matrix whose entries This later case is for example realized in models with consta c low for a class of models in which the mass eigenvalues in full generality and we cannot guaran is absent, in which case the function Let us discuss the constraint We are thus left with and the mass matrix takes the form or, if we want the matter coupling ues of this subsection JHEP04(2016)027 are ). l ¯ (2.28) (2.24) (2.25) (2.23) ¯ k i 2.2 R ure of the = 0 under S all matter fields ¯ ¯ Φ) (2.27) Φ) (2.26) ζ , , (Φ (Φ A B e matter fields positive, e fourth derivative of the ¯ ∼ ∼ Φ during inflation. Such curvature stabilizers were he vanishing of a certain ¯ ¯ S Φ s. At the level of the masses tential since the Christoffel S ontributions here but rather -antiholomirphic part of the wing form e matter masses squared get l four directions in els described by eq. ( Φ The curvature tensor in such e of the potential Kähler and ¯ ¯ SS S ctional curvature and if during S oice of the non-linear terms in S to potential the the Kähler term . . . R ¯  R l ¯ ¯ ¯ ı ¯ ¯ m  F . ¯  ¯ Γ i U l ¯ = 0 i g ¯ m F m ik k m to reach the minimum at = ⇒ − ⇒ − ¯ SU ¯ g Γ F i l ¯ S S m ik ¯ F ¯ k Γ 2 i i X R − ¯ , Φ) l ¯ ζ l ¯ and − – 8 – − ¯ k ¯ k − j i i = 2 F ) K ¯ K  = (Φ i ıj ¯ ¯ S ¯ g S = U = S M S l ¯ l ¯ K = ¯ k ∆ ¯ k ı i ¯ ¯ ¯ i Φ) . So for positive and sufficiently large ∆ Φ)( 2 , , F R R | are both non-vanishing. A bisectional curvature of the kind (Φ (Φ ), ¯ S (Φ) A B W F f i | − − ζ K = = + and + ¯ S Φ forces the scalar field field 2 i S − K µ ¯ ¯ S W Φ F i ∆ ∂ ]. For example it is known that the negative sectional curvat ¯ K SS → ( 3 S 2 ∆ 2 i , R µ 2 K/ : e ζ ≡ i can be constructed to enforce the condition that Φ = F ¯ Φ In general, the curvature of the moduli space is related to th Φ ¯ S S the potential Kähler curvature stabilizers are implemented via the following ch R It seems inif principle the always underlying manifold possible Kähler hasinflation to a these matter corresponding make fields bise vanish. the In masses particular, if of we add th symbols vanish because theyare are therefore related linear to in a one third vanishing derivativ moduli such that the correspondinga moduli vanish case during coincides inflation. with the fourth derivative of the po Kähler moduli space then this does not affectshifted any by of our conclusions except that th the condition that ¨he potentialKähler as follows The typical choice of the curvature as a stabilizer is when al proposed in [ In the inflationary sector consisting of two superfields Φ and where During inflation themodulus curvature by of creating a the wallthis in moduli effect the space potential comes for from enforces amass the given t matrix fact modulu has that a the correction diagonal due holomorphic to the curvature of the follo 2.2 Stabilization using moduli space curvature will have a positive massdiscuss squared. the We conditions will for not the consider absence such of c tachyons in the mod JHEP04(2016)027 . at f ij i ¯ fA U (2.29) (2.30) (2.33) (2.34) (2.32) (2.35) (2.31) ). = , to f ij ¯ g 2.19 ) is satisfied. ¯ − fA is sufficient for + = 1.5 ¯ ′ Φ. In such case ] ) ¯ g We conclude that g ij 12 − – A ) )( 10 k ¯ Φ, which includes our k Φ Φ , g∂ yons in the matter sector g∂ to the matter fields # + hogonal nilpotent multiplets = 1 at Φ = , + ) . ′ y. For models with a non-flat ′ S tential [ 2 0 g g 0 K . ( ( Φ 2 > ¯ Φ ) k rmined by equation ( > | − . Φ 2 Φ ¯ 2 g

Φ) = 0 at Φ = 2

| ∂ . | K (Φ) = 0. In this case the off-diagonal Φ g ¯ Φ g )(1 | 0 ± | ¯ g Φ) K + Φ 2 2 1 i − Φ g ij Φ > ± | − λ K ) of the potential. Kähler Also a general ∂ ( ± i (1 A i 2 i − ) i λ ), then the off-diagonal entries in the mass = ] has λ (Φ ¯ g λ 1.5 λ 3 k 2 1 1 is positive during inflation.

