Cosmology and supergravity

International Journal of Modern Physics A Vol. 31, No. 25 (2016) 1630044 (19 pages) c World Scientific Publishing Company DOI: 10.1142/S0217751X16300441 Cosmology and supergravity∗ Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. S. Ferrara Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland INFN-Laboratori Nazionali di Frascati, Via Enrico Fermi 40, 00044 Frascati, Italy Department of Physics and Astronomy, U.C.L.A., Los Angeles, CA 90095-1547, USA [email protected] A. Kehagias Physics Division, NTU Athens, 15781 Zografou, Athens, Greece [email protected] A. Sagnotti Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7 56126 Pisa, Italy [email protected] Published 8 September 2016 Abdus Salam was a true master of 20th Century Theoretical Physics. Not only was he a pioneer of the Standard Model (for which he shared the Nobel Prize with S. Glashow and S. Weinberg), but he also (co)authored many other outstanding contributions to the field of Fundamental Interactions and their unification. In particular, he was a major contributor to the development of supersymmetric theories, where he also coined the word “Supersymmetry” (replacing the earlier “Supergauges” drawn from String Theory). He also introduced the basic concept of “Superspace” and the notion of “Goldstone Fermion” (Goldstino). These concepts proved instrumental for the exploration of the ultraviolet properties and for the study of spontaneously broken phases of super Yang– Mills theories and Supergravity. They continue to play a key role in current developments in Early-Universe Cosmology. In this contribution we review models of inflation based on Supergravity with spontaneously broken local supersymmetry, with emphasis on the role of nilpotent superfields to describe a de Sitter phase of our Universe. Keywords: Supersymmetry; supergravity; cosmology; inflation; string theory. ∗ Based on an invited talk given at the Memorial Meeting for Nobel Laureate Prof. Abdus Salam’s 90th Birthday, 25–28 January 2016, NTU, Singapore. 1630044-1 S. Ferrara, A. Kehagias & A. Sagnotti Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. 1. Introduction Supergravity1 combines Supersymmetry with General Relativity (GR). This brings about scalar fields, some of which can play a natural role in the Early Universe. Nowadays it is well established that inflationary Cosmology is accurately described via the evolution of a single real scalar field, the inflaton, in a Friedmann, Lemaı̂tre, Robertson, Walker (FLRW) geometry.2 A scalar field associated with the Higgs particle was also recently discovered at LHC,3 confirming the interpretation of the Standard Model as a spontaneously broken phase (BEH mechanism) of a nonAbelian Yang–Mills theory.4 There is thus some evidence that Nature is inclined to favor, both in Cosmology and in Particle Physics, theories with scalar degrees of freedom, albeit in diverse ranges of energy scales. Interestingly, there is also a cosmological model where inflaton and Higgs fields are identified: this is the Higgs inflation model of Ref. 5, which rests on a nonminimal coupling h2 R of the Higgs field h to gravity. Another well-known example rests on an R + R2 extension of General Relativity (GR). This is the Starobinsky model of inflation,6,7 which is also conformally equivalent to GR coupled to a scalar field, the scalaron,8 with the special scalar potential  √ 2 2 (1) V = V0 1 − e− 3 φ , V0 ∼ 10−9 in Planck units . These two models (and also a more general class) give identical predictions9 for the slow-roll parameters ǫ and η, which are determined by the potential according to  2 V ′′ M2 V ′ , η = MP2 . (2) ǫ= P 2 V V The spectral index of scalar perturbations (scalar tilt) and the tensor-to-scalar ratio turn out to be ns = 1 − 6ǫ + 2 η ≃ 1 − 2 , N r = 16 ǫ ≃ 12 , N2 (3) where N= 1 MP2 Z φ φend V dφ V′ (4) is the total number of e-folds of inflation. An interesting modification of the Starobinsky potential, suggested by its embedding in R + R2 Supergravity,10,11 involves a deformation parameter α and reads11,12  √ 2 2 Vα = V0 1 − e− 3α φ . (5) It gives the same result of Eq. (3) for ns , but the tensor-to-scalar ratio is now r≃ 12α . N2 1630044-2 (6) Cosmology and supergravity This family of models provides an interpolation between the Starobinsky model (for α = 1) and Linde’s chaotic inflation model13 with a quadratic potential (in the limit α → ∞). The chaotic inflation model leads again to the scalar tilt (3), but now the tensor-to-scalar ratio becomes Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. r≃ 8 . N (7) The recent 2015 data analysis from Planck14 and BICEP215 favors ns ≈ 0.97 and r < 0.1, and thus the Starobinsky model, which lies well within the allowed parameter space due to the additional 1/N suppression factor r present in Eq. (3) as compared to Eq. (7). The form (5) for Vα can be further generalized, allowing for an arbitrary, mono tonically increasing function f tanh √ϕ6α , such that  ϕ Vα = V0 f tanh √ 6α 2  ϕ , ϕ → ∞ : f tanh √ 6α  → 1 − e− √ 2 3α φ + ··· . (8) These modifications led to the concept of α-attractors.12 This contribution is organized as follows. In Sec. 2 we describe the single-field inflation in Supergravity, in Sec. 3 we discuss inflation and supersymmetry breaking and in Sec. 4 we present some minimal Supergravity models of inflation. Nilpotent superfields and sgoldstino-less models are reviewed in Sec. 5, in Sec. 6 we discuss higher-curvature Supergravity and its dual standard Supergravity description, in Sec. 7 orthogonal nilpotent superfields are explored and Sec. 8 contains our conclusions and outlooks. Finally in App. A we briefly review constraint superfields which preserve N = 1 supersymmetry. 