Maximal Supersymmetry and B-Mode Targets
MMaximal Supersymmetry and B-Mode Targets
Renata Kallosh , Andrei Linde , Timm Wrase , Yusuke Yamada SITP and Department of Physics, Stanford University, Stanford, California 94305, USA Institute for Theoretical Physics, TU Wien, A-1040 Vienna, Austria
Abstract
Extending the work of Ferrara and one of the authors [1], we present dynamical cosmological mod-els of α -attractors with plateau potentials for 3 α = 1 , , , , , ,
7. These models are motivated bygeometric properties of maximally supersymmetric theories: M-theory, superstring theory, and max-imal N = 8 supergravity. After a consistent truncation of maximal to minimal supersymmetry in aseven-disk geometry, we perform a two-step procedure: 1) we introduce a superpotential, which stabi-lizes the moduli of the seven-disk geometry in a supersymmetric minimum, 2) we add a cosmologicalsector with a nilpotent stabilizer, which breaks supersymmetry spontaneously and leads to a desirableclass of cosmological attractor models. These models with n s consistent with observational data, andwith tensor-to-scalar ratio r ≈ − − − , provide natural targets for future B-mode searches. Werelate the issue of stability of inflationary trajectories in these models to tessellations of a hyperbolicgeometry. [email protected], [email protected], [email protected], [email protected] a r X i v : . [ h e p - t h ] A p r ontents N = 1 supersymmetric minimum . . . . . . . . . . . . . . 135.2 Step 2: Introducing a cosmological sector . . . . . . . . . . . . . . . . . . . . . . 15 Introduction
We would like to specify targets for the future B-mode detectors for the ratio of the tensor toscalar fluctuations, r = A t A s . We propose models of inflation with definite values of r which canbe validated/falsified either by the detection of primordial gravity waves, or by the improvedbounds on r . At present, the bound is considered to be r ≤ × − [2] at 95% confidencelevel. There are many inflationary models consistent with this bound, see for example, [3–5]where the future CMB observations are described.Here we will focus on inflationary α -attractor models [6–12], based on the hyperbolic geom-etry of a Poincar´e disk. Such a disk is beautifully represented by Escher’s picture Circle LimitIV with radius squared R = 3 α . The hyperbolic geometry of the disk has the following lineelement ds = dx + dy (cid:16) − x + y α (cid:17) , (1.1)which describes a disk with a boundary so that x + y < α . More details on this are given inSec. 2. The original derivation of this class of models was based on superconformal symmetryand its breaking.At present one can view α -attractor cosmological models with a plateau potential as provid-ing a simple explanation, due to the hyperbolic geometry of the moduli space, of the equationrelating the tilt of the spectrum n s to the number of e-folding of inflation N : n s ≈ − N . (1.2)This equation is valid in the approximation of a large number of e-foldings N and it is in a goodagreement with the data. In addition to providing an equation for n s , the hyperbolic geometryleads to the B-mode prediction for r = A t A s : r ≈ α N = R N , (1.3)for reasonable choices of the potential and for α not far from 1. General expressions for n s and r with their full dependence on N and on α are also known for a large class of models [8]. Theywere derived in the slow roll approximation and they are more complicated than expressionsin (1.2), (1.3). Both CMB observables n s and r follow from the choice of the geometry and arenot very sensitive to the changes in the potential due to attractor properties of these models.The experimental value of the scalar tilt suggests that n s ≈ − pN with p = 2 is a goodfit to the data. Here p controls the order of the pole in the kinetic term of the inflaton and p = 2 corresponds to a second order pole, see (1.1). Various considerations leading to thiskind of relation between n s and N were suggested in the past. For example, in [13], using theequation of state analysis, it was argued that robust inflationary predictions can be definedby two constants of order one, p and q , so that at large N , n s = 1 − pN and r = 24 qN p . We3xplain in Sec. 3 why in hyperbolic geometry p = 2 and 6 q = R . Related ideas were developedin [14–16]. Specific examples of such models with plateau potentials and α = 1 include theStarobinsky model [17], Higgs inflation [18] and conformal inflation models [6].The α -attractor models in N = 1 supergravity, starting with [8], may have any value of α and, therefore any value of r . An example of an N = 1 supergravity model with a very lowlevel of 3 α = 1 / α -attractor models [6–12], can be related to string theory in the followingsense: the effective supergravity model is based on two superfields, one is the inflaton, theother one is often called a stabilizer. It is a nilpotent superfield which is present on the D3brane [20, 21].When the geometry of these models is associated with half-maximal N = 4 supergravity [22]and the maximal N = 4 superconformal theory [23], one finds [10] that the lowest value of 3 α in these models is 1. It corresponds to a unit size Escher disk with R = 3 α = 1 . (1.4)Note that the relevant value of r is three times smaller than that of the Starobinsky model,Higgs inflation model and conformal inflation models, corresponding to α = 1, and provides awell motivated B-mode target r ∼ − , as explained in [10].In [1] it was shown that starting with gravitational theories of maximal supersymmetry,M-theory, string theory and maximal supergravity, one finds a seven-disk manifold defined byseven complex scalars. Two assumptions were made in [1]:The first assumption was that there exists a dynamical mechanism which realizes someconditions on these scalars, given in eq. (4.17) in [1]. If these conditions can be realizeddynamically, it would mean that in each case the kinetic term of a single remaining complexscalar is defined by a hyperbolic geometry with R = 3 α = 1 , , , , , , . (1.5)The second assumption in [1] was that these models can be developed further to produce theinflationary potential of the α -attractor models with a plateau potential in a way consistentwith the constraints. This would make the models proposed in [1] legitimate cosmologicalmodels, with specific predictions for n s in (1.2) and r in (1.3), (1.5).The purpose of this paper is to show how to construct such dynamical models, thereby val-idating the assumptions made in [1]. This will explain how starting from from the seven-diskmanifolds of maximal N = 8 supersymmetry models to derive the minimal N = 1 supersym-metric cosmological models with B-mode targets scanning the region of r between 10 − and10 − . 4 Capturing infinity in a finite space: plateau potentials
Escher was inspired by islamic tilings in Alhambra and he produced beautiful art using atessellation of the flat surface. The line element of it is ds = dXd ¯ X = dx + dy , X = x + i y . (2.1)A tessellation is the tiling of a plane using one or more geometric shapes, called tiles, withno overlaps and no gaps. Consider a simple example when in the plane the whole surface iscovered with equilateral triangles, as in Fig. 1.Figure 1: A tessellation of the flat surface plane covered with equilateral triangles. The shiftof the whole plane over the distance AB, will cover the underlying pattern again. This is a translation of the plane. We can also turn the duplicate through 60 degrees about the pointC, and we see that it covers the original pattern exactly. This is a rotation . Also after a reflection in the line PQ, the pattern remains the same.