1 2 (1 –

c − 2.3 + 1 " ) − + V + e 2 + k – 9 – = | 2 2 i | Φ g c V log | g λ | g∂ α 4 3 ) are ¯ + = 4 3 Φ) = 6= 0 in order to have a non-trivial inflationary sector. 2 3 , + i 2 ′ + f V + − µ g (Φ 2.13 )). The masses squared in the matter sector are then ( V = k V k 2 i Φ = are real and all masses squared of the matter fields are positive and there = µ ∂ 2.17 ¯ i ] adds more examples in which the condition ( Φ) = 2 i ) 2 i ¯ Φ , 3 λ µ µ Φ , = 0, i.e. there is no coupling of 1.5 2 (Φ K ( k ) where during inflation f ij = 0 ) simplifies, since A (Φ) = 0 = ( f ij 2.2 = 0 in ( g f ij c ¯ 2.19 fA ¯ Note, that the condition of flatness of the potential Kähler fA = + ′ ′ ) ) g ij g ij A A ] also satisfies the flatness condition ( ( ( 2 c 2 23 c , + In all these models we also find that during inflation e g ij 22 A 1 ¨he potentialKähler the stability of the matter sector is dete the absence of tachyons in the matter sector but not necessar the quadratic level in the superpotential in ( as well as the simple case hyperbolic geometry models with shift symmetric po Kähler A new addition to thein class of [ inflationary models based on ort are no tachyons. We look for models ( matrix all vanish ( given by c 2.4 Models with class of models studied in [ for flat potentials Kähler Another class of models that was studied in [ It is important here that the the mass formula ( 2.3 Cases with flat potential Kähler and restricted couplings: In these models we necessarily have or in the form So we see thatduring this and is after another inflation. class of models without any tach If we choose however entries in the mass matrix in eq. ( This can be rewritten in the form JHEP04(2016)027 ] ). + to 24 ) = 2 – i 1.5 ) (3.1) W ¯ (2.37) (2.36) (2.38) (2.39) S Φ 22 ∂ S . ( , S, U ) = is taken to i (Φ . ,U W ]. However, it g ij W Φ (Φ 30 . D gA = 0 this amounts 0 i . Such a class of g mat g ij U > = 2¯ W A 2 g ij ) = 0 the potential is | tial is to generate them i g ∝ , for gA )) U g ij (Φ) + adratic in let like the ones in [ ± | f ij g atrix i + 3¯ A ] that these models have no 2.19 λ ¯ ] in application to the models r potential by quantum effects. fA be coupled to matter. In these g ij ( ization, one can reproduce the = 0, er geometry, which can be large 27 ∝ ) = k A 30 i + ]. Consequently, it was difficult to ) S is either nilpotent, or very strongly f ij ]. ¯ ′ g , ) 27 + e ,U ¯ S Φ, 29 − g ij ¯ equal to zero when we work with the fA V , ) . (Φ A (Φ) k k ( 2 g ij 28 + 2 Φ g Φ DZ c ′ ) = 3 f | ) gA g∂ g∂ g − | g ij 2 + + A ′ ′ = 3¯ ( ± g g – 10 – (Φ) and 2 ( f ij i 2 symmetry of the scalar potential that amounts in c k λ = f ( Φ ¯ , 2 fA i ∂ = Z W λ ¯ Φ k g , so it satisfies the flatness Kähler condition in eq. ( Φ mat Φ V 2 D W K + e ¯ 3 Φ) ], where they take the superpotential to be 2 | √ − = ( 27 g and avoid tachyons in the models proposed in [ | = ij f k (Φ S 2 1 ¯ fA f + e − mat = 0. Lastly, the potential Kähler in the inflationary sector + V ′ ′ ). In this case we have ) ) S = g ij ij g ¯ ,W Φ) = 3 2 i , A A √ µ ( = 0, which can be achieved by adding higher order terms such as 2 (Φ (Φ) c S k g 3 + )(1 + √ i g ij are absent in the inflaton potential. For Φ = A vanishes along the inflationary trajectory. In our language 1 ,U c ] required deviation from the rules formulated in [ K K To obtain these results, we assumed that the field One way to produce the stabilizing terms in the poten Kähler (Φ Φ (Φ) = 27 f DZ and the potential. Kähler Indeed,results in obtained the in limit the of theory strong of stabil nilpotent fields [ Furthermore the authors imposed a to setting terms proportional to f This implies stabilized at W ∂ our notation to ( Next we consider a particular case for which models was proposed by in [ 3 Models with orthogonalInflationary