2. Single-Field Inflation in Supergravity We can now describe how N = 1 Supergravity can accommodate these “singlefield” inflationary models, explaining how to embed the inflaton ϕ in a general Supergravity theory coupled to matter in an FLRW geometry and the role of its superpartners. Under the assumption that no additional Supersymmetry (N ≥ 2) is restored in the Early Universe, the most general N = 1 extension of GR is obtained by coupling the graviton multiplet (2, 3/2) to a certain number of chiral multiplets (1/2, 0, 0), whose complex scalar fields are denoted by z i , i = 1, . . . , Ns /2 and to (gauge) vector multiplets (1, 1/2), whose vector fields are denoted by AΛ µ (Λ = 1, . . . , NV ). These multiplets can acquire supersymmetric masses, and in this case the massive vector multiplet becomes (1, 2(1/2), 0), eating a chiral multiplet in the supersymmetric version of the BEH mechanism. For Cosmology, the relevant part of the Lagrangian16,17 is the sector that couples the scalar fields to the Einstein–Hilbert action, described by L = −R − ∂i ∂̄ KDµ z i Dν z̄ ̄ g µν − V (z, z̄) + · · · , 1630044-3 (9) S. Ferrara, A. Kehagias & A. Sagnotti where K is the Kähler potential of the σ-model scalar geometry and the “dots” hide fermionic terms and gauge interactions. The scalar covariant derivative is Dµ z i = ∂µ z i + δΛ z i AΛ µ, (10) i where the δΛ z are Killing vectors. This term allows to write massive vector multiplets à la Stueckelberg. The scalar potential is   1 V (z i , z̄ ī ) = eG Gi G̄ (G−1 )i̄ − 3 + (Re fΛΣ )−1 DΛ DΣ , (11) 2 where, in terms of the superpotential W (z i ), Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. G = K + log |W |2 , Gij̄ = ∂i ∂j̄ K . (12) The first and third non-negative terms in Eq. (11) are usually referred to as “F” and “D” term contributions: together with the second, negative term, they encode the option of attaining unbroken Supersymmetry in Anti-de Sitter space. Alternatively, the potential can be recast in the more compact form V (z i , z̄ ı̄ ) = Fi F i + DΛ DΛ − 3|W |2 eK , (13) where
K Fi = e 2 W K,i + W,i , DΛ = G,i δΛ z i . (14) The D-term potential can endow a vector multiplet with a supersymmetric mass term, and can also give rise to a de Sitter phase, thanks to its non-negative contribution to the potential. Only F-breaking terms can thus give AdS phases. The (field dependent) matrices Re fΛΣ , Im fΛΣ provide the normalization of the terms quadratic in Yang–Mills curvatures. Their role in Cosmology deserves to be investigated further, since they give direct couplings of the inflaton to matter, which are relevant for the epoch of reheating. 3. Inflation and Supersymmetry Breaking In a given phase, unbroken Supersymmetry requires Fi = D Λ = 0 , (15) V = −3|W |2 eK . (16) so that These are Minkowski or AdS phases depending on whether or not W vanishes. On the other hand, supersymmetry is broken if at least one of the Fi or DΛ does not vanish. In phases with broken Supersymmetry one can have maximally symmetric AdS, dS or Minkowski vacua, so that one can accommodate both the inflationary phase (dS) and the subsequent Particle Physics (Minkowski) phase. However, it is not trivial to construct corresponding models, since the two scales are very different if Supersymmetry is at least partly related to the Hierarchy problem. 1630044-4 Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. Cosmology and supergravity In view of the negative term present in the scalar potential (11), it might seem impossible (or at least not natural) to retrieve a de Sitter phase for large values of a scalar field to be identified with the inflaton. The supersymmetric versions of the R + R2 (Starobinsky) model show how this puzzle is resolved: either the theory has (with F-terms) a no-scale structure, which makes the potential positive along the inflationary trajectory,18 or the potential is a pure D-term and is therefore positive.19 These models contain two chiral superfields (T, S),20,21 as in the old minimal version of R + R2 Supergravity,18 or one massive vector multiplet,10,11 as in the new minimal version, and attain unbroken Supersymmetry in a Minkowski vacuum at the end of inflation. In the framework of nilpotent superfield inflation,22 some progress was recently made23,24 on the problem of embedding two different supersymmetry breaking scales in the inflationary potential. The multiplet S, which does not contain the inflaton (T multiplet), is replaced by a nilpotent superfield satisfying S2 = 0 . (17) This condition eliminates the sgoldstino scalar from the theory, but its F-component still drives inflation, or at least participates in it. This mechanism was first applied to the Starobinsky model, replacing the S field by a Volkov–Akulov nilpotent field22 and then to general F-term induced inflationary models.25 Although the examples are so far restricted to the N = 1 → N = 0 breaking in four-dimensional supergravity, these types of construction are potentially very instructive for String Theory, where one readily looses control of the vacuum in the presence of broken supersymmetry.26 Orientifold vacua27 provide a natural and interesting entry point into this intricate dynamics, via the phenomenon of “brane SUSY breaking”.28 This rests on non-BPS combinations of branes and orientifolds that are individually BPS, and its simplest ten-dimensional setting was related to non-linear supersymmetry in Ref. 29. Recent work, starting from Ref. 25, linked it more clearly to the superHiggs effect in Supergravity,16 and also to the KKLT scenario of Ref. 30. Let us conclude this section, however, by recalling that a first attempt to make use of the nilpotent Volkov–Akulov multiplet in Cosmology, identifying the inflaton with the sgoldstino, was made in Ref. 31. 4. Minimal Models for Inflation and Supergravity This class includes models where the inflaton is identified with the sgoldstino and only one chiral multiplet T is used. However, the f (R) Supergravity models32 yield potentials that either have no plateau or, when they do, lead to AdS rather than to dS phases.34,35 This also reflects a no-go theorem.33 A way out of this situation was recently found with “α-scale Supergravity”:36 adding two superpotentials W+ + W− which separately give a flat potential along the inflaton (Re T ) direction can result in a de Sitter plateau for large Re T . 