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Inflatino-Less Cosmology John Joseph M Inflatino-less Cosmology John Joseph M. Carrasco, Renata Kallosh, Andrei Linde To cite this version: John Joseph M. Carrasco, Renata Kallosh, Andrei Linde. Inflatino-less Cosmology. Physical Review D, American Physical Society, 2015, 93 (6), 10.1103/PhysRevD.93.061301. cea-01426635 HAL Id: cea-01426635 https://hal-cea.archives-ouvertes.fr/cea-01426635 Submitted on 4 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Inflatino-less Cosmology John Joseph M. Carrasco,1, 2 Renata Kallosh,3 and Andrei Linde3 1Institut de Physique Th´eorique,CEA/DSM/IPhT, CEA-Saclay, 91191 Gif-sur-Yvette, France 2School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, USA 3SITP and Department of Physics, Stanford University, Stanford, CA 94305 USA We construct inflationary models in the context of supergravity with orthogonal nilpotent su- perfields [1]. When local supersymmetry is gauge-fixed in the unitary gauge, these models describe theories with only a single real scalar (the inflaton), a graviton and a gravitino. Critically, there is no inflatino, no sgoldstino, and no sinflaton in these models. This dramatically simplifies cosmological models which can simultaneously describe inflation, dark energy and SUSY breaking. Introduction. In this letter we show that the re- unitary gauge, fixing local supersymmetry, there is only cently constructed models of supergravity with orthog- one real scalar φ(x), a massless graviton and a massive onal nilpotent superfields [1] significantly simplify con- gravitino. There is no sgoldstino, no sinflaton and no struction of cosmological models which simultaneously inflatino. An essential property of these models is that describe inflation, dark energy and SUSY breaking. Now the form of the potential is different from the standard one can achieve this goal using the absolutely minimal supergravity potentials, as shown in [1]: number of ingredients: graviton, gravitino, and a single K 2 2 real scalar, the inflaton. Our methods apply to a broad V = e (jDSW j − 3W ) : (1) ¯ ¯ class of inflationary theories, but they are especially suit- S=S=Φ−Φ=0 able for describing α-attractors [2], which provide a very First, the 3 scalars S, S¯ and Φ − Φ¯ vanish, there is no good fit to the latest cosmological data [3]. need to stabilize them. Secondly, the terms which would Orthogonal nilpotent superfields. Global supersym- normally be present in the potential, quadratic and linear in DΦW , are absent, despite the fact that DΦW can be metry models with orthogonal nilpotent superfields were φ studied in [4, 5]. Their generalization to local supergrav- arbitrary. This is because the auxiliary field F from the ity interacting with orthogonal nilpotent multiplets was inflaton multiplet is fermionic as a consequence of the presented in [1]. The models depend on two constrained orthogonality constraint SB = 0. chiral superfields1, S and Φ. A stabilizer S has a nilpo- Here we will study supergravity models with con- tency of degree two, Sn = 0 for n ≥ 2. This constraint strained superfields removes the complex scalar S(x), sgoldstino, from the ¯ ¯ 2 bosonic spectrum. The chiral inflaton multiplet Φ has as K(S ; S; Φ; Φ);W = f(Φ)S+g(Φ); S = SB = 0: (2) a first component a complex scalar Φ(x) = φ(x) + ib(x). The consistency of these models with the constraints was 1 ¯ One can form a real superfield B ≡ 2i Φ − Φ with the studied in [1] where it was shown that the superfield first component b(x), the sinflaton, and impose the or- K¨ahlerpotential can be also brought to the form thogonality constraint SB = 0. As a result, fields in the K(S ; S¯; Φ; Φ¯) = S S¯ + h(A)B2 : (3) inflaton multiplet are no longer independent: the sinfla- φ φ ton b(x), inflatino χ and the auxiliary F become func- This means, in particular that the K¨ahler potential in s tions of the spin 1/2 field χ in the S-multiplet. All of supergravity vanishes when the bosonic constraints are arXiv:1512.00546v1 [hep-th] 2 Dec 2015 s these fields vanish in the unitary gauge χ = 0, which imposed: fixes the local supersymmetry of the action. It follows from S2 = 0 and SB = 0 that B is nilpo- K(SS¯;Φ; Φ)¯ = 0 : (4) ¯ ¯ tent of degree 3, Bm = 0 for m ≥ 3. The second real S=S=Φ−Φ=0 superfield which can be formed from the chiral inflaton Note, however, that the bosonic constraints S = S¯ = 1 ¯ Φ − Φ¯ = 0, when deriving the supergravity action, have superfield is A = 2 Φ + Φ . It starts with one real inflaton scalar