nilpotent models multiplets basedprovide on another an interesting orthogonal class of nilpotentmodels models the multip that terms can in easily the scalar potential that are linear or qu of [ strongly stabilize the field is not necessary to generateThese higher terms order are terms related in to thealready curvature Kähle at invariants of the the tree Kähl level. by quantum effects. Implementation of this method in [ tachyons in the matter sector. This leads to the following off-diagonal entries in the mass m 2.5 ‘Sgoldstino-less’ models with Therefore the eigenvalues in this case are given by (see eq. ( Thus we reproduced in our language the conclusion of [ be canonical JHEP04(2016)027 ∼ m (3.6) (3.8) (3.5) (3.7) (3.4) (3.2) (3.9) (3.3) W at the , such as S S , i.e. where S the inflatino, . 0 > ly acceptable models 2 uplings to the fermion sector of

. Therefore in unitary | s the goldstino has to be g . , | χ ilpotent multiplets is that 1 2 f ij . Fortunately, our matter ons and the analysis of the do not couple to i . ± W ¯ ¯ u ring and after inflation. Note g , n matter and inflaton, i fA i i tter fields with the inflaton in χ does depend on Φ, i.e. even if U − λ U +

f the fermion sector at the min- is not orthogonal to . = 0 . D g ij = i g ij = 0 with complicated non-linear + W multiplet u A ¯ g =0 2 Φ i χ = 0 | v ¯ gA S U g − D ) = 0

| + − ) φ =0 k 4 3 s k χ W Φ = χ W i Φ | + + Φ U v g∂ f ij g∂ V D K + W ¯ φ fA + ′ + S χ ′ g ⇒ – 11 – ) = ( + g D | ( + ′ s W ¯ g Φ ) i k χ Φ s g ij U Φ W ± | ∂ χ K ∂ A S 2 K i ( ¯ Φ 2 e D λ = ∼ = Φ s ( c φ i 2 χ = K c W + λ χ i v 2 = g ij U K + e 1 2 D | c ¯ gA . This does not change the diagonal entries in the mass matrix = 0 gauge, or in a gauge = 0. The mass matrix eigenvalues are then given by the same g = | 2.3 − s v f ij 2.1 + χ A V = , may lead to instabilities. = 0 at the minimum of the potential, the true goldstino is proportional to the fermion of the 2 i j s µ φ U χ , in the i χ U ) but simplifies the off-diagonal entries as follows W 6= 0 one has to deal either with the mixing of the gravitino with f ij Φ ) D ¯ 2.13 Φ , does not lead to instabilities in these models. However, co SA φ 6= 0. This may simplify the construction of phenomenological j χ ′ − = Another interesting feature of the models with orthogonal n If we now consider models where the matter fields Thus, the models with orthogonal multiplets have simplified U ) µ i g ij ψ Φ m U ( µ A formula as in subsection the inflationary models.supplemented However, by now additional when terms matter was added, quadratic level, then W ( of this type in casesthe where superpotential. there is a direct interaction of ma Φ inflatino couplings. vanishes: in eq. ( In principle, terms like thatimum could since destroy the the simplicity gravitino o is mixed with matter fermions general case in section γ couplings are such that In this case there arethat also no this tachyons result in the is matter valid sector in du this class of models even if S The massive gravitino isreheating decoupled is relatively from simple. other In spin models where 1/2 the fermi inflaton This means, for example, that a direct cubic coupling betwee since gauge, where the inflatino and JHEP04(2016)027 a f the (4.3) (4.2) (4.1) to the 2 ) and the U , 2.1 ) = 0, which 2 ¯ Φ 4 g − + Φ 2 ( ¯ . S H Y, sion, the mass eigen- Y ). We couple this infla- = 2.2 ¯ 3 U, nteresting to check what . The last term in this one can confirm stability 2 + with orthogonal nilpotent U S ≥ m 2.29 ¯ U he spin 1/2 fermions of the 2 + U = oupling the inflationary sector ds, ¯ S | + 1 l in ( S 2 g ¯ Φ = 0 during inflation. | λ ¯ S + − ± S to the potential. Kähler The result is ) + m . The mass eigenvalues of the complex 2 ¯ UY, (Φ) in the superpotential ( U ¯ U Φ g ) 2 ) with 2 ), which provides additional stabilization , m 2 + ¯ − SU 2 ¯ ¯ Φ Φ) 2 + 4.2 2 U 2.33 g 4 Φ ζS 2 − )(1 3 2 ¯ m Φ) 2 − U during inflation. − Φ ), ( – 12 – Φ + (Φ) and 2 )(1 2 , which we introduce according to the explanation M (1 2 f and one adds a term proportional to Φ − − V H Y Φ 3 2.31 (Φ) + (1 (1 = − ≈ 2 = 0. and Sf (Φ) + (1 const s m V U log χ Sf + α log 3 2 m (Φ) + ¯ | Φ in these models where Φ 0 without loss of generality. Here the inflaton superfield Φ is α − g (Φ) = g 3 2 g is a nilpotent superfield, both belong to the inflaton sector o (Φ) + = = ± | − g 0 by absorbing its phase in S 2 K | W to the mass eigenvalues ( = = g 1, 0, as explained for the general case in section | 2 | m, M > K m > W < + ζ > ¯ (Φ) Φ V are then f | = U ζ 2 µ Whereas matter fields are already stable in this model, it is i One can also check that, in agreement with our general discus that it adds further simplifies the study. The model is theory. The inflaton is Φ + will happen, if one adds a higher order term where we can take confirming the general case in eqs. ( superpotential. The example below is oneof of the inflationary the trajectory first and andto absence simplest matter. of models tachyons We when where considered c nilpotent and orthogonal superfiel 4 Examples without tachyons 4.1 A simple model with a single matter multiplet disk variable, Φ hyperbolic geometry inflaton independent potentia Kähler tionary model with matter fields of the fields for 4.2 A modelOur with next two example matter has multiplets generic functions above where we can take theory is preserved atmultiplets the in minimum the of unitary the gauge potential in models Therefore the decoupling of the massive gravitino from all t values remain positive if matter field equation takes into account that JHEP04(2016)027 ], ]. 12 35 (5.1) – (4.4) – 32 10 (Φ) = 0 ). Since g ij nvalues for A 2.3 Φ ). These two ∂ . As before, all in this equation. ), which can also 2.32 M h terms quadratic ± ). When all three in eq. ( ± 2.32 2 , 1.5 ij , 0 M A 2 as the following features: > + ithout creating tachyonic chaotic inflation in super- -attractor models [ one can easily add matter | ial. In this paper we have 2 2 α

rticular, that under certain 2 g sent. | ector will not destabilize an | ) it means that m n the potential Kähler and in during inflation: y trajectory and even without hat there are no tachyons. The ed in eq. ( ± √ the inflationary sector in the Kähler 2 (Φ) 2.3 ly: inflation may remain driven as in eq. ( 2 ing general models with a canon- ]. Flatness of the potential Kähler / H M 3 3 13 ± m being | ≈ 2 2 1 2 V m avity cosmology, since under this condition the of the cosmological gravitino problem [ λ trongly suppressed. This leads to a smaller value ± + i (Φ) is the inflaton-dependent mass of the g 2 λ

and M 1 ≡ + λ have canonical potentials Kähler and quadratic , all matter fields quickly reach their minima at – 13 – 2 p | 2 ). Finally the models have to have a flat Kähler Y (Φ) H 2 (Φ) + / 1.3 2 3 2 / | and 3 m g 4 | m U 3 | 3 4 + + V either directly, or using the general formula ( V ], as well as the advanced version of = Y 3 = 2 i – are related to the Φ-independent coupling 1 µ 2 i i In terms of the superpotential in eq. ( µ and λ 6 U . In this relatively simple case, we can compute the mass eige during inflation, 4 correspond to four different combinations of the signs , 2 W 3 , H 2 3 , ≈ = 1 (Φ) = 0. The matter part of the superpotential has to start wit V i f ij A The simplest version of the models protected from tachyons h The two matter multiplets, Note that the condition that matter has no direct coupling to = 0 during inflation. 