1630044-5 S. Ferrara, A. Kehagias & A. Sagnotti The problem with these models is that the inflaton trajectory is unstable in the Im T direction, but only for small inflaton field: modifications to the superpotential are advocated to generate a satisfactory inflationary potential. For single-field models and related problems, see also Ref. 37. R + R2 Supergravity, D-term inflation,11,38 α-attractor scenarios,39 no-scale inflationary models,20 and α-scale models36 have a nice SU (1, 1)/U (1) hyperbolic geometry for the inflaton superfield, with Rα = − 2 , 3α ns ≃ 1 − 2 , N r≃ 12 α , N2 (18) Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. where Rα is the curvature of the scalar manifold. 4.1. D-term inflation An appealing and economical class of models allows to describe any potential of a single scalar field which is the square of a real function:11 V (ϕ) = g2 2 P (ϕ) . 2 (19) These are the D-term models, which describe the self-interactions of a massive vector multiplet whose scalar component is the inflaton. Up to an integration constant (the Fayet–Iliopoulos term), the potential is fixed by the geometry, since the Kähler metric is
2 ds2 = dϕ2 + P ′ (ϕ) da2 . (20) After gauge fixing, the field a is absorbed by the vector, via da + gA, giving rise
2 2 to a mass term g2 P ′ (ϕ) A2µ (BEH mechanism). In particular, the Starobinsky model corresponds to √2 P (ϕ) = 1 − e − 3 ϕ , (21) but in all these examples there is no superpotential and only a de Sitter plateau is possible. At the end of inflation ϕ = 0, D = 0 and Supersymmetry is recovered in Minkowski space, since V = 0. 4.2. R + R2 supergravity There are two distinct classes of models, depending on the choice of auxiliary fields: old and new minimal models. The off-shell degrees of freedom contain the 6(= 10 − 4 diff) degrees of freedom of the graviton gµν and the 12(= 16 − 4 diff) degrees of freedom of the gravitino ψµ . The nB = nF off-shell condition requires six more bosons. There are two choices for the latter, which reflect the two minimal supegravity multiplets of the N = 1 theory: • old minimal: Aµ , S, P (6 DOF’s) • new minimal: Aµ , Bµν (6 DOF’s due to gauge inv. δBµν = ∂µ bν − ∂ν bµ ). 1630044-6 Cosmology and supergravity These 12B + 12F degrees of freedom must fill massive multiplets like Weyl2 : (2, 2(3/2), 1) , 2 Rold : 2(1/2, 0, 0) , 2 Rnew : (1, 2(1/2), 0) . (22) Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. After superconformal manipulations, these two theories can be turned into standard Supergravity coupled to matter. The new minimal gives D-term inflation as described before, while the old minimal gives F-term inflation with the two chiral superfields T (inflaton multiplet) and S (sgoldstino multiplet). The T submanifold is SU (1, 1)/U (1) with scalar curvature R = −2/3, and the no-scale structure of the Kähler potential is responsible for the universal expression  √ 2 2 2 (23) 1 − e− 3 ϕ , V = M 2 MPl along the inflationary trajectory where FS 6= 0, FT = 0, which identifies S with the sgoldstino. 4.3. Other models Several examples exist with two chiral multiplets of the same sort, for which FS leads to a de Sitter plateau with FT = 0, while at the end of inflation FS = FT = 0 and Supersymmetry is recovered. A class of models (α attractors) modify the superpotential but not the Kähler geometry of the original R+R2 theory, which now reads:12 W (S, T ) = S f (T ) , (24) 2 with scalar curvature Rα = − 3α . Along the inflationary trajectory the potential is positive since V ∼ |f |2 ≥ 0 . (25) An alternative class of models with opposite role for Kähler potential and superpotential rest on the choice of Eq. (24), combined however with the trivial Kähler geometry corresponding to
2 1 Φ + Φ̄ + S S̄ . (26) K= 2 The inflaton is now identified with ϕ = Im Φ, thus avoiding the dangerous exponential factor eK in the supersymmetric potential. Along the inflationary trajectory V (ϕ) ∼ |f (ϕ)|2 , (27) so that the inflaton potential is fully encoded in the superpotential shape. 5. Nilpotent Superfields and Sgoldstino-less Models In all the models reviewed so far it is difficult to exit inflation with Supersymmetry broken at a scale much lower than the de Sitter plateau (Hubble scale during inflation). A way to solve this problem is to introduce a nilpotent (Volkov–Akulov) 1630044-7 S. Ferrara, A. Kehagias & A. Sagnotti multiplet S satisfying40–43 the constraint of Eq. (17), so that the goldstino lacks its scalar partner, which is commonly called the sgoldstino. This solves the stabilization problem and gives rise to a de Sitter plateau. The first cosmological model with a nilpotent sgoldstino multiplet was a generalization of the Volkov–Akulov–Starobinsky supergravity,22 where V = e K(T ) KS−1 |f (T )|2 . S̄ W (S, T ) = S f (T ) , (28) Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. Two classes of models which incorporate separate scales of Supersymmetry breaking during and at the exit of inflation were then proposed. They rest on a trivial (flat) Kähler geometry K(Φ, S) =
2 1 Φ + Φ̄ + S S̄ , 2 (29) but differ in their supersymmetry breaking patterns during and after inflation. • In the first class of models23
W (Φ, S) = M 2 S 1 + g 2 (Φ) + W0 , (30) where g(Φ) vanishes at Φ = 0 and the inflaton ϕ is identified with its imaginary part. Along the inflaton trajectory Re Φ = 0, and the potential reduces to
