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 Théorique,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 superfields [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 (|DSW | − 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- Kählerpotential 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 Kähler 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,√θ). For example, the chiral superfield√ 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

Kähler 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 Kählerpotential either vanishes along the inflaton direction, or can be represented m3/2(φ) = g(φ) . (10) in such form after some Kählertransformations, 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 Kähler equal to potentials. The simplest Kählerpotential with a flat direction describing a canonically normalized inflaton field Λ = f 2(0) − 3g2(0) . (11) φ = Re Φ is given by [7, 8] The condition√ 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 Kählerpoten- 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 Kähler 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 Kählerpo- f 2(φ) − 3g2(φ) is valid for all choices of the Kählerpo- tentials include tentials described above, the field φ in the theories with the Kählerpotentials (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 ϕ as follows: ϕ It describes hyperbolic geometry in disk variables. The φ = tanh √ . (12) 6α same geometry can be described in half-plane variables by the Kählerpotential Meanwhile for the theory (8) one has

2 √ 3 (Φ + Φ)¯ − 2 ϕ 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 ϕ, depends on describing the Kähler geometry of α-attractors. See 2 the choice of the Kählerpotential. 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 Kählerpotentials with the this context. term SS¯ under the logarithm. In all of these cases, the ¯ Kählerpotential 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 Kähler 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. It is related to (7) and (8) by a change of dance with a string landscape scenario. The gravitino√ variables [10–12], see also Appendix. mass is m3/2 = b, which nearly coincides with a/ 3. For 3 b ∼ 10−15 one can have the gravitino mass in the of- α-attractor model of the canonically normalized field ϕ: ten discussed TeV range. To have a proper amplitude of M 2 ϕ scalar perturbations one should have M ∼ 10−5 a, b. V = tanh2 √ + a2 − 3b2 . (20) 2 6α If we consider a model with the simplest canonical Kählerpotential (6), the potential (14) is quartic with re- The gravitino mass is m3/2 = b. spect to the canonically normalized inflaton field, which q q rules out this simple model. M 2 2 2 m2 2 2 Model 4: f(φ) = 2 φ + a , g(φ) = 2 φ + b The situation instantly improves in the theory with the logarithmic Kählerpotential (7), which yields the In this model one has following potential in terms of the canonically normalized M 2 − 3m2 V = φ2 + a2 − 3b2 . (21) field ϕ: 2 ϕ ϕ V = M 2 tanh4 √ + 2aM tanh2 √ + a2 − 3b2 . (16) In the theory with the logarithmic Kähler potential (7) 6α 6α the potential of a canonically normalized inflaton field becomes This is the typical T-model α attractor potential [2]. In- ϕ 2 2 flation occurs at the plateau where tanh √ ≈ 1. In this M − 3m 2 ϕ 2 2 6α V = tanh √ + a − 3b . (22) regime the second term in (16) is much smaller than the 2 6α first term, and both terms are much greater than Λ, so This model is very similar to Model 3, but the gravitino inflation is described by the quartic T-model potential q m2 2 2 mass is φ-dependent, m3/2 = φ + b . ϕ 2 V = M 2 tanh4 √ . (17) 6α Model 5: f(φ) = pF 2(φ) + a2, g(φ) = pG2(φ) + b2 The observational predictions of this model for α . 