6 i gravitino, and the in matter fields or higher, as in eq. ( U where and all mass eigenvalues are greater than 3 potential, strictly independent ofconditions the are satisfied, inflaton one direction, underlying can successful model guarantee of that inflation,mass the which eigenvalues matter also for means s these t modelsbe during given inflation in are the deriv form by a single field even if many interacting matterthe fields matter are has pre no directthe coupling superpotential. to the inflationary sector i shown that this classfields of to models these has modelsaffecting an without the additional destabilizing inflationary advantage: the evolutionconditions inflationar at outlined all. in thisinstabilities This or paper, means, forcing in one many pa can fields add to matter evolve simultaneous fields w based on the hyperbolic geometry ofin the these moduli space models [ helps to ensure flatness of the inflaton potent the matter fields computations agree, the relevant values of dependence in where 4 eigenvalues of the mass matrix are greater than In this paper we consideredgravity some with potential of Kähler with the a most flatical popular potential direction, Kähler models includ [ of 5 Conclusions potential and in thedecay superpotential rate of is the often inflaton field usedof to in matter the during supergr reheating reheating is temperature, s which simplifies the solution JHEP04(2016)027 ]. 36 (A.1) . The ] in the . Some 2.5 27 2.4 (Φ) in the infla- ysics of Stanford g also supported by s of our theorems, ate publication [ ) may depend on the haler for enlightening ). Also we need that . Stabilization in some hanism can be used in ¯ i 1.3 U, 2.3 also noticed in [ (Φ) where also argued to with flat potential Kähler (Φ) and U U e combining the MSSM or le, the squark and slepton l the conditions except the g lues. In principle, the cur- ositivity of masses: we still f e 3 + ails given in section √ g non-vanishing vev’s, see ap- bility in the models violating ¯ k of RK and AL is supported S and matter in the superpoten- and Holography’. The work of ork was initiated. itions. Here we explained these is paper. e creates a wall in the potential in eq. ( ing of the inflationary sector to S s with orthogonal nilpotent mul- + ijk # B (Φ) = ) = 0 in eq. ( f 2 f ij Φ . 2 2 A − ¯ Φ) Φ (Φ) = 0, was presented in section )(1 g 2 mSU − why the moduli space curvature presents a uni- – 14 – Φ in the superpotential. In particular, we take the (1 . − 2.2 S -coupled matter (Φ) + (1 A.2 ] and for which S g . Various successful examples of stabilization are given " 3 27 and log α = 0 which means (Φ) + 2 3 A.1 ij Sf − A = = S ∂ (Φ) = 0. But cubic terms with K ′ W ) g 1.3 ] are given in section = 0 or . Examples of unsuccessful models, violating the condition 24 . Further investigation of such models will appear in a separ ij 4 – A Φ 22 A.3 ∂ We are grateful to J.J. Carrasco, S. Ferrara, D. Roest and J. T To develop these results towards a more realistic model, lik We stress that our conditions in models with independent We have also explained in section University for the hospitality during a visit in whichA this w Examples with tachyons A.1 Tachyons in models with discussions and collaboration onby related the projects. SITP, and The bythe the wor Templeton NSF foundation Grant grant PHY-1316699. ‘Inflation,TW the The is Multiverse, work supported of by AL COST is MP1210. TW thanks the Department of Ph context of their models,multiplets that with we vanishing can vev’s. nowsome introduce, of Note for our examp that conditionspendix may tachyonic play insta a constructive role, creatin simplified features of matter modulitiplets stabilization [ in model in section can be found in appendices inflationary sector fields, and the same for higherother terms cases in that th were studied in [ either versal mechanism for moduli stabilization.for Such the a moduli