The symmetry elements of the tessellation there include translations, rotations and re-flections , for example X → X + a, ¯ X → ¯ X + ¯ a , X → Xe i β , ¯ X → ¯ Xe − i β , X → − X , (2.2)and combinations of these. Escher has reached a perfection in his tessellations of the flat surface,see for example Fig. 2 where he had to cut a repeating pattern to fit it into a finite space ofthe picture. For a long time Escher struggled to produce an infinitely repeating pattern in afinite figure. His desire was to capture infinity in a finite space .It was Coxeter who gave Escher the idea for the Poincar´e disk. When Escher saw the figureof the tessellation of the hyperbolic plane by triangles produced by Coxeter in [24], see Fig. 3,left, he realized that this solves his problem. The figure’s hyperbolic tiling, with triangular tilesdiminishing in size and repeating (theoretically) infinitely within the confines of a circle, wasexactly what Escher had been looking for in order to capture infinity in a finite space . Thisallowed him to produce his well-known Circle Limit woodcuts, see the Angels and Devils CircleIV, in Fig. 3, right. 5igure 2:
Left: Escher’s tessellation of the flat surface plane with Angels and Devils design. Right: Escher’stessellations of the flat surface plane with Lizard/Fish/Bat design.
Figure 3:
Left: Coxeter’s tessellation of a hyperbolic plane by triangles (Poincar´e disk model). Right: Escher’stessellation for the hyperbolic tiling for the woodcut Angels and Devils, “Circle Limit IV”.
The complex disk coordinates Z describing the hyperbolic geometry are particularly suitablefor providing a mathematical meaning to the concept of capturing infinity in a finite space .Namely, a line element of the Poincar´e disk of radius squared 3 α = 1 can be given as [12] ds = dx + dy (1 − x − y ) = dZd ¯ Z (1 − Z ¯ Z ) = 12 dϕ + dθ cos ( √ θ ) , Z = tanh (cid:16) ϕ + i θ √ (cid:17) . (2.3)The angular variable θ is periodic but the variable ϕ is unrestricted. x + y < ⇐⇒ tanh (cid:18) ϕ √ (cid:19) < − ∞ < ϕ < + ∞ , < θ < π . (2.4)We have a map from a finite variable x + y < −∞ < ϕ < + ∞ , realized by the fact that tanh (cid:0) ϕ/ √ (cid:1) <
1: the origin of the plateaupotential for inflation in Fig. 4 can be traced to Escher’s concept of capturing infinity in afinite space .We will see that in our cosmological models the angular variable θ will be quickly stabilizedat θ = 0 whereas the inflaton field ϕ will have a plateau type potential V ∼ tanh (cid:0) ϕ/ √ (cid:1) incanonical variables, corresponding to a simple potential in the disk variables V ∼ Z ¯ Z .6 φ V Figure 4: A plot of V ( ϕ ) = tanh (cid:0) ϕ/ √ (cid:1) .The Cayley transform relates the upper half plane coordinate X , Im X >
0, to the interiorof the disk coordinate Z , | Z | <
1. For cosmological models this relation was studied in [25]: X = ˜ x + i˜ y = i 1 + Z − Z , ˜ y > , Z = X − i X + i , Z ¯ Z < . (2.5)Tessellation of the hyperbolic half-plane are defined by its symmetries, by M¨obius transfor-mations . The line element in half-plane variables, see the left part of Fig. 5, with ˜ y > ds hp = − dXd ¯ X ( X − ¯ X ) = d ˜ x + d ˜ y y . (2.6)It corresponds to the unit size disk geometry ds d = dZd ¯ Z (1 − Z ¯ Z ) = dx + dy (1 − x − y ) = ds hp . (2.7)Figure 5: Left: Escher’s tessellation of a hyperbolic half-plane with “Angels and Devils” design. Right: Cayleytransform of upper complex half-plane to a hyperbolic disk
The symmetries in half plane include: translation of the real part of X , dilatation ofthe entire plane, inversion, and reflection of the real part of X : X → X + b , ¯ X → ¯ X + b , (2.8)7 → a X , (2.9) X → − /X , (2.10) X + ¯ X → − ( X + ¯ X ) . (2.11)The first three separate transformations can be also given in the form X → aX + bcX + d , ∆ ≡ ad − bc (cid:54) = 0 , (2.12)where a, b, c, d are real parameters. The first one, the shift of the real part, is the case of c = 0, a = d = 1, the second one, rescaling, is b = c = 0, ad = 1, the third one, inversion, is a = d = 0, b/c = − R = 3 α the metric of the disk and half plane become, respectively ds d = 3 α dZd ¯ Z (1 − Z ¯ Z ) = dx + dy (cid:16) − x + y α (cid:17) , (2.13) ds hp = − α dXd ¯ X ( X − ¯ X ) = d ˜ x + 3 α d ˜ y y . (2.14)The curvature of the moduli space associated with the metric in (2.13) or (2.14) is given by R = − α . (2.15) In the form (2.14) it is particularly clear what is the origin of the n s and r equations in (1.2)and (1.3). For a constant axion ˜ x the hyperbolic geometry line element is ds hp | ˜ x = c = 3 α d ˜ y ˜ y . (3.1)It was explained in [9] that in general, if one starts with a kinetic term for scalars in the form L kin = a p dρ ρ p , (3.2)and assumes that inflation takes place near ρ = 0, so that the potential is V ∼ V (1 − cρ + . . . ) , c > , (3.3)8hen one finds that at large N (assuming p > n s = 1 − pp − N , r = 8 c p − p − a p − p ( p − pp − N pp − . (3.4)Note the following features of the general pole inflation models [9] described above • For p = 2 the model displays an attractor behavior, where the dependence on c in thepotential (3.3) is absent (without absorbing such a dependence into a redefinition of theresidue of