10 In this model practically coincide with the predictions of the simpler 2 ϕ 2 2 2 2 p 2 2 model V = M 2 tanh √ , for the same number of e- V = F (φ) − 3G (φ) + a − 3b , m3/2 = G (φ) + b . 6α foldings N [2]. However, at the end of inflation in the (23) model (17) the inflaton field begins to oscillate in the ap- Because of the freedom of choice of the holomorphic func- proximately quartic potential ∼ ϕ4. The average equa- tions F and G, one can have a wide variety of potentials tion of state during this stage is the same as of the hot fitting all observational data even if the fields φ is canon- plasma, p = ρ/3, as if reheating finishes immediately ically normalized, with the Kählerpotential (6), see e.g. after inflation. This increases the required number of e- [14]. Meanwhile in the theories with the Kählerpotential foldings by ∆N ∼ 3 [13]. In its turn, this leads to a slight (7) one finds a family of T-model α-attractors with increase of the spectral index n , which may provide even ϕ ϕ s V = F 2(tanh √ ) − 3G2(tanh √ ) + a2 − 3b2 . (24) better fit to the recent Planck data. 6α 6α Model 2: f(φ) = Mφ2 + a, g(φ) = mφ2 + b For a wide range of functions F and G, these theories have universal cosmological predictions for α 10 The potential is . and any given number of e-foldings: ns = 1 − 2/N, 2 V = (M 2 − 3m2)φ4 + 2(Ma − 3m b)φ2 + a2 − 3b2 . (18) r = 12α/N [2]. However, by a proper choice of the func- tion F one can modify the required number of e-foldings N, which can be useful for tuning the predictions for n . This model is very similar to the previous one, but s there is one potentially interesting difference: The grav- p 2 2 2 Model 6: f(φ) = (1 − φ) + a , g(φ) = b itino mass depends on the inflaton, m3/2 = mφ + b. It is a particular version of Model 5 for F (φ) = M(1 − φ) q M 2 2 2 and G(φ) = 0. This yields Model 3: f(φ) = 2 φ + a , g(φ) = b 2 2 2 2 The potential is V = M (1 − φ) + Λ, Λ = a − 3b , m3/2 = b. (25)

2 Using the half-plane Kählerpotential (8) and the relation M 2 2 2 √ V = φ + a − 3b . (19) − 2 ϕ 2 φ = e 3α (13) one finds √ 2 The potential for φ is exactly quadratic, plus a cosmo- 2 − 2 ϕ V = M 1 − e 3α + Λ . (26) logical constant. In the theory with the logarithmic Kählerpotential This represents the family of E-model α-attractors [2, 9], (7) this potential becomes a potential for the simplest which reduces to the Starobinsky model for α = 1, Λ = 0 4

¯ and m3/2 = 0. Meanwhile our class of theories describes try ZZ < 1, E-models with arbitrary α, Λ and m3/2. " ¯ 2 # Conclusions. As one could see from the previous sec- 3 (1 − ZZ) ¯ K = − α log 2 + SS, tion, it is very easy to formulate and analyze models with 2 (1 − Z2)(1 − Z ) orthogonal nilpotent fields. Previously, it was a much W = A(Z) + SB(Z) , (27) more complicated task because of certain constraints imposed on inflationary models with nilpotent fields, see e.g. or a half-plane geometry, T + T¯ > 0, [15–17] and the more advanced models presented in [12]. These constraints where required for simplification of the 3 (T + T¯)2 K = − α log + SS,¯ fermionic sector, but they are no longer required in the 2 4T T¯ new class of models where the fermionic sector is trivial W = G(T ) + SF (T ) . (28) because the inflatino disappears in the unitary gauge. As a result, we have lots of flexibility in finding economical or a Killing adapted geometry models containing only inflaton, graviton and gravitino, h Φ − Φ¯ i and yet capable of simultaneously describing inflation, K = −3α log cosh √ + SS,¯ dark energy and SUSY breaking. 