\( \overline{D3} \) induced geometric inflation

Journal of High Energy Physics, Jul 2017

Effective supergravity inflationary models induced by anti-D3 brane interaction with the moduli fields in the bulk geometry have a geometric description. The Kähler function carries the complete geometric information on the theory. The non-vanishing bisectional curvature plays an important role in the construction. The new geometric formalism, with the nilpotent superfield representing the anti-D3 brane, allows a powerful generalization of the existing inflationary models based on supergravity. They can easily incorporate arbitrary values of the Hubble parameter, cosmological constant and gravitino mass. We illustrate it by providing generalized versions of polynomial chaotic inflation, T- and E-models of α-attractor type, disk merger. We also describe a multi-stage cosmological attractor regime, which we call cascade inflation.

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\( \overline{D3} \) induced geometric inflation

HJE D3 induced geometric in ation Renata Kallosh 0 1 3 Andrei Linde 0 1 3 Diederik Roest 0 1 2 Yusuke Yamada 0 1 3 0 Nijenborgh 4 , 9747 AG Groningen , The Netherlands 1 382 Via Pueblo Mall , Stanford, CA 94305 , U.S.A 2 Van Swinderen Institute for Particle Physics and Gravity, University of Groningen 3 Stanford Institute for Theoretical Physics and Department of Physics, Stanford University E ective supergravity in ationary models induced by anti-D3 brane interaction with the moduli elds in the bulk geometry have a geometric description. The Kahler function carries the complete geometric information on the theory. The non-vanishing bisectional curvature plays an important role in the construction. The new geometric formalism, with the nilpotent super eld representing the anti-D3 brane, allows a powerful generalization of the existing in ationary models based on supergravity. They can easily incorporate arbitrary values of the Hubble parameter, cosmological constant and gravitino mass. We illustrate it by providing generalized versions of polynomial chaotic in ation, Tand E-models of -attractor type, disk merger. We also describe a multi-stage cosmological attractor regime, which we call cascade in ation. Cosmology of Theories beyond the SM; D-branes; Supergravity Models - 1 Introduction 2.1 2.2 2.3 3.1 3.2 3.3 3.4 2 Geometric in ation features D3-brane induced geometry Curvature invariants Stability analysis 3 Model building paradise Polynomial in ation T-models E-models Two-disk merger models 3.4.1 3.4.2 3.4.3 E-model T-model Cascade in ation 3.5 Seven-disk merger model 4 Discussion 1 Introduction one may ask a question: what kind of interaction between S and (T i; T i) would lead to phenomenological supergravity models of in ation, including the exit stage, that are compatible with the data? Here we will construct what we call D3 induced geometric in ation models. In these models, once one decides about the potential V(T i; T i), it is easy to nd the corresponding { 1 { S- eld geometry GSS(T i; T i) in the supergravity Kahler function G, and one is guaranteed to reproduce the desired potential during in ation. However, one has still to check the stability of each model and show the absence of tachyons. The bisectional curvature of these geometric models will play a role in the stability analysis. We will develop a general class of D3 induced geometric in ation with multiple moduli in CY bulk interacting with D3 nilpotent multiplet S. It is important that the D3 induced geometric in ation models have a non-vanishing gravitino mass | W does not vanish during and at the exit from in ation. In this case, one can use the advantage of a geometric Kahler function formalism where G K + log W + log W ; V = eG (G G G 3) and study various interesting application of the new models. Here the index includes the directions S and T i. The role of the Kahler function G was recognized starting with [29, 30] when supergravity models interacting with matter were rst constructed. It was shown there that the action is fully determined by the Kahler function. However, in some cosmological models, for example in D-term in ation [31], or in models in [32], during the evolution the superpotential might vanish. For these models it was more useful to employ the Kahler potential and the superpotential W since the Kahler function G has a singularity at W = 0. Meanwhile, the analysis of non-supersymmetric Minkowski and metastable de Sitter vacua with spontaneously broken supersymmetry was based mostly on the analysis using the Kahler function G, see for example, [33{37]. Comparative to this analysis, the new ingredient