the pole as studied in [26] ). In such a case (3.4) simplifies to n s = 1 − N , r = a p N . (3.5)This case is realized in hyperbolic geometry with 2 a p = R Es . It is interesting that theattractor features of cosmological models starting with the hyperbolic geometry followfrom one of the symmetries shown in (2.9). Namely, the kinetic term in (3.2) for p = 2 isinvariant under the change of the parameter c in the potential since if and only if p = 2we have for cρ = ρ (cid:48) that dρ ρ = d ( ρ (cid:48) ) ( ρ (cid:48) ) . The corresponding symmetry of the hyperbolic halfplane is the dilatation. • This dilatation of the half plane in eq. (2.9), leading to the attractor property of thisparticular pole inflation, will be shown below to also lead to the stabilization of theinflationary trajectory, once this symmetry as well as the inversion symmetry in (2.10)are implemented as symmetries of the K¨ahler potential.
The potential in N = 1 supergravity depends on a K¨ahler potential K and a superpotential W V = e K ( | DW | − | W | ) . (4.1)Here we explain the choices of the K¨ahler frame, following [11], emphasizing the relation tothe elements of the tessellation and the stability of the inflaton directions. In [1] the standardform of the K¨ahler potential was used describing a unit size Poincar´e disk K O = − ln( − i( τ − ¯ τ )) . (4.2)The corresponding line element/kinetic term for the scalar ds = dτ d ¯ τ ( τ − ¯ τ ) (4.3) In the models of pole inflation in [26] a slightly different framework was proposed. It was suggested to set c = 1 via a rescaling of ρ followed by a change of the kinetic term a p → ˜ a p = c − p a p . τ (cid:48) = aτ + bcτ + d , ad − bc (cid:54) = 0 , (4.4)where a, b, c, d are real parameters. The axion-dilaton pair, using notation in [11], is for 3 α = 1 τ = χ − i e −√ ϕ . (4.5)We are not in space-time anymore but in the moduli space of scalars fields, therefore insteadof X = ˜ x + i˜ y in eq. (2.6) we are using the holomorphic variable τ ( x ), where x ≡ x µ denotesthe space-time dependence of τ .Note that the tessallation of the hyperbolic plane in τ variables consists of a few independentoperations (subgroups of the M¨obius group). The hyperbolic line element in (2.6), (4.3) isinvariant under all transformations of this group. In terms of the axion field χ ( x ) and thescalar field ϕ ( x ) in (4.5) the relevant symmetries in (2.8) are a shift of the axion by a constant and a reflection of the axion in (2.11) χ ( x ) → χ ( x ) + b , (4.6) χ ( x ) → − χ ( x ) . (4.7)Let us look at the relevant tessellations of the half plane in Fig. 5 left and compare it with thetessellations in Fig. 1 of the flat full plane. These two symmetries in (4.6) and (4.7) are theobvious ones. According to (4.6) one can shift the Angels and Devils to the right (for positive b ) or to the left (for negative b ), the same way as it is done in Fig. 1 of the flat full plane.There the whole plane, not only an upper half of it, is shifted to the right by the distance AB,and the pattern covers the plane again. According to (4.7) we can choose a vertical line, like inFig. 1 it is a line PQ, and we can make a reflection in this line, preserving the pattern, see alsoFig. 2 left, which shows the Angels and Devils away from the boundary of a half plane, whereone can see clearly the existence of such PQ lines. Altogether, this is a convincing argument togive the symmetries of the hyperbolic plane in (4.6) and (4.7) a name: Tessellation Set 1 .The other two symmetries of the geometry in (4.3) are the inversion and the scaling τ → − /τ , (4.8) τ → a τ . (4.9)These two symmetries are absent in a full plane, see for example Fig. 2 left. However, in thehalf plane which has a boundary, these symmetries control the fact that near the boundarythe Angels and Devils in Fig. 5 left are getting smaller, still preserving the pattern. These arerather non-trivial tessellations inherited from the finite size hyperbolic disk tessellations in Fig.3. These are symmetries responsible for capturing infinity in a finite space. We will give thesymmetries of the hyperbolic plane in (4.8) and (4.9) the name: Tessellation Set 2 . The K¨ahler potential in (4.2) is invariant under the axion shift symmetry (4.6) and axionreflection (4.7), i.e. under Tessellation Set 1. However, it breaks the remaining symmetries f the geometry: inversion symmetry (4.8) and the scaling symmetry (4.9), it is not invariantunder Tessellation Set 2. In case when the axion Re τ is an inflaton field, and the K¨ahler potential is of the form e K = 1(Im τ ) α , (4.10)the potential tends to have a run-away factor depending on the sinflaton, Im τ : V = 1(Im τ ) α ( | DW | − | W | ) . (4.11)This is known in string theory as the K¨ahler moduli problem. In the string theory/supergravitycontext the KKLT construction [27] is one way to stabilize these type of moduli; another one isLVS [28]. In the axion monodromy inflation [29], this problem was addressed in models withoutsupersymmetry, where the 2-form axion does not have a susy partner.For α -attractor models a new K¨ahler frame was proposed in [11] where it was argued thatin case that the pseudo-scalar is a heavy field and the scalar field is a light one, it is relativelyeasy to stabilize the axion. The corresponding K¨ahler potential takes the form K new = −