6α Φ Φ The absence of the inflatino also helps us argue that W = A(tanh √ ) + SB(tanh √ ) 6α 6α there there is no problem with the unitarity bound during √ √ 2 Φ 2 Φ inflation in this class of models. The effective cutoff in su- = G(e 3α ) + SF (e 3α ) . (29) pergravity is the scale at which scattering amplitudes vio- late unitarity bound. In the theories with nilpotent√ fields, In all of these models the Kählerpotential and the super- 2 2 1/4 this cutoff is expected at Λ ' (H + m3/2) > H, in potential, separately, are related by a change of variables units Mp = 1 [1, 5, 16,√ 18]. During inflation with H 1, the UV cut-off Λ > H is much higher than the typi- √ 1 + Z Φ 2 Φ T = ,Z = tanh √ ,T = e 3α . (30) cal energy of inflationary quantum fluctuations ∼ H. In 1 − Z 6α general, there could be some additional contributions to scattering due to gravitino-inflatino mixing, but in the Now we would like to impose the orthogonality constraint theory that we consider there is no inflatino, and there- on our superfields. The Killing-adapted variable Φ in (29) fore no violation of the unitarity bound is expected dur- is an unconstrained superfield whose scalar part is not ing inflation at sub-Planckian energy density. restricted by the boundaries. Therefore we may start by We hope to return to this issue in the future, simulta- imposing the orthogonality and the nilpotency constraint neously with investigation of reheating in the new class of in the form models. In particular, we expect that the absence of the S2 = 0, S(Φ − Φ¯) = 0, (Φ − Φ¯)n = 0, n ≥ 3 , (31) inflatino should significantly simplify the theory of nonthermal gravitino production by an oscillating inflaton Using the relation between these variables, one can de- field [18–21]. rive the related constraints for the disk and half-plane variables Z, and T , respectively. We are grateful to S. Ferrara and J. Thaler for many enlightening discussions and collaboration. JJMC is sup- S(Φ−Φ¯) = 0 ⇒ S(Z−Z¯) = 0 ⇒ S(T−T¯) = 0. ported by the European Research Council under ERC- STG-639729, ‘Strategic Predictions for Quantum Field It can be shown also that the Kählerpotentials in (27), Theories’. The work of RK and AL is supported by the (28), (29), take the form of eq. (3) due to orthogonality SITP, and by the NSF Grant PHY-1316699. The work of constraint. AL is also supported by the Templeton foundation grant ‘Inflation, the Multiverse, and Holography.’

[1] S. Ferrara, R. Kallosh and J. Thaler, “Cosmology with orthogonal nilpotent superfields,” to appear. [2] R. Kallosh and A. Linde, “Universality Class in Confor- Appendix A. Hyperbolic geometry models with mal Inflation,” JCAP 1307, 002 (2013) [arXiv:1306.5220 orthogonal nilpotent superfields [hep-th]]. S. Ferrara, R. Kallosh, A. Linde and M. Por- rati, “Minimal Supergravity Models of Inflation,” Phys. Rev. D 88, no. 8, 085038 (2013) [arXiv:1307.7696 [hep- Three equivalent versions of α-attractor models with th]]. R. Kallosh, A. Linde and D. Roest, “Superconfor- 2 S (x, θ) = 0 and hyperbolic geometry of the inflaton mal Inflationary α-Attractors,” JHEP 1311, 198 (2013) moduli space [9, 10] are given either by a disk geome- [arXiv:1311.0472 [hep-th]]. 5

[3] P. A. R. Ade et al. [Planck Collaboration], “Planck 2015 713, 365 (2012) [arXiv:1111.4397 [hep-ph]]. results. XX. Constraints on inflation,” arXiv:1502.02114 [14] R. Kallosh, A. Linde and A. Westphal, “Chaotic Inflation [astro-ph.CO]. P. A. R. Ade et al. [BICEP2 and in Supergravity after Planck and BICEP2,” Phys. Rev. Keck Array Collaborations], “BICEP2 / Keck Array D 90, no. 2, 023534 (2014) [arXiv:1405.0270 [hep-th]]. VI: Improved Constraints On Cosmology and Fore- [15] R. Kallosh and A. Linde, “Inflation and Uplifting with grounds When Adding 95 GHz Data