curvatur whichvature we can would make like these toaddition restrict moduli to to other fields stabilization vanishing stable va tools and which we heavy. studied This in th mec lead to a successful couplingresults to matter, as under a different particular cond case of our general argument with det decoupling of the matter sector from Let us start out by looking at an example in which we satisfy al ¨he andKähler superpotential special models, for example, the ones with tionary superpotential are sufficientmatter, for but a they successful may coupl notthe be decoupling necessary. requirement For between example, thetial in inflationary may models sector be relaxed withoutneed affecting that our in main eq. result ( for the p NMSSM models with an inflationary model, we will note here, as JHEP04(2016)027 ot (A.2) (A.5) (A.6) (A.7) (A.4) (A.3) tachyons. . larger than M U 0, we find the = ) with decoupled λ , 1.2 during inflation and | ormation of the kind M > g )˜ U in ( α ) with 3 − ¯ ¯ ss of the potential Kähler is . U, U, 2.32 teed and in particular at the . abilizes the matter fields. coupling between the inflaton 2 e field ¯ U U 2 U, U U U + (1 + + ¯ S, ′ 2 α . 2 ¯ ¯ M . 3 S S + and, taking g S 2 ] which in half-flat variables can be − S S | ¯ , S + 9 ¯ Φ)˜ + g U Φ 2 – | S + + (Φ) 7 g 2 U ¯ + M Φ + 2 2 | (Φ) 2 )). If, on the other hand, one makes the M (Φ + , the gravitino mass squared. Thus we see + g ¯ 2 ¯ ¯ Φ Φ) Φ) ¯ ¯ Φ Φ / Φ + mf 2 3 ± | | ) allows for tachyons in the matter sector, i.e. (Φ) + 2.21 2 4Φ 4Φ m against becoming tachyonic due to the presence (Φ) Φ + – 15 – M Sf ± (Φ) + g log (Φ) + (Φ + (Φ + 2.19 = ), ( U g α = 0. α V 2 2 3 Sf 3 | M log g k U = − | log log 2.20 α α 2 3 (Φ) + 2 3 α α = Φ = (Φ) + µ + e − 2 2 3 3 -attractor models [ Sf 2 | K − − Sf W α = Φ = ˜ g 0 and | = = = = k K W ≈ K K W W + e V V ¯ Φ, i.e. the potential Kähler is not flat during inflation. = (Φ). One can check that the eigenvalue with the minus sign is n 2 g ) is present and in this case we do not expect a protection from µ α 2 3 ) that in this model it is impossible to make both masses for 2.22 ]. This leads to W A.2 10 6= 0 at Φ = Φ (Φ) = Φ g K Now we add the interaction with matter according to our rules Now reconsider the same model by performing a transf Kähler from eq. ( we find We see that the positivityend of of the masses inflation squared we is have not guaran with ˜ of the inflationary sector (cf. eqs. ( eigenvalues of the mass matrix during inflation to be This model has a flat potential Kähler and but it the also matter has in anot direct the sufficient superpotential. to protect In the such matter case, field the flatne simplest choice it cannot be excluded in general that the inflaton sector dest matter in presented as follows It is straightforward to calculate the masses squared for th then one finds that the masses are positive and given by eq. ( We can compute the masses of the complex field Here the gravitino mass in the state with A.2 Tachyons in modelsConsider the with early a versions non-flat potential of Kähler positive definite so the general formula ( The term in ( made in [ JHEP04(2016)027 φ , and (A.8) (A.9) c metry ± ommons = 0, but = v u = u = 0, odels; it is known v , = 6= 0. However, during eaking in the standard φ 2 chyons during inflation. ¯ U, v e minima AdS, which is v a 3. Then during inflation 2 U / be done to make it realistic. turn to a discussion of this u 1 on can be very complicated, = 0. With a decrease of 2 e for spontaneous symmetry redited. + y models where a tachyonic ddress these issues. However, λ = 0, ¯ u S = 0. Note that this expression . g, the universe becomes divided e end the universe falls into one S u c λ < + 4 ± g or after inflation could be useful + 2 u, v and 1, = ) v ) 2 2 u c ¯ Φ c < 2 − , these two minima merge, and two other − 2 . ¯ Φ) = 0, v λ c = 0 becomes unstable, and the system may Φ v ]. )(1 √ − 2 v 2 c − 2 = 38 – 16 – = Φ u − , ( φ (1 u − 2 ]. λ , one finds the potential 37 below U 36 + (1 i v φ 2 λ + φ + u log ), which permits any use, distribution and reproduction in 2 2 α = Φ v 6= 0. In the end of the process the fields fall into one of the 2 3 separated by domain walls, which is a cosmological disaster + U 2 S − u , c u e α ± = = 6 ϕ √ = = K = 0, W , the minimum at CC-BY 4.0 2 V v u c This article is distributed under the terms of the Creative C 2 + = tanh √ φ λ √ is extremely small. This can be avoided in more complicated m 1 the potential has a stable dS minimum for the matter fields at c breaking ≈ Of course, this is just a toy model, and several things should Thus we see that the post-inflationary cosmological evoluti As a simplest example, consider the model φ for describing spontaneous symmetrypossibility breaking. in We a hope separate to publication re [ it is not obviousundesirable. whether Therefore one this toy can modelthis should break example be it indicates modified without to that a tachyonic making instabilities thes durin First of all, in the processinto of domains spontaneous symmetry with breakin of the two Minkowski minima. with two different stages of tachyonic instability, but in th for example that domainmodel. walls Yet another do issue not is appear that SUSY after is symmetry unbroken in br the minim unless two supersymmetric Minkowski minima the subsequent decrease of the field Writing Φ = Open Access. any medium, provided the original author(s) and source are c (if it has enough time) fall to one of the two minima at minima appear, at Attribution License ( A.3 Models with tachyons and stable Minkowski vacuum with sym where one can check that Im(Φ) = 0 is a minimum for is always positive or zero. We will assume that at In this paper weBut concentrated tachyons on are not models alwaysbreaking. where bad, one as Also, they can hybrid are avoidinstability often inflation ta can responsibl is be an used constructively example [ of inflationar also two deep separatefrom Minkowski 1 minima to at large JHEP04(2016)027 , , , , Sov. Phys. , (2013) 002 , , , 07 (2015) 914 ]. , ]. 16 e the best fit to the JCAP Microwave Background , SPIRE ]. , -Attractors SPIRE IN ]. α IN ][ ][ ]. ]. SPIRE ]. Improved Constraints on (2016) 031302 IN SPIRE IN ][ SPIRE SPIRE ][ ]. 116 SPIRE Hyperbolic geometry of cosmological IN IN IN ]. ][ ][ ]. Beginning inflation in an inhomogeneous ]. ]. ][ Minimal Supergravity Models of Inflation SPIRE Natural chaotic inflation in supergravity IN [ SPIRE Comptes Rendus Physique -Attractors: Planck, LHC and Dark Energy SPIRE SPIRE SPIRE arXiv:1504.05557 IN , [ α arXiv:1506.00936 Cosmological Attractors and Initial Conditions for IN [ IN IN Planck 2015 results. XX. Constraints on inflation Planck intermediate results. XLI. A map of ][ On the effective interactions of a light gravitino with – 17 – ][ ][ ][ hep-th/9709111 [ ]. ]. Phys. Rev. Lett. arXiv:1412.7111 , Superconformal Inflationary [ General inflaton potentials in supergravity Chaotic Inflation in Supergravity Chaotic inflation of the universe in supergravity hep-ph/0004243 arXiv:1307.7696 arXiv:1011.5945 From Linear SUSY to Constrained Superfields [ [ [ SPIRE SPIRE ]. IN ]. ]. IN [ [ ]. ]. (2015) 041301 (1997) 001 (2015) 063519 Escher in the Sky Universality Class in Conformal Inflation New models of chaotic inflation in supergravity arXiv:1512.02882 collaborations, P.A.R. Ade et al., (2015) 030 11 , SPIRE SPIRE SPIRE SPIRE SPIRE arXiv:1008.3375 D 92 arXiv:0907.2441 arXiv:1506.01708 arXiv:1311.0472 IN IN IN [ [ [ [ 02 D 92 IN IN (2000) 3572 [ ][ ][ (1984) 27 ][ (2013) 085038 (2011) 043507 JHEP 