12 ln (cid:16) − ( τ − ¯ τ ) τ ¯ τ (cid:17) . (4.12)In terms of symmetries corresponding to a set of a hyperbolic plane tessellations, it is compli-mentary to the K¨ahler potential in (4.2). The K¨ahler potential in (4.12) is invariant under the inversion symmetry (4.8) and the scal-ing symmetry (4.9), i.e. under Tessellation Set 2. However, it breaks the remaining symmetriesof the geometry: the axion shift symmetry (4.6) and the axion reflection symmetry (4.7). Itis not invariant under Tessellation Set 1 . The K¨ahler potential has a shift symmetry for adilaton, complemented by a rescaling of the axion [11] χ → ad χ , ϕ → ϕ + 1 √ ad . (4.13)At χ = 0 one finds that e K new | τ = − ¯ τ = e − ln( − ( τ − ¯ τ )24 τ ¯ τ ) | τ = − ¯ τ = 1 . (4.14)and there is no run-away behavior of the potential. Good choices of the superpotentials producedesirable cosmological models with V new = | DW new | − | W new | . (4.15)Thus, in the new frame, the K¨ahler potential has an inflaton flat direction, which is lifted bythe superpotential. 11learly, the relation between the old frame (4.2) and the new frame (4.12) satisfies the K¨ahlersymmetry K new → K − ln Φ ¯Φ , W new → W · Φ , (4.16)and one theory can be related to the other. However, the potential in (4.15) is expected to besmall to describe slow roll inflation, it will produce small deviation of the flatness of the theoryin the inflaton direction. Therefore the new choice of the frame (4.12) has an advantage overthe choice in (4.2) with regard to the stabilization of the inflationary trajectory. Starting with K O one has to look for a W O which will make the inflationary potential approximately flatdespite the fact that we started with the strong run-away potential in (4.11). Instead, we startwith K new with a flat inflaton potential broken by a small W new . In technical terms, it was aflip of one set of a hyperbolic plane tessellation in (4.6), (4.7), which we called Tessellation Set1, to another set in hyperbolic plane tessellation in (4.8), (4.9), which we called TessellationSet 2, which has created a desirable stabilization effect. The seven disk model in [1] has the following K¨ahler potential K = − (cid:88) i =1 log( T i + ¯ T i ) . (5.1)Here we use for each of the seven unit size disks the following variables T i = i τ i and T i = e −√ ϕ i + i χ i . It was argued in [1] that such a K¨ahler potential for the seven disks can be derivedfrom maximally supersymmetric models with 8 Majorana spinors, M-theory, superstring theory,maximal supergravity, by a consistent truncation to a minimal supersymmetric model with asingle Majorana spinor.Following [11] and as explained in the previous section, we make a choice of the K¨ahlerframe where the K¨ahler potential has an inflaton shift symmetry: we use the following K¨ahlerpotential K = − (cid:88) i =1 log ( T i + ¯ T i ) T i ¯ T i . (5.2)An equivalent form can be given in disk variables K = − (cid:88) i =1 log (1 − Z i ¯ Z i ) (1 − Z i )(1 − ¯ Z i ) . (5.3)At this level the seven-disk theory has an unbroken N = 1 supersymmetry and no potential.Our first step is to find a superpotential and scalar potential depending on the disk variables T i N = 1 supersymmetric minimum producing the constraints on the moduli whichwere imposed in [1], namely:3 α = 7 : T = T = T = T = T = T = T ≡ T α = 6 : T = T = T = T = T = T ≡ T , T = const3 α = 5 : T = T = T = T = T ≡ T , T = T = const3 α = 4 : T = T = T = T ≡ T , T = T = T = const3 α = 3 : T = T = T ≡ T , T = T = T = T = const3 α = 2 : T = T ≡ T , T = T = T = T = T = const3 α = 1 : T ≡ T , T = T = T = T = T = T = const (5.4)Let us explain for example how the single α -attractor model with 3 α = 7 case is achievedhere. The kinetic terms for the seven complex moduli originally is L kin = − (cid:88) i =1 dT i d ¯ T i ( T i + ¯ T i ) . (5.5)The K¨ahler potential of each T i corresponds to the α -attractor with 3 α = 1. When the conditionthat T = T = T = T = T = T = T ≡ T (5.6)is enforced the kinetic term becomes L kin = − dT d ¯ T ( T + ¯ T ) . (5.7)This is an α -attractor model with 3 α = 7 with regard to the kinetic term. In the following, wewill show how the above identifications (5.4), or equivalently, 3 α = 1 , . . . , N = 1 supersymmetric minimum We would like to dynamically enforce that all seven fields (or a subset thereof, see below) movesynchronously during inflation so that T i − T j = 0. This can be done via a superpotential thatgives a very large mass to the combinations T i − T j : W = M (cid:88) ≤ i 6. For the kinetic terms, restrictingto the saxions (the axions work in the same way), we have L kin = − (cid:88) i ∂ µ u i ∂ µ u i = − (cid:16) ∂ µ u∂ µ u + (cid:88) j =1 ∂ µ ˜ u j ∂ µ ˜ u j (cid:17) . (5.11)The canonical masses in these new variables at the supersymmetric minimum are then givenby m u = m a = 0 , m u j = m a j = 3136 e − √ u M . (5.12)Thus we have kept the ‘diagonal’ directions massless, while making all the ‘transverse’ directionsarbitrarily heavy.Analogously we can ‘identify’ any number 1 ≤ n < W = M (cid:16) (cid:88) ≤ i 7, critical point equations are satisfiedfor S = 0, T i = T j , i, j ∈ { , . . . , n } , Re( T k ) = c , k ∈ { n + 1 , . . . , } , Im( T i ) =Im( T k ) = 0, and f ( n Re( T )) f (cid:48) ( n Re( T )) = 0. In this case the F-term for S is simply D S W = f ( n Re( T ). α = 7 case For n = 7 the superpotential is given by W = M (cid:88) ≤ i We begin with models in half-plane variables with K = − α T + ¯ T ) T ¯ T + S ¯ S , (6.1)and with the simple superpotential W = mS (1 − T ) . (6.2)It is convenient to switch to the new variables T = e − √ α ϕ (1 + i √ a ). During inflation, thefield a is stabilized at a = 0, whereas the field ϕ is the canonically normalized inflaton field18ith the plateau potential V = m (cid:16) − e − √ α ϕ (cid:17) . (6.3)This is the simplest representative of E-models introduced in [7, 8, 10]. The theory of initialconditions for inflation in similar theories of inflation with plateau potentials was developedin [12, 32]. The amplitude of the scalar perturbations in such models matches the Plancknormalization for m ≈ − √ α [33]. The set of the E-model potentials for 3 α = 1 , , , ..., N . By solving field equations in the leading approximation in 1 /N , onefinds [35] ϕ N ≈ (cid:114) α N α . (6.4)For α -attractors with a more general class of potentials V = m (cid:0) − e − √ α ϕ (cid:1) n one has ϕ N ≈ (cid:114) α N n α . (6.5) φ V Figure 6: E-model potentials for 3 α = 1 , , , ..., 7. The central (blue) line corresponds to 3 α = 1. Thethird line from the center (green) corresponds to the supergravity generalization of the Starobinskymodel with α = 1 [7, 34]. The outer line shows the potential with 3 α = 7. The value of the inflatonfield ϕ is shown in Planck units M p = 1; the height of the potential is shown in units of m . Now we will consider α -attractors in disk variables. The simplest model is described by K = − α − Z ¯ Z ) (1 − Z )(1 − ¯ Z ) + S ¯ S , W = m S Z . (6.6)This leads to the T-model potential of the inflaton field [8, 10] V = m tanh ϕ √ α . (6.7)19he set of the T-model potentials for 3 α = 1 , , , ..., - - φ V Figure 7: T-model potentials for 3 α = 1 , , , ..., 7. The central (blue) line corresponds to the diskwith 3 α = 1. The third line from the center (green) corresponds to the conformal inflation model with α = 1 [6]. The outer line shows the potential with 3 α = 7. The value of the inflaton field ϕ is shownin Planck units M p = 1; the height of the potential is shown in units of m . In the leading approximation in 1/N, the predictions of E-models and T-models for theobservational parameters n s and r coincide with each other for any given α . However, the valueof the inflaton field corresponding to the moment when the remaining number of e-foldings ofinflation becomes equal to some number N is slightly different, because of the different shapeof the potential at small ϕ [36]: ϕ N ≈ (cid:114) α N α . (6.8)For T-models with more general potentials V = m tanh n ϕ √ α one has ϕ N ≈ (cid:114) α N n α . (6.9) In the two-disk case, we have just 4 moduli and would like to see the stabilization of 3 of them.By making some simple choices of the superpotential function f ( T ) we can show that therelevant potentials acquire a plateau shape describing α -attractors with 3 α = 1 or 2. A morecomplete dynamical picture will be revealed since we will be able to study not only the massesof the non-inflaton stabilized moduli near the inflaton trajectory, but the global properties ofthe models.Here we start with K = − 12 log ( T + ¯ T ) T ¯ T − 12 log ( T + ¯ T ) T ¯ T + S ¯ S . (6.10)20tarting with a two-disk model, each a unit size one, we have two options: One can freezedynamically one of the directions, e.g. T , by stabilizing it at some point T = c , and get3 α = 1 for inflation driven by the field T . Alternatively, one can enforce T = T and getinflation with 3 α = 2. T = 1, 3 α = 1 As an example of the model of two-disk model with 3 α = 1, we will study the theory with thesuperpotential W = mS (1 − T ) + M (1 − T ) . (6.11)Here m ∼ − is the inflaton mass scale, up to a factor O (1), and M (cid:29) m is the stabilizingmass parameter. During inflation at S = 1 − T = 0 supersymmetry is unbroken in the T , T directions, but broken in the S direction: D T W = − mS + K T (cid:0) mS (1 − T ) + M (1 − T ) (cid:1) (cid:12)(cid:12)(cid:12) S =1 − T =0 = 0 ,D T W = 2 M (1 − T ) + K T (cid:0) mS (1 − T ) + M (1 − T ) (cid:1) (cid:12)(cid:12)(cid:12) S =1 − T =0 = 0 ,D S W = m (1 − T ) + ¯ S ( (cid:0) mS (1 − T ) + M (1 − T ) (cid:1) (cid:12)(cid:12)(cid:12) S =1 − T =0 = m (1 − T ) (cid:54) = 0 . (6.12)As before, we will use the new variables u i and a i , where T i = e −√ u i (1 + i √ a i ). Thevariables u i and a i become canonical in the limit of small a i . For M = 0 and a = a = 0, thepotential depends only on the field u . One can easily check that for any u and M (cid:54) = 0, thefields u , a and a vanish at the (local) minimum of the potential. The potential of the field u for u = a = a = 0 is the standard potential (6.3) of the E-model α -attractor with 3 α = 1: V = m (cid:16) − e −√ u (cid:17) . (6.13)The masses of the fields a i are always greater than the Hubble constant, so they are stronglystabilized at a = a = 0. The same is true for the field u , which is strongly stabilized at u = a = a = 0 for M > m √ . The inflationary potential in terms of u and u for a i = 0 isshown in Fig. 8. 