field, φ(x). to be applied only after the relevant derivatives over S and S¯ and Φ and Φ¯ are taken. The unusual property of these models is that in the For models with orthogonal nilpotent superfields, the inflaton action is very simple. Taking into account (4) and with a proper normalization of S, we find 1 We use bold face letters for superfields which in global case are ¯ functions of (x; θ;pθ). For example, the chiral superfieldp S means −1 ¯ 2 2 s 2 s φ 2 φ e Linfl = −KΦ;Φ¯ @Φ@Φ + [3g (Φ) − f (Φ)] : (5) that S = S(x) + 2θχ + θ F , and Φ = Φ(x) + 2θχ + θ F . Φ=Φ¯ 2 K¨ahler potentials in models with S2 = SB = 0. The new generation of models is different in two respects. Many successful inflationary models in supergravity are First of all, it describes a non-vanishing gravitino mass based on theories where the K¨ahlerpotential either van- ishes along the inflaton direction, or can be represented m3=2(φ) = g(φ) : (10) in such form after some K¨ahlertransformations, see for example [6{12, 14]. In models with S2 = SB = 0, where Additionally, it may also describe non-vanishing vacuum B = (Φ − Φ¯)=(2i), this requirement is naturally satisfied energy (cosmological constant) at the minimum of the (3), (4). potential. Without any loss of generality one may assume that the minimum of the potential corresponding to our Here we study the cosmological models with orthogo- vacuum state is at φ = 0. The cosmological constant is nal nilpotent superfields (2) over several different K¨ahler equal to potentials. The simplest K¨ahlerpotential with a flat di- rection describing a canonically normalized inflaton field Λ = f 2(0) − 3g2(0) : (11) φ = Re Φ is given by [7, 8] The conditionp that φ = 0 is a minimum implies that 1 0 0 K = (Φ − Φ)¯ 2 + SS:¯ (6) f (0) = 3g (0); up to small corrections vanishing in the 4 limit Λ ! 0. Here the geometry of the moduli space is flat. These conditions, plus the requirement that the func- We will be especially interested in the K¨ahlerpoten- tions f(φ) and g(φ) are holomorphic, leave lots of freedom tials for a broad class of cosmological attractors describ- to describe observational data. Indeed there are many ing Escher-type hyperbolic geometry [9, 10] of the in- ways to do so, depending on the choice of the K¨ahler flaton moduli space. Compatibility of the constraints potential. S2 = SB = 0 with the hyperbolic geometry is demon- Even though the expression of the potential V = strated in the Appendix. Examples of such K¨ahlerpo- f 2(φ) − 3g2(φ) is valid for all choices of the K¨ahlerpo- tentials include tentials described above, the field φ in the theories with the K¨ahlerpotentials (7) and (8) is not canonically nor- " ¯ 2 # 3 (1 − ΦΦ) ¯ malized. In the theory (7) the field φ is related to the K = − α log 2 + SS: (7) 2 (1 − Φ2)(1 − Φ ) canonically normalized inflaton field ' as follows: ' It describes hyperbolic geometry in disk variables. The φ = tanh p : (12) 6α same geometry can be described in half-plane variables by the K¨ahlerpotential Meanwhile for the theory (8) one has 2 p 3 (Φ + Φ)¯ − 2 ' K = − α log + SS:¯ (8) φ = e 3α : (13) 2 4ΦΦ¯ Thus, the potential V = f 2(φ) − 3g2(φ), expressed in These two versions correspond to equivalent ways of terms of a canonically normalized field ', depends on describing the K¨ahler geometry of α-attractors. See 2 the choice of the K¨ahlerpotential. In the next section refs. [10{12] for a detailed discussion of this issue. we will describe several realistic inflationary models in One may also consider these K¨ahlerpotentials with the this context. term SS¯ under the logarithm. In all of these cases, the ¯ K¨ahlerpotential vanishes for Φ = Φ, and KS;S¯ = 1 or can Inflationary models. be brought to K ¯ = 1 by a holomorphic transformation S;S 2 defined in [1]. The inflaton action is given by (5), and Model 1: f(φ) = Mφ + a; g(φ) = b. the inflaton potential is given by a simple expression The potential in this model is 2 2 V = f (φ) − 3g (φ) : (9) V = M 2φ4 + 2aMφ2 + a2 − 3b2 : (14) This result is similar to the expression V = f 2(φ) for The cosmological constant in this model, and all other the family of models with W = Sf(Φ) developed in [8]. models we present here, is equal to Λ = a2 − 3b2: (15) In realistic models we should have Λ ∼ 10−120 due to an 2 The third equivalent choice corresponds to the choice of K¨ahler 2 2 potential made in (6) when the nilpotency constraint B3 = 0 is almost precise cancellation between a and 3b in accor- taken into account.
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