85 , JCAP , B 139 D 88 D 83 Phys. Rev. (2010) 011 (2009) 066 (2015) 147 (2013) 198 (1984) 930 [ Does the first chaotic inflation model in supergravity provid Phys. Rev. arXiv:1511.05143 , collaboration, P.A.R. Ade et al., collaboration, P.A.R. Ade et al., , , 11 09 10 11 59 arXiv:1510.09217 arXiv:1503.06785 arXiv:1306.5220 universe matter fermions JHEP [ Planck lensing-induced B-modes [ Planck arXiv:1502.02114 BICEP2, Keck Array Inflation JHEP Phys. Rev. JHEP attractors [ Phys. Rev. Phys. Lett. Phys. Rev. Lett. JCAP JETP Planck data? Data with Inclusion of 95GHz Band Cosmology and Foregrounds from BICEP2 and Keck Array Cosmic [9] R. Kallosh, A. Linde and D. Roest, [6] A. Linde, [7] R. Kallosh and A. Linde, [8] S. Ferrara, R. Kallosh, A. Linde and M. Porrati, [4] A.S. Goncharov and A.D. Linde, [5] A.B. Goncharov and A.D. Linde, [2] R. Kallosh and A. Linde, [3] R. Kallosh, A. Linde and T. Rube, [1] M. Kawasaki, M. Yamaguchi and T. Yanagida, [18] A. Brignole, F. Feruglio and F. Zwirner, [19] Z. Komargodski and N. Seiberg, [16] [17] W.E. East, M. Kleban, A. Linde and L. Senatore, [13] R. Kallosh and A. Linde, [14] [15] [11] J.J.M. Carrasco, R. Kallosh and A. Linde, [12] J.J.M. Carrasco, R. Kallosh and A. Linde, [10] J.J.M. Carrasco, R. Kallosh, A. Linde and D. Roest, References JHEP04(2016)027 , ]. ]. ]. ]. , ]. SPIRE SPIRE (2016) 101 IN , SPIRE IN [ Decoupling IN SPIRE ][ , SPIRE 02 IN ][ IN [ ][ , JHEP ]. , , , (1991) 38 SPIRE hep-ph/9503210 IN ]. ]. , ]. ][ ]. astro-ph/9307002 nd T. van der Aalst, arXiv:1108.2278 [ on. B 259 [ SPIRE SPIRE arXiv:1511.07547 [ SPIRE IN IN SPIRE UV Corrections in Sgoldstino-less IN IN ][ ][ ][ ]. ][ Orthogonal Nilpotent Superfields from After Primordial Inflation (1994) 748 (2013) 038 Phys. Lett. , Cambridge University Press (1990). Inflation Can Save the Gravitino , ]. SPIRE 03 Inflatino-less Cosmology arXiv:1405.0270 Minimal scalar-less matter-coupled supergravity IN [ D 49 – 18 – (2016) 025012 ][ Chaotic Inflation in Supergravity after Planck and ]. The goldstone and goldstino of supersymmetric From linear to nonlinear supersymmetry via ]. ]. ]. SPIRE Cosmology with orthogonal nilpotent superfields JCAP IN [ , D 93 Is It Easy to Save the Gravitino? arXiv:1512.00545 arXiv:1512.00546 SPIRE SPIRE [ [ arXiv:1504.05958 SPIRE SPIRE [ On sgoldstino-less supergravity models of inflation Constrained superfields in Supergravity IN arXiv:1509.06345 Matrix Analysis IN [ [ ]. [ IN IN Phys. Rev. [ [ , (2014) 023534 SPIRE Phys. Rev. arXiv:1411.2605 IN [ , D 90 (2015) 001 ][ (1982) 59 (1983) 30 (1984) 265 (2016) 263 (2016) 043516 (2016) 061301 10 arXiv:1603.02661 , Axions in inflationary cosmology Hybrid inflation B 118 B 127 B 138 B 752 D 93 D 93 (2014) 172 Effects of the gravitino on the inflationary universe arXiv:1601.03397 JHEP Phys. Rev. , , , 12 arXiv:1512.02158 Phys. Lett. Phys. Lett. Inflation Phys. Lett. functional integration BICEP2 JHEP Phys. Rev. [ inflation Phys. Lett. Phys. Rev. limits in multi-sector supergravities Linear Models [32] J.R. Ellis, A.D. Linde and D.V. Nanopoulos, [37] A.D. Linde, [38] A.D. Linde, [34] M. Yu. Khlopov and A.D. Linde, [35] T. Moroi, [36] R. Kallosh, A. Linde, D. Roest and T. Wrase, in preparati [31] R.A. Horn and C.R. Johnson, [33] D.V. Nanopoulos, K.A. Olive and M. Srednicki, [28] R. Kallosh, A. Karlsson and D. Murli, [29] R. Kallosh, A. Karlsson, B. Mosk and D.[30] Murli, E. Dudas, L. Heurtier, C. Wieck and M.W. Winkler, [26] R. Kallosh, A. Linde and A. Westphal, [27] G. Dall’Agata and F. Zwirner, [23] J.J.M. Carrasco, R. Kallosh and A. Linde, [24] G. Dall’Agata and F. Farakos, [25] A. Achucarro, S. Hardeman, J.M. Oberreuter, K. Schalm a [21] G. Dall’Agata, S. Ferrara and F. Zwirner, [22] S. Ferrara, R. Kallosh and J. Thaler, [20] Y. Kahn, D.A. Roberts and J. Thaler,