21igure 8: Potential V ( u i ) in the theory defined via equations (6.10) and (6.11). For M = 10 m we have a heavy and stabilized u . The remaining field u plays the role of the inflaton with theE-model α -attractor potential V = m (cid:0) − e −√ u (cid:1) corresponding to 3 α = 1. The fields are shownin Planckian units M p = 1; the height of the potential is shown in units of m . α = 1 / α = 2 / Now we study a model illustrating the dynamical merger of two α -attractors with α = 1 / α -attractor with α = 2 / 3. We will consider the superpotential W = mS (cid:18) − T + T (cid:19) + M ( T − T ) . (6.14)As we will see, the corresponding potential V is very different from the one studied in theprevious subsection.We will investigate the potential V using variables u i and a i , where T i = e −√ u i (1 + i √ a i ).One finds that the critical point equations ∂ a i V = 0 are solved for a = a = 0 and this solutioncorresponds to a minimum of the potential in these two directions. For large M , the two fields u i are merged during inflation into one canonically normalized field ϕ = ( u + u ) / √ χ = ( u − u ) / √ M (cid:29) m , ϕ > log √ Mm and χ = 0 the field χ acquires a tachyonic mass, which leads to a tachyonicinstability for the χ direction.The nature of this effect is illustrated by Figs. 9 and 10. The colored area in Fig. 9 showsthe part of the potential with V < m ; the red area in the upper right corner shows an infinitelylong inflationary plateau asymptotically approaching V = m . The fields tend to roll downfrom this plateau towards the narrow gorge, along which the fields u and u coincide. But atthe early stages of this process the fields fall towards one of the two stable inflaton directions22hown by the two blue valleys in Fig. 9 along which one of the fields u i remains nearly constant,see the field flow diagram in Fig. 10.Figure 9: The potential of the canonical fields u i for M = 500 m . The height of the potential isshown in units m . The colored area shows the part of the potential with V < m ; the red areain the upper right corner shows the inflationary plateau asymptotically approaching V = m . Fromthere on, inflation continues when the field rolls towards the narrow gorge along which the fields u and u coincide. This gorge is seen as a narrow diagonal cut beginning at the center of the figure.For ϕ = ( u + u ) / √ > log √ Mm , the gorge is wide, as seen in the right upper corner of thefigure. The stable inflaton directions are shown by the blue valleys along which one of the u i remainsnearly constant. The potential along each such direction asymptotically behaves as the α -attractorpotential with 3 α = 1. The merger of these two attractors, which occurs when ϕ becomes smallerthan log √ Mm , corresponds to the phase transition to 3 α = 2. Note that this process is very slow to develop because the tachyonic mass of this field χ at the inflationary plateau with M (cid:29) m and ϕ > log √ Mm is exponentially small, muchsmaller than the Hubble constant, just like the inflaton mass. Thus, the field χ , as well as theinflaton field, will experience inflationary fluctuations. But the general evolution of these fieldsis dominated by their classical rolling, as shown in Fig. 10.The potential along each of the two blue valleys in Fig. 9 asymptotically behaves as the α -attractor potential with α = 1 / 3. When ϕ becomes smaller than log √ Mm , the tachyonic massof the field χ vanishes, this field becomes stable at χ = 0, and the two inflaton directions mergeinto the inflationary gorge with χ = 0. This corresponds to symmetry restoration between u and u . The potential along the bottom of this gorge is V = m (1 − e − ϕ ) , which correspondsto α = 2 / χ for a while remain large,until the positive mass squared of the field χ becomes greater than the Hubble constant squared H ∼ V / 3. This happens at ϕ < log √ Mm . This concludes the merger of the two inflatondirections and the phase transition from α = 1 / α = 2 / u u Figure 10: Gradients of the inflationary potential. In the area where the slow-roll regime is possible(colored areas in Fig. 9), these gradients describe the inflationary slow-roll evolution of the fields u i .The fields starting their motion at the inflationary plateau in the right upper corner in Figs. 9 and Fig.10 typically fall down towards one of the two streams, each of which can be approximately describedas an α -attractor with 3 α = 1. Then these streams merge into one stream corresponding to 3 α = 2. It is important to discuss the necessary condition that the effective trajectory 3 α = 2 lasts for N e-foldings. The two axions have a positive mass O ( H ) even for large ϕ , so we do not find anyfurther constraints from their stabilization. However, as mentioned above, the χ -direction mayacquire a tachyonic mass for sufficiently large ϕ , and there the inflaton trajectory bifurcatesinto two trajectories with 3 