V = M 4 |g(Φ)|2 2 + |g(Φ)|2 + V0 , V0 = M 4 − 3 W02 . (31) Assuming V0 ≃ 0, one finds H m3/2 = √ , 3 1 ESB = |FS | 2 = p HMPl > H , V = FS F S − 3W02 , while FΦ = 0 during inflation (Re Φ = 0). • In the second class of models24 the superpotential is  √  W (Φ, S) = f (Φ) 1 + 3 S , (32) (33) which combines nilpotency and no-scale structure. Here the function f (Φ) satisfies the conditions f¯(Φ) = f (−Φ̄) , f ′ (0) = 0 , f (0) 6= 0 . √ The scalar potential is of no-scale type, and letting Φ = (a + iϕ)/ 2, √ 2 2 FS F S = 3ea |f (Φ)|2 , V (a, ϕ) = F Φ FΦ = e a |f ′ (Φ) + a 2f (Φ)|2 . (34) (35) The field a is stabilized at a = 0, since f is an even function of a. During inflation a gets a mass O(H) without mixing with Φ and is rapidly driven to a = 0, so that the inflationary potential reduces to 2  iϕ V (a = 0, ϕ) = f ′ √ , V (0, 0) = 0 . (36) 2 1630044-8 Cosmology and supergravity These models lack the fine-tuning of the previous class (V0 = 0), and it is interesting to compare the supersymmetry breaking patterns. Here FS never vanishes, and at the end of inflation F S FS = 3 eG(0,0) = 3 m23/2 . (37) In particular, hF S iΦ=0 = √ 3 f¯(0) , m3/2 = |f (0)| , (38) Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. and the inflaton potential vanishes at the end of inflation. A choice that reproduces the Starobinsky potential is f (Φ) = λ − i µ1 Φ + µ2 e i √23 Φ . (39) Interestingly, ma and m3/2 depend on the integration constant λ, but V is independent of it, and hence the same is true for mϕ . 6. Higher-Curvature Supergravity and Standard Supergravity Duals Work in this direction started with the R + R2 Starobinsky model, whose supersymmetric extension was derived in the late 80s18,19 and was recently revived in view of the new CMB data.10,11,20,34 Models dual to higher-derivative theories give more restrictions than their bosonic counterparts or standard Supergravity duals. Theories with unconstrained superfields also include the Supergravity embedding of R2 duals, whose bosonic counterparts describe standard Einstein gravity coupled to a massless scalar field in de Sitter space. These theories were recently resurrected in Refs. 44 and 45. The R2 higher curvature Supergravity was recently obtained in both the old and new minimal formulations.46 In the old-minimal formulation, the superspace Lagrangian is α R R̄ D − β R2 F , (40) where R= Σ(S̄0 ) , S0 D̄α̇ R = 0 (41) is the scalar curvature multiplet, with Weyl and chiral weights (w = 1, n = 1). The dual standard Supergravity has Kähler potential and superpotential K = −3 log(T + T̄ − α S S̄) , W = T S − β S3 , (42) and the Kählerian manifold is SU (2, 1)/U (2). Note the rigid scale invariance of the action under T → e2 λ T , S → eλ S , 1630044-9 S0 → e−λ S0 . (43) S. Ferrara, A. Kehagias & A. Sagnotti If α = 0 S is not dynamical, and integrating it out gives an SU (1, 1)/U (1) σ-model with Kähler potential and superpotential 2 K = −3 log(T + T̄ ) , W = 2T 3 √ . 3 3β (44) Higher-curvature supergravities can be classified by the nilpotency properties of the chiral curvature R. Such nilpotency constraints give rise to dual theories with nilpotent chiral superfields.22 In particular, the constraint Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. R2 = 0 , (45) in R + R2 generates a dual theory where the inflaton chiral multiplet T (scalaron) is coupled to the Volkov–Akulov multiplet S S2 = 0 , D̄α̇ S = 0 . (46) For this theory (the V-A-S Supergravity), the Kähler potential and superpotential are K = −3 log(T + T̄ − S S̄) , W = M S T + f S + W0 , (47) respectively, and due to its no-scale structure the scalar potential is semi-positive definite V = |M T + f |2 . 3 (T + T̄ )2 In terms of the canonically normalized field r √2 2 SU (1, 1) φ T = e 3 + ia , (φ, a) ∈ , 3 U (1) the potential Eq. (48) becomes √ 2 2 M 2 √2 M2  1 − e− 3 φ + e − 2 3 φ a2 . V = 12 18 (48) (49) (50) Here a in the axion, which is much heavier than the inflaton during inflation M 2 −2 √ 2 φ0 M2 3 e . (51) m2φ ≃ ≪ m2a = 9 9 There are then only two natural supersymmetric models with genuine single-field φ inflation. One is the new-minimal R + R2 theory, where the inflaton has a massive vector as bosonic partner, and the V-A-S (sgoldstino-less) Supergravity just described. Another interesting example is the sgoldstino-less version of the RR̄ theory described before. This is obtained imposing the same constraint R2 = 0 as for the V-A-S Supergravity,47 and is dual to the latter with f = W0 = 0 . 1630044-10 (52) Cosmology and supergravity The corresponding potential V = M2 |T |2 M2 M 2 − 2√ 2 φ 2 3 = a , + e 12 18 3 (T + T̄ )2 (53) is positive definite and scale invariant. This model results in a de Sitter vacuum geometry with a positive vacuum energy M2 4 M . (54) 12 Pl On the other hand, the Volkov–Akulov model coupled to Supergravity involves two parameters, and its vacuum energy has an arbitrary sign. The pure V-A theory coupled to Supergravity has indeed a superfield action determined by22 Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. V (a = 0) = K = 3 S S̄ , W = f S + W0 , S2 = 0 . (55) Moreover, the cosmological constant turns out to be 1 2 |f | − 3|W0 |2 . (56) 3 The full-fledged component expression of the model, including all fermionic terms, was recently worked out.48,49 The higher-curvature supergravity dual50,51 is the standard (anti-de Sitter) supergravity Lagrangian augmented with the nilpotency constraint  2 R − λ = 0. (57) S0 Λ= This is equivalent to adding to the action the term 2  R − λ S03 F , σ S0 (58) where σ is a chiral Lagrange multiplier. A superfield Legendre transformation and the superspace identity     (Λ + Λ̄)S0 S̄0 D = ΛRS02 F + h.c. , (59) which holds up to a total derivative for any chiral superfield Λ, turn indeed the action into the V-A superspace action coupled to standard Supergravity with f = λ − 3 W0 . (60) Hence, supersymmetry is broken whenever 3 W0 6= λ 6= 0 . (61) In the higher-derivative formulation, the goldstino G is encoded in the Rarita– Schwinger field. At the linearized level around flat space   λ λ 3 γ µν ∂µ ψν − γ µ ψµ , δG = ǫ . (62) G=− 2λ 2 2 1630044-11 S. Ferrara, A. Kehagias & A. Sagnotti The linearized equation of motion for the gravitino reads   λ λ 1 γ µνρ ∂ν ψρ − γ µν ψν − γ µν ∂ν − γ µ G = 0 , 6 3 2 (63) and is gauge invariant under δψµ = ∂µ ǫ + λ γµ ǫ . 6 (64) Both the γ-trace and the divergence of the equation of motion yield Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. γ µν ∂µ ψν − γ µ ∂µ G = 0 , (65) so that gauging away the Goldstino G one recovers the standard formulation of a massive gravitino. Tables 1–3 summarize the various dualities linking higher-curvature supergravities in the old-minimal and new-minimal formulations with standard Supergravity.