From Keck Array,” Nilpotent Superfields,” JCAP 1501, no. 01, 025 (2015) arXiv:1510.09217 [astro-ph.CO]. [arXiv:1408.5950 [hep-th]]. [4] Z. Komargodski and N. Seiberg, “From linear SUSY to [16] G. Dall’Agata and F. Zwirner, “On sgoldstino-less su- constrained superfields,” J. High Energy Phys. 09 (2009) pergravity models of inflation,” JHEP 1412, 172 (2014) 066 [arXiv:0907.2441 [hep-th]]. [arXiv:1411.2605 [hep-th]]. [5] Y. Kahn, D. A. Roberts and J. Thaler, “The goldstone [17] R. Kallosh, A. Linde and M. Scalisi, “Inflation, de Sit- and goldstino of supersymmetric inflation,” JHEP 1510, ter Landscape and Super-Higgs effect,” JHEP 1503, 111 001 (2015) [arXiv:1504.05958 [hep-th]]. (2015) doi:10.1007/JHEP03(2015)111 [arXiv:1411.5671 [6] A. S. Goncharov and A. D. Linde, “Chaotic Inflation Of [hep-th]]. A. B. Lahanas and K. Tamvakis, “Inflation The Universe In Supergravity,” Sov. Phys. JETP 59, 930 in no-scale supergravity,” Phys. Rev. D 91, no. 8, (1984) [Zh. Eksp. Teor. Fiz. 86, 1594 (1984)]. A. B. Gon- 085001 (2015) [arXiv:1501.06547 [hep-th]]. R. Kallosh charov and A. D. Linde, “Chaotic Inflation in Supergrav- and A. Linde, “Planck, LHC, and α-attractors,” Phys. ity,” Phys. Lett. B 139, 27 (1984). A. Linde, “Does the Rev. D 91, 083528 (2015) [arXiv:1502.07733 [astro- first chaotic inflation model in supergravity provide the ph.CO]]. best fit to the Planck data?,” JCAP 1502, no. 02, 030 [18] R. Kallosh, L. Kofman, A. D. Linde and A. Van Proeyen, (2015) [arXiv:1412.7111 [hep-th]]. “Superconformal symmetry, supergravity and cosmol- [7] M. Kawasaki, M. Yamaguchi and T. Yanagida, “Natural ogy,” Class. Quant. Grav. 17, 4269 (2000) [Class. Quant. chaotic inflation in supergravity,” Phys. Rev. Lett. 85, Grav. 21, 5017 (2004)] [hep-th/0006179]. 3572 (2000) [hep-ph/0004243]. [19] R. Kallosh, L. Kofman, A. D. Linde and A. Van Proeyen, [8] R. Kallosh and A. Linde, “New models of chaotic “Gravitino production after inflation,” Phys. Rev. D 61 inflation in supergravity,” JCAP 1011, 011 (2010) (2000) 103503 [hep-th/9907124]. [arXiv:1008.3375 [hep-th]]. R. Kallosh, A. Linde and [20] G. F. Giudice, I. Tkachev and A. Riotto, “Nonthermal T. Rube, “General inflaton potentials in supergravity,” production of dangerous relics in the early universe,” Phys. Rev. D 83, 043507 (2011) [arXiv:1011.5945 [hep- JHEP 9908, 009 (1999) [hep-ph/9907510]. G. F. Giudice, th]]. A. Riotto and I. Tkachev, “Thermal and nonthermal pro- [9] R. Kallosh and A. Linde, “Escher in the Sky,” duction of gravitinos in the early universe,” JHEP 9911, arXiv:1503.06785 [hep-th]. 036 (1999) [hep-ph/9911302]. [10] J. J. M. Carrasco, R. Kallosh, A. Linde and D. Roest, [21] H. P. Nilles, M. Peloso and L. Sorbo, “Nonthermal pro- “The Hyperbolic Geometry of Cosmological Attractors,” duction of gravitinos and inflatinos,” Phys. Rev. Lett. 87, Phys. Rev. D 92 (2015) 4, 041301 [arXiv:1504.05557 051302 (2001) [hep-ph/0102264]. H. P. Nilles, M. Peloso [hep-th]]. and L. Sorbo, “Coupled fields in external background [11] J. J. M. Carrasco, R. Kallosh and A. Linde, “Cosmo- with application to nonthermal production of gravitinos,” logical Attractors and Initial Conditions for Inflation,” JHEP 0104, 004 (2001) [hep-th/0103202]. H. P. Nilles, Phys. Rev. D 92, no. 6, 063519 (2015) [arXiv:1506.00936 K. A. Olive and M. Peloso, “The Inflatino problem in su- [hep-th]]. pergravity inflationary models,” Phys. Lett. B 522, 304 [12] J. J. M. Carrasco, R. Kallosh and A. Linde, “α- (2001) [hep-ph/0107212]. Attractors: Planck, LHC and Dark Energy,” JHEP 1510, 147 (2015) [arXiv:1506.01708 [hep-th]]. [13] F. L. Bezrukov and D. S. Gorbunov, “Distinguishing between R2-inflation and Higgs-inflation,” Phys. Lett. B