α = 1.Let us estimate how large M is required to be in order to stabilize the field χ along thetrajectory 3 α = 2 for the last N e-foldings. According to (6.4), the inflaton field value as afunction of the e-folding number N is given by ϕ N ≈ (cid:113) α log N α for the E-models we study.Thus in our example with 3 α = 2 we find ϕ N = log(2 N ) , and the mass of χ is m χ = 2 e − ϕ (128 M − m e ϕ + m e ϕ ) = 32 M − m N + m N N . (6.15)From this expression, we obtain the following condition for M : M > N / m N . For N = 55, this constraint becomes M (cid:38) m . To producethe observational result for the amplitude os scalar perturbations we need m ∼ − in Planckunits, so M should be greater than O (10 − ). This implies that the inflationary trajectory χ = 0,which corresponds to α -attractor with α = 2 / 3, becomes stable for M (cid:38) − GeV, whichis well below the Planck scale and may correspond to the string/GUT scale, which is quitenatural in our context. 24t is also useful to know the condition that the mass of χ becomes larger than H , so that χ fluctuations are suppressed during the last N e-folds. From Eq. (6.15), the condition m χ > H can be satisfied for M > N √ m. (6.17)For N = 55, this constraint reads M (cid:38) m , and this can also be satisfied naturally, if M isa Planck/string/GUT scale parameter.Note that all the way until the field ϕ becomes smaller than log √ Mm , the mass squaredof the field χ remains much smaller than H . In this regime, classical evolution of all fieldstypically brings them towards one of the two valleys corresponding to α = 1 / 3, but details ofthis evolution may be somewhat affected by quantum fluctuations of the field χ ; see a discussionof a very similar regime in [37]. However, for M > N √ m the evolution of the universe duringthe last N e-foldings is described by the standard single field α -attractor theory with α = 2 / In this section, we discuss the two-disk model in the disk variables Z i with K = − (cid:88) i =1 log (1 − Z i ¯ Z i ) (1 − Z i )(1 − ¯ Z i ) + S ¯ S , W = m S ( Z + Z ) + M ( Z − Z ) . (6.18)As in the case with half-plane variables T i , the stabilization of inflationary trajectory at Z = Z leads to the α -attractor potential with 3 α = 2. We use the parametrization Z i =tanh √ ( ϕ i + i θ i ) such that ϕ i and θ i become canonical variables on the inflationary trajectory θ i = 0. As in the E-model discussed in the previous section, for a large M , χ = √ ( ϕ − ϕ )becomes heavy and is stabilized at the origin. There, the direction ϕ = √ ( ϕ + ϕ ) becomesthe inflaton. Note that the axionic directions θ i always have positive masses and are stabilizedat θ i = 0. The scalar potential in this T-model is given by V = m tanh ϕ and is shown inFigs. 11 and 12. 25igure 11: A view from above on the inflationary potential. The red area shows the inflationaryplateau asymptotically rising to V = m . The blue lines show valleys corresponding to inflationary at-tractors with α = 1 / 3. After the merger, they form a narrow gorge with the potential V = m tanh ϕ ,which corresponds to the T-model α -attractor with α = 2 / ϕ , which corresponds to the merger point, depends on the parameter M . For sufficientlylarge M , the last 60 e-folds of inflation are described by the single attractor with α = 2 / Figure 12: A part of the Figure 11 shown in the same range of ϕ as in Fig. 11, but in a verynarrow range of χ near the inflaton direction χ = 0. One can easily recognize the T-model potential V = m tanh ϕ with 3 α = 2, which is produced by merging of the inflaton directions with 3 α = 1shown by the blue lines in the Figure 11. To make the merger last for N e-foldings, the mass parameter M should be sufficiently large,as in the case of the E-model studied in the previous section. The axionic directions do not26cquire tachyonic mass even for large ϕ but the field χ does. On the 3 α = 2 trajectory, thelast e-folding number N and the value of ϕ satisfy the same relation as in eq. (6.8), and m χ isgiven by m χ ≈ M − m N N (6.19)in the leading order in large N .The corresponding stability constraints on M in this model are the same as the constraints(6.16), (6.17) in the model studied in the previous section. The mass squared of the field χ ispositive during the last N e-foldings for M > N / m . For N = 55, this constraint becomes M (cid:38) m . The mass of the field χ becomes greater than the Hubble scale for M > N √ m .For N = 55, this is achieved for M (cid:38) m . For m ∼ − , the required value of the massparameter M should be greater than 10 − − − in the Planck mass units. In the previous two sections we explored the effect of the merger of two disks, and found thatfor sufficiently strong stabilization described by the parameter M one can easily stabilize theinflationary directory in such a way that during the last 50 - 60 e-foldings of inflation insteadof two independent α attractors with α = 1 / α = 2 / f = m (1 − n ( T + . . . + T n )), with n ≤ 7. Then, in analogy with equations(6.16), (6.17), one finds that at N (cid:29) M > N / mn , (7.1)and the strong stabilization with all of these masses greater than the Hubble constant is achievedfor M > N m √ n . (7.2)This result coincides with the result for the two-disk merger (6.16), (6.17) for n = 2. It alsoshows that the merger of n disks is easier to achieve for large n . In particular, for n = 7 thestability condition (7.1) during the last N = 55 e-foldings of inflation is satisfied for M (cid:38) . m ,and the strong stabilization condition (7.2) of the α -attractor regime with α = 7 / N (cid:46) M (cid:38) m . 