. Old-minimal dualities: −ΦS0 S̄0 |D + W S03 |F , Φ = exp − K 3 Table 1. Higher Curvature Supergravity  ΦH = 1 − h SR , 0   R WH = W S R̄ S̄0  ΦS = 1 + T + T̄ − h(S, S̄) WS = T S − W (S) 0 ΦH = 1 WH = W ΦH = WH = Standard Supergravity ΦS = 1 + T + T̄  R S0  WS = −SW ′ (S) + W (S) −α SR S̄R̄ 0 0 T =−W ′ (S) ΦS = T + T̄ − αS S̄ 3 −β R 3 S0 WS = T S − βS 3 Table 2.  Nilpotent old-minimal dualities: −ΦS0 S̄0 |D + W S03 |F ,  . Φ = exp − K 3 Higher Curvature Supergravity ΦH = 1 − 1 R R̄ M 2 S0 S̄0 WH = W0 + ΦH = − M12 WH = ξ SR 0 + ΦS = T + T̄ − S S̄ 2 σR 2 S0 WS = M T S + f S + W0
S 2 = 0, f = ξ − 12 R R̄ S0 S̄0 ΦS = T + T̄ − S S̄ 2 σR 2 S0 WS = M T S (S 2 = 0) ΦH = 1 WH = W0 + σ Standard Supergravity ΦS = 1 − S S̄  R S0 −λ 2 WS = f S + W0 (S 2 = 0, f = λ − 3W0 ) 1630044-12 Cosmology and supergravity Table 3.   New-minimal dualities: Φ = exp − K . 3 Higher Curvature Supergravity   L log S LS̄  0 0 D   Wα S LS̄ W α S LS̄ 0 0 0 0 Standard Supergravity ΦS = −U exp U Wα (U )W α (U ) F ΦS = (T + T̄ ) exp V Wα  L S0 S̄0  Wα  L S0 S̄0  Wα (V )W α (V ) F Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. 7. Orthogonal Nilpotent Superfields We have seen so far that simple models of inflation, and in particular the supersymmetric version of the Starobinsky model, rest on a pair of chiral multiplet, the sgoldstino multiplet S and the inflaton multiplet T . Sgoldstino-less models are 2 obtained by replacing S by a nilpotent superfield (SN L = 0), which is the local version of the V-A multiplet. This setting should correspond to a linear model where the scalar partners of the goldstino are infinitely heavy, so that the sgoldstino becomes a non-dynamical composite field. Following Refs. 43 and 52, other types of constraints can be imposed, which remove other degrees of freedom from the T multiplet. The most interesting of them is the orthogonality constraint53–55 SN L (TON L − T̄ON L ) = 0 , 2 (SN L = 0) , (66) which also implies TON L − T̄ON L
3 = 0. (67) This constraint removes the inflatino (spin-1/2 partner of the inflaton), as well as the sinflaton (spin-0 partner of the inflaton), so that this description should correspond to a regime where the inflatino and the sinflaton are both infinitely heavy. The new aspect of these “non-chiral orthogonality constraints” is that the T auxiliary field FT becomes nilpotent, and therefore fails to contribute to the scalar potential, which takes the form V (ϕ = Re T ) = f 2 (ϕ) − 3g 2 (ϕ) , (68) for a quadratic Kähler potential and a superpotential of the form W (SN L , TON L ) = SN L f (TON L ) + g(TON L ) . (69) The potential V in Eq. (68) may or may not reproduce the inflaton trajectory for models with a “linear T multiplet”. This setting presents an advantage with respect to the linear T model, because it eliminates the sinflaton, thus bypassing the problems related to its stabilization. It also avoids goldstino-inflatino mixing, which makes matter creation in the Early Universe very complicated. In the unitary gauge, the inflatino field simply vanishes, since it is proportional to the goldstino.53,56 1630044-13 S. Ferrara, A. Kehagias & A. Sagnotti 2 Orthogonality constraints with SN L (SN L = 0). Table 4. SN L (TON L − T̄ON L ) = 0 (implies (TON L − T̄ON L ′ SN L T̄ON L (implies (implies sgoldstino-less, inflatino-less, sinflaton-less = 0) = chiral ′ SN L D̄ȧ TON L ′′ SN L TON L )3 sgoldstino-less, inflatino-less = 0) =0 ′′ 3 (TON L) sgoldstino-less, scalar-less = 0) Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. SN L Wα (VON L = 0 sgoldstino-less, gaugino-less In Table 4 we collect the various orthogonality constraints. The supergravity model for a matter multiplet T corresponding to the constraint ST = 0 was derived in Ref. 56. This model has been recently shown57 to describe the effective dynamics of a fermion, other than the N = 1 goldstino, which lives on a D̄3-brane world volume. 8. Conclusions and Outlook The orthogonality constraints in Eq. (66) and the resulting scalar potential in Eq. (68) allow the construction of MSIM (minimal supersymmetric inflationary models), which accommodate, with appropriate fine tuning, dark energy (cosmological constant Λ), the supersymmetry breaking scale m3/2 , and the inflationary Hubble scale H.54 A simplified class of models is obtained with (in MPl units) g(ϕ) = g0 = m3/2 , f (ϕ) = H fI (ϕ) + f0 , (70) where ϕ is the appropriate canonically normalized scalar field, whenever the Kähler potential is not quadratic but has the more general form as in Refs. 53 and 54. Here, fI (ϕ) is a function with the property fI (ϕ) → 1, (for ϕ large) while at the extremum of the potential ϕ = 0, fI (ϕ) = 0. Hence, the scalar potential satisfies V (ϕ = 0) = f02 − 3m23/2 = Λ, while for large ϕ, V (ϕ) → H 2 (as ϕ → ∞), for values of the parameters such that Λ ≈ 10−120 , m3/2 ≈ 10−16 and H = 10−5 , where we took the SUSY breaking scale at the end of inflation (approximate Minkowski spacetime) to be at the TeV scale as a minimal value, which is inspired by the current LHC results. Finally, we would like to note that microscopic models which may yield in suitable limits the non-linear realisations considered so far have been proposed in Refs. 58–60 and matter couplings to the inflation sector, with and without nonlinear superfields, were considered in Refs. 23, 24, 50, 61 and 62. This review reflects the lecture presented by SF at the 2016 Memorial Meeting for Prof. Salam, and overlaps in part with Ref. 63. 1630044-14 Cosmology and supergravity Acknowledgments We are grateful to I. Antoniadis, G. Dall’Agata, E. Dudas, F. Farakos, P. Fré, R. Kallosh, A. Linde, M. Porrati, A. Riotto, A. Sorin, J. Thaler, A. Van Proeyen, T. Wrase and F. Zwirner for useful discussions and collaboration on related projects. SF is supported in part by INFN-CSN4-GSS, while AS is supported in part by Scuola Normale Superiore and by INFN-CSN4-STEFI. Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. Appendix A. Constrained Superfields and N = 1 Supersymmetry Non-linear constraints involving a pair of chiral superfields (X, Wα ), (X, Uȧ ) can have solutions that differ sharply from the V-A case. Here Wα , Uȧ are the chiral superfields Wα = 1 2 D̄ Dα V , 4 Uȧ = D̄ȧ L , gauge field-strength multiplet , (A.1) L linear (or tensor) multiplet, D2 L = D̄2 L = 0 . (A.2) The (chiral) constraints in question are 1) X 2 = 0 , XWα = 0 , 2) X 2 = 0 , XUȧ = 0 , (A.3) (or XL = chiral) . (A.4) 2 N = 1 supersymmetry is broken solving X = 0, with the V-A solution  2  G , Gα , F , X = XN L = S = 2F (A.5) and then solving the second portions of Eqs. (A.3), (A.4), XN L Wα = 0 , XN L Uȧ = 0 . Using the fact that Wα , Uȧ have components   i µ ν β β β ȧ Wα = λα , Lα = δα D − (σ σ̄ )α Fµν , ∂αȧ λ̄ , 2   µ Uȧ = χ̄ȧ , Λȧβ = σȧβ (Hµ + i∂µ φ), ∂αȧ χ̄a , (A.6) (A.7) (A.8) where 1 ǫµνρσ ∂ ν bρσ , (A.9) 3! the constraints leave free the bosonic fields but express the gaugino (and tensorino) in terms of the V-A G goldstino43 according to Fµν = ∂µ Aν − ∂ν Aµ , Hµ = Gβ + O(G2 ) , λα = i Lαβ √ 2F (A.10) Gβ χ̄ȧ = − i Λα̇β √ + O(G2 ) . 2F (A.11) 1630044-15 S. Ferrara, A. Kehagias & A. Sagnotti The full solution can be obtained from the last component of the constraints by iteration. The constraint on the linear multiplet was considered in Ref. 60, and has the effect of leaving (φ, bµν ) in the spectrum. However, there is another solution to the constraints, where instead the chiral multiplet X is not the V-A multiplet but the constraints in Eqs. (A.3), (A.4) can be used to express X in terms of Wα (or Uȧ ). This is the case of the supersymmetric Born–Infeld and the non-linear tensor multiplet constraints of Bagger and Galperin63–66 where Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. 1) X = 2) X = W α Wα , m − D̄2 X̄ U α̇ Uα̇ . m − D̄2 X̄ (A.12) (A.13) The resulting Lagrangians, which are simply the F-components of X, describe a non-linear theory with N = 2 spontaneously broken to N = 1. References 1. D. Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D 13, 3214 (1976); S. Deser and B. Zumino, Phys. Lett. B 62, 335 (1976); For a review see: D. Z. Freedman and A. Van Proeyen, Supergravity (Cambridge University Press, Cambridge, 2012). 2. For recent reviews see: V. Mukhanov, Physical Foundations of Cosmology (Cambridge University Press, 2005); S. Weinberg, Cosmology (Oxford University Press, 2008). 