27or α = 7 / 3, the Planck normalization for m is m ≈ (cid:113) × − , which leads to the stabilitycondition M (cid:38) × − in the Planck mass units. In this paper we have used the relative simplicity of the general class of α -attractor models, [6]-[9], to propose cosmological models with discrete values of the α -parameter3 α = R E = 1 , , . . . , . (8.1)These models realize the suggestion in [1] that the consistent truncation of theories with max-imal supersymmetry (M-theory, superstring theory, N = 8 supergravity) to minimal N = 1supersymmetry models, leads to cosmological models with seven discrete values for the squareof the radius of the Escher disk in moduli space.It is instructive to remind us here that, if one would assume that the maximal supersymmetrymodels are first truncated to half-maximal supersymmetry models, for example, N = 4 super-gravity [22] and maximal N = 4 superconformal models [23], one would recover the hyperbolicgeometry with a single unit size disk. This would mean that 3 α = 1.In the first part of the paper we have described the advantage of using the hyperbolicgeometry of the moduli space to explain the relation between the tilt of the spectrum n s andthe number of e-foldings N , n s ≈ − N , which is supported by the observational data. We alsoprovided a geometric reason for α -attractor models with plateau potentials, and explained therelation of hyperbolic geometry to the Escher’s concept of ‘capturing infinity’ in a finite space.Finally, we have shown that understanding ‘tessellations’ of the hyperbolic disk, or equivalentof the half-plane geometry, is useful for the choice of the K¨ahler frame providing the stabilityof cosmological models. To derive the dynamical cosmological models supporting the case (8.1)we employ a two-step procedure:As the first step, in Sec. 5.1, we introduce a superpotential W ( T i ) in (5.13) where T i arecoordinates of the seven-disk manifold. It is consistent with N = 1 supersymmetry, such that ithas a supersymmetric minimum realizing dynamically the conditions on seven complex moduli(5.4), which were postulated in [1]. The superpotential depends on the parameter m , controllingthe inflaton potential, and the parameter M , which is responsible for the dynamical merger ofthe disks in a state where some of the moduli T i coincide.As a second step, in Sec. 5.2, we introduce the cosmological sector of α -attractor models.In addition to seven complex disk moduli we introduce a stabilizer superfield S which can beeither a nilpotent superfield, associated with the uplifting anti-D3 brane in string theory, orthe one with a very heavy scalar, which during inflation serves the purpose of stabilizing thenon-inflaton directions. The corresponding K¨ahler potential and superpotential are given ineqs. (5.16) and (5.17) respectively, with W given in eqn. (5.13).28he original models contain a rather large number of moduli: seven complex scalars anda stabilizer. We have studied these cosmological models close to the inflationary trajectoryand we have found the conditions where the masses of all non-inflaton directions are positive.A more detailed study was performed in Sec. 6 in a toy model where the starting point isa two-disk manifold, where the regimes with 3 α = 1 or 3 α = 2 are possible. We studiedthese models both in half-plane coordinates and in disk coordinates. The global analysis ofthe cosmological evolution was performed, not just near the inflationary trajectory, and it waspossible to evaluate the value of the parameter M providing more than N = 55 e-folds ofinflation in the regime with 3 α = 2. Depending on the choice of W we have found modelswith M (cid:38) − − − M P l , describing inflation either with 3 α = 1, or 3 α = 2. A cosmologicalphase transition between these two regimes is possible for smaller values of M . The analogousconsiderations for the seven-disk models in Sec. 7 show that the cosmological stability of themaximally symmetric regime with 3 α = 7 requires M (cid:38) m ∼ × − M P l .Thus, in this paper we provided a dynamical realization of a new class of cosmological α -attractors motivated by maximally supersymmetric theories, such as M-theory, superstringtheory, and maximal N = 8 supergravity [1]. These models suggest a set of discrete targetsfor the search of tensor modes in the range 10 − (cid:46) r (cid:46) − . In particular, the maximallysymmetric model 3 α = 7 and r ≈ − becomes an interesting realistic target for relativelyearly detection of B-modes. The case with 3 α = 1, r ≈ − remains a well motivated longerterm goal.For decades, one of the main goals of inflationary cosmology was to use observations toreconstruct the inflation potential [38]. 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