3. ATLAS Collab. (G. Aad et al.), Phys. Lett. B 716, 1 (2012), arXiv:1207.7214 [hep-ex]; CMS Collab. (S. Chatrchyan et al.), Phys. Lett. B 716, 30 (2012), arXiv:1207.7235 [hep-ex]. 4. F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964); P. W. Higgs, Phys. Rev. 145, 1156 (1966); G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 44, 189 (1972). 5. F. L. Bezrukov and M. Shaposhnikov, Phys. Lett. B 659, 703 (2008), arXiv:0710.3755 [hep-th]. 6. A. A. Starobinsky, Phys. Lett. B 91, 99 (1980). 7. V. F. Mukhanov and G. V. Chibisov, J. Exp. Theor. Phys. Lett. 33, 532 (1981) [ibid. 33, 549 (1981)]. 8. B. Whitt, Phys. Lett. B 145, 176 (1984). 9. A. Kehagias, A. M. Dizgah and A. Riotto, Phys. Rev. D 89, 043527 (2014), arXiv:1312. 1155 [hep-th]. 10. F. Farakos, A. Kehagias and A. Riotto, Nucl. Phys. B 876, 187 (2013), arXiv:1307. 1137 [hep-th]. 11. S. Ferrara et al., Phys. Rev. D 88, 085038 (2013), arXiv:1307.7696 [hep-th]; J. Cosmol. Astropart. Phys. 1311, 046 (2013), arXiv:1309.1085 [hep-th]. 12. R. Kallosh, A. Linde and D. Roest, Phys. Rev. Lett. 112, 011303 (2014), arXiv:1310. 3950 [hep-th]; J. High Energy Phys. 1311, 198 (2013), arXiv:1311.0472 [hep-th]. 13. A. D. Linde, Phys. Lett. B 129, 177 (1983). 14. Planck Collab. (P. A. R. Ade et al.), Planck 2015 results. XX. Constraints on inflation, arXiv:1502.02114 [astro-ph.CO]. 15. BICEP2 and Planck Collabs. (P. A. R. Ade et al.), Phys. Rev. Lett. 114, 101301 (2015), arXiv:1502.00612 [astro-ph.CO]. 1630044-16 Cosmology and supergravity 16. 17. 18. 19. 20. 21. 22. 23. Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. E. Cremmer et al., Nucl. Phys. B 212, 413 (1983). J. Bagger and E. Witten, Phys. Lett. B 118, 103 (1982). S. Cecotti, Phys. Lett. B 190, 86 (1987). S. Cecotti et al., Nucl. Phys. B 306, 160 (1988). J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. Lett. 111, 111301 (2013) [Erratum: ibid. 111, 129902 (2013)], arXiv:1305.1247 [hep-th]. R. Kallosh and A. Linde, J. Cosmol. Astropart. Phys. 1306, 028 (2013), arXiv:1306. 3214 [hep-th]. I. Antoniadis et al., Phys. Lett. B 733, 32 (2014), arXiv:1403.3269 [hep-th]. R. Kallosh and A. Linde, J. Cosmol. Astropart. Phys. 1501, 025 (2015), arXiv:1408. 5950 [hep-th]. G. Dall’Agata and F. Zwirner, J. High Energy Phys. 1412, 172 (2014), arXiv:1411. 2605 [hep-th]. S. Ferrara, R. Kallosh and A. Linde, J. High Energy Phys. 1410, 143 (2014), arXiv:1408.4096 [hep-th]. W. Fischler and L. Susskind, Phys. Lett. B 171, 383 (1986); Phys. Lett. B 173, 262 (1986); E. Dudas et al., Nucl. Phys. B 708, 3 (2005), arXiv:hep-th/0410101; N. Kitazawa, Phys. Lett. B 660, 415 (2008), arXiv:0801.1702 [hep-th]; R. Pius, A. Rudra and A. Sen, J. High Energy Phys. 1410, 70 (2014), arXiv:1404.6254 [hep-th]. A. Sagnotti, Open strings and their symmetry groups, in Non-perturbative Quantum Field Theory, Cargese, ’87, eds. G. Mack et al.(Pergamon Press, 1988), p. 521, arXiv:hep-th/0208020; G. Pradisi and A. Sagnotti, Phys. Lett. B 216, 59 (1989); P. Horava, Nucl. Phys. B 327, 461 (1989); Phys. Lett. B 231, 251 (1989); M. Bianchi and A. Sagnotti, Phys. Lett. B 247, 517 (1990); Nucl. Phys. B 361, 519 (1991); M. Bianchi, G. Pradisi and A. Sagnotti, Nucl. Phys. B 376, 365 (1992); A. Sagnotti, Phys. Lett. B 294, 196 (1992), arXiv:hep-th/9210127; For reviews see: E. Dudas, Class. Quantum Grav. 17, R41 (2000), arXiv:hep-ph/0006190; C. Angelantonj and A. Sagnotti, Phys. Rept. 371, 1 (2002) [Erratum: ibid. 376, 339 (2003)], arXiv:hep-th/ 0204089. S. Sugimoto, Prog. Theor. Phys. 102, 685 (1999), arXiv:hep-th/9905159; I. Antoniadis, E. Dudas and A. Sagnotti, Phys. Lett. B 464, 38 (1999), arXiv:hep-th/9908023; C. Angelantonj, Nucl. Phys. B 566, 126 (2000), arXiv:hep-th/9908064; G. Aldazabal and A. M. Uranga, J. High Energy Phys. 9910, 024 (1999), arXiv:hep-th/9908072; C. Angelantonj et al., Nucl. Phys. B 572, 36 (2000), arXiv:hep-th/9911081. E. Dudas and J. Mourad, Phys. Lett. B 514, 173 (2001), doi:10.1016/S0370-2693(01) 00777-8, arXiv:hep-th/0012071. S. Kachru et al., Phys. Rev. D 68, 046005 (2003), doi:10.1103/PhysRevD.68.046005, arXiv:hep-th/0301240. L. Alvarez-Gaume, C. Gomez and R. Jimenez, Phys. Lett. B 690, 68 (2010), arXiv:1001.0010 [hep-th]; J. Cosmol. Astropart. Phys. 1203, 017 (2012), arXiv:1110. 3984 [astro-ph.CO]. S. Ketov,AIP Conf. Proc. 1241, 613 (2010), arXiv:0910.1165 [hep-th]; Phys. Lett. B 692, 272 (2010), arXiv:1005.3630 [hep-th]; PTEP 2013, 123B04 (2013), arXiv:1309. 0293 [hep-th]; S. V. Ketov and S. Tsujikawa, Phys. Rev. D 86, 023529 (2012), arXiv:1205.2918 [hep-th]; S. V. Ketov, On the supersymmetrization of f(R) gravity, arXiv:1309.0293 [hep-th]. J. Ellis, D. V. Nanopoulos and K. A. Olive, J. Cosmol. Astropart. Phys. 1310, 009 (2013), arXiv:1307.3537 [hep-th]. 1630044-17 Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. S. Ferrara, A. Kehagias & A. Sagnotti 34. S. Ferrara, R. Kallosh and A. Van Proeyen, J. High Energy Phys. 1311, 134 (2013), arXiv:1309.4052 [hep-th]. 35. S. Ferrara, A. Kehagias and M. Porrati, Phys. Lett. B 727, 314 (2013), arXiv:1310. 0399 [hep-th]. 36. D. Roest and M. Scalisi, Phys. Rev. D 92, 043525 (2015), arXiv:1503.07909 [hep-th]. 37. S. V. Ketov and T. Terada, J. High Energy Phys. 1412, 062 (2014), arXiv:1408.6524 [hep-th]. 38. S. Ferrara, P. Fre and A. S. Sorin, Fortsch. Phys. 62, 277 (2014), arXiv:1401.1201 [hep-th]; J. High Energy Phys. 1404, 095 (2014), arXiv:1311.5059 [hep-th]. 39. J. J. M. Carrasco et al., Phys. Rev. D 92, 041301 (2015), arXiv:1504.05557 [hep-th]. 40. M. Rocek, Phys. Rev. Lett. 41, 451 (1978). 41. U. Lindstrom and M. Rocek, Phys. Rev. D 19, 2300 (1979). 42. R. Casalbuoni et al., Phys. Lett. B 220, 569 (1989). 43. Z. Komargodski and N. Seiberg, J. High Energy Phys. 0909, 066 (2009), arXiv:0907. 2441 [hep-th]. 44. C. Kounnas, D. Lüst and N. Toumbas, Fortsch. Phys. 63, 12 (2015), arXiv:1409.7076 [hep-th]. 45. L. Alvarez-Gaume et al., Fortsch. Phys. 64, 176 (2016), arXiv:1505.07657 [hep-th]. 46. S. Ferrara, A. Kehagias and M. Porrati, J. High Energy Phys. 1508, 001 (2015), arXiv:1506.01566 [hep-th]. 47. S. Ferrara, M. Porrati and A. Sagnotti, Phys. Lett. B 749, 589 (2015), arXiv:1508. 02939 [hep-th]. 48. E. A. Bergshoeff et al., Phys. Rev. D 92, 085040 (2015) [Erratum: ibid. 93, 069901 (2016)], arXiv:1507.08264 [hep-th]; arXiv:1602.01678 [hep-th]. 49. F. Hasegawa and Y. Yamada, Phys. Rev. D 92, 105027 (2015), arXiv:1509.04987 [hep-th]. 50. E. Dudas et al., J. High Energy Phys. 1509, 217 (2015), arXiv:1507.07842 [hep-th]. 51. I. Antoniadis and C. Markou, Eur. Phys. J. C 75, 582 (2015), arXiv:1508.06767 [hep-th]. 52. A. Brignole, F. Feruglio and F. Zwirner, J. High Energy Phys. 9711, 001 (1997), arXiv:hep-th/9709111; Phys. Lett. B 438, 89 (1998), arXiv:hep-ph/9805282. 53. S. Ferrara, R. Kallosh and J. Thaler, Phys. Rev. D 93, 043516 (2016), arXiv:1512. 00545 [hep-th]. 54. J. J. M. Carrasco, R. Kallosh and A. Linde, Phys. Rev. D 93, 061301 (2016), arXiv:1512.00546 [hep-th]. 55. G. Dall’Agata and F. Farakos, J. High Energy Phys. 1602, 101 (2016), arXiv:1512. 02158 [hep-th]. 56. G. Dall’Agata, S. Ferrara and F. Zwirner, Phys. Lett. B 752, 263 (2016), arXiv:1509. 06345 [hep-th]. 57. B. Vercnocke and T. Wrase, Constrained superfields from an anti-D3-brane in KKLT, arXiv:1605.03961 [hep-th]. 58. R. Kallosh et al., Orthogonal nilpotent superfields from linear models, arXiv:1603. 02661 [hep-th]. 59. S. Ferrara et al., J. High Energy Phys. 1604, 065 (2016), arXiv:1603.02653 [hep-th]. 60. G. Dall’Agata, E. Dudas and F. Farakos, On the origin of constrained superfields, arXiv:1603.03416 [hep-th]. 61. S. Ferrara and M. Porrati, Phys. Lett. B 737, 135 (2014), arXiv:1407.6164 [hep-th]. 1630044-18 Cosmology and supergravity Int. J. Mod. Phys. A 2016.31. Downloaded from www.worldscientific.com by 95.224.132.72 on 05/19/17. For personal use only. 62. R. Kallosh, A. Linde and T. Wrase, J. High Energy Phys. 1604, 027 (2016), arXiv:1602.07818 [hep-th]. 63. S. Ferrara and A. Sagnotti, Some pathways in non-linear supersymmetry: Special geometry Born–Infeld’s, cosmology and dualities, arXiv:1506.05730 [hep-th]; Supersymmetry and inflation, PoS PLANCK 2015, 113 (2015), arXiv:1509.01500 [hep-th]; Fortsch. Phys. 64, 371 (2016). 64. S. Ferrara, A. Sagnotti and A. Yeranyan, J. High Energy Phys. 1505, 051 (2015), arXiv:1503.04731 [hep-th]. 65. J. Bagger and A. Galperin, Phys. Rev. D 55, 1091 (1997), arXiv:hep-th/9608177. 66. J. Bagger and A. Galperin, Phys. Lett. B 412, 296 (1997), arXiv:hep-th/9707061. 1630044-19