here is the fact that the S super eld is nilpotent and that we will use it for developing in ationary models with the exit to de Sitter minima. Our Hermitian Kahler function will be of the form (1.1) HJEP07(21)5 G(T i; T i; S; S) = G0(T i; T i) + S + S + GSS(T i; T i)SS ; which we will show will describe the general case of supergravity models with one nilpotent multiplet and non-vanishing superpotentials. We will show below that, in general, from the knowledge of the potential V(T i; T i) and the T -dependent Kahler function G0(T i; T i) it is possible to recover the S- eld geometry GSS(T i; T i)dSdS: Whereas the complete formula will be given below in eq. (2.12), here we would like to point out that under certain conditions the relation between the S- eld geometry and the potential simpli es signi cantly. If the gravitino mass is constant throughout in ation at S = 0, and supersymmetry is unbroken in the T i directions, i.e. during in ation we have eG(T i;T i) = jm3=2j2 = const ; GT i (T i; T i) = 0 ; one nds the following simple relation between the in ationary potential and the geometry: conditions (1.4) will be satis ed during and after in ation. Examples of models with non-trivial Hermitian function GSS(T; T ) include warped Calabi-Yau throats [18{23], in which the Kahler potential takes the form ln(T + T SS) = ln(T + T ) + SS T + T ; GSS = 1 T + T Another instance of a non- at geometry of the S- eld are the `axion stabiliser' terms [38, 39] in the Kahler potential metric of the kind SS(T i T i)2A(T i; T i) ; GSS = (T i T i)2A(T i; T i): (1.6) (1.7) The inclusion of these terms in some models is necessary for the stability of the in ationary trajectory. Finally, a new class of models where GSS(T; T ) is a general Hermitian function was proposed in [40]. A number of nice and interesting examples were studied, starting with W = M S + W0 and shift symmetric canonical Kahler potentials, where also stability issues were studied, or with Poincare half-plane geometries in Kahler potentials. An important feature of the D3 induced geometric in ation models with one modulus is that the bisectional curvature is non-vanishing, RT T SS 6= 0 during in ation and at the exit, at the minimum of the potential. This is the consequence of the fact that the metric GSS is not a product of a holomorphic F (T ) function times an anti-holomorphic function F (T ). In the latter case it can be removed by a holomorphic change of the Kahler manifold coordinates F (T )S ! S0 which leads to a at geometry of the nilpotent super eld. This case includes models with canonical geometry for the nilpotent eld, GSS = 1 and some general superpotentials W = g(T i) + f (T i)S. For these models the Kahler geometry of the nilpotent eld S is at, and hence RijSS = 0. A nice feature of our examples is that all of them during in ation, in case of a single modulus, have no tachyons without any assumption. At the minimum of the potential we do not have a general argument of stability, however, a priori these models allow a way to associate geometry with the good choices of the potentials which have a minimum at the exit from in ation. The same argument refers to multiple moduli models. A choice of the potentials is possible such that the desirable relations between moduli can be implemented as a requirement of the minimum of the potential, as a result we end up with single modulus models which have a stable in ationary trajectory. Comparatively to other model building we used before, we have found various advantages, which we dubbed as a `model building paradise', based on a geometry of the D3-brane and associated nilpotent multiplet interacting with moduli of the Calabi-Yau manifolds. In particular, we have a parameter of supersymmetry breaking independent of the Hubble parameter and the models are simple. { 3 { 2.1 D3-brane induced geometry We will explain here that the most general Kahler invariant Kahler function G depending on multiple Calabi-Yau moduli and on a nilpotent multiplet S can be reduced to the form we show in eq. (1.2). An equivalent form is to use K(T i; T i; S; S) = K0(T i; T i) + S + S + GSS(T i; T i)SS ; W = W0 ; where the gravitino mass, in general is given by the following expression: jm3=2(T i; T i)j2 = eG0(T i;T i) = eK0(T i;T i)jW0j2 : HJEP07(21)5 The linear terms in the Kahler function and potential are directly related to spontaneous SUSY breaking and hence an integral aspect of our set-up. We will start with the observation [41, 42] that the most general supergravity theory with a number of unconstrained chiral multiplets T i and a single nilpotent super eld S is given by K = K0(T i; T i) + KS(T i; T i)S + KS(T i; T i)S + GSS(T i; T i)SS ; W = g(T i) + f (T i)S; where KS; KS and GSS are non-holomorphic functions while f and g are holomorphic. These are the most general Taylor expansions of the Kahler and superpotential due to the nilpotency condition S2 = 0. These general expressions can be simpli ed without loss of generality by a number of rede nitions. First of all, one can use a Kahler transformation acting as W ! W F ; K ! K log jF j2 ; to set W = W0 by choosing F = W0=(g + f S). The resulting Kahler potential is given by K0 = K00(T i; T i) + KS0(T i; T i)S + K0S(T i; T i)S + GSS(T i; T i)SS; in terms of the rede ned variables K00 = K0 + 2 log(jgj=jW0j), and KS0 = KS + f =g. In this frame, the supersymmetry breaking is set by KS0 which we assume to be non-vanishing due to the nilpotency of S. We can subsequently use the eld rede nition KS0S = S. Note that KS0 is not holomorphic, and hence this eld rede nition breaks the complex manifold structure. However, at least in the bosonic part of the theory1 this is not a problem for the following reason. The geometry spanned by the physical scalars is given by the Kahler manifold with a projection S = 0, since the bosonic component of S is a fermion bilinear, i.e., ds2 = Gij dT idT j jdS=S=0: 1The fermionic action will be a ected by this change only in the part depending on the goldstino, a fermion in the nilpotent multiplet, S. But in the unitary gauge, S = 0, in which this fermion is absent, fermions in T i multiplets and the gauge with S = 0 is simple. there will be no changes. In our models with GT i = 0 in this gauge the gravitino decouples from the { 4 { (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) Therefore, the eld rede nition dS0 = KS0dS + @iKS0SdT i + @j KS0SdT j does not change the Kahler manifold of physical scalars. It only a ects the nilpotent part of the Kahler potential, which now has a metric form of the total potential where V (T i; T i) vanishes at the minimum: This completes the argument that the most general supergravity theory can be brought to the form (1.2) or (2.1) (omitting all primes), when evaluated at dS = S = 0. In models satisfying our condition (1.4) and with a positive CC, we can use the following HJEP07(21)5 V(T i; T i) = V (T i; T i) + ; jFSj2 3jW0j2 : This gives us an alternative form of the D3 geometry GSS(T i; T i) = eG(T i;T i) V(T i; T i) + 3eG(T i;T i) G T iT i GT i GT i eG(T i;T i) : This geometry directly gives any phenomenologically favored potential. { 5 { where the measure of supersymmetry breaking at T i = T due to the D3-brane is set by i Note that the above metric explicitly includes an independent Hubble, SUSY breaking and dark energy scale. In the absence of the nilpotent eld, this model has a SUSY AdS solution with at least one at direction amongst the T i moduli that will provide the in aton. The inclusion of the D3-brane yields the uplift term. When including a constant S term to the superpotential, or equivalently a constant metric GSS, this uplifts to a non-SUSY vacuum with arbitrary CC and a at direction. The subsequent introduction of an in ationary pro le can be performed either by means of a holomorphic function f in the superpotential, or more generally by means of an moduli-dependent metric for the S- eld, leading to the D3-brane induced geometry (1.3). Also in more general models that do not satisfy (1.4), we can reconstruct any desired potential V(T i; T i) starting from the Kahler function G(T i; T i). In supergravity, the scalar potential and geometry are related as follows, assuming that GS = 1: V(T i; T i) = eG(T i;T i)(GSS(T i; T i) + G T iT i GT i GT i 3): V(T i; T i), we nd the proper choice of GSS is This relation is invertible with respect to GSS. In order to realize the desired potential (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) In case of one modulus T , this geometry is determined by two curvature invariants that will characterize the cosmological parameters. In addition to the full Ricci scalar, one can also de ne the Ricci scalar of the submanifold de ned by S = 0, as the only allowed coordinate rede nitions on this Kahler geometry preserve the nilpotency condition. This will be referred to as the sectional curvature and is given by Rsec = G T T GT T (GT T T T GT T T G T T GT T T ) : The importance of this geometric quantity for in ationary model building has been stressed in various places. For example in the case of the hyperbolic disk relevant for -attractors, one has K = 3 ln(T + T ) ; Rsec = 2 3 ; where the latter is of course independent of the Kahler frame. The new ingredient in the D3 induced geometric in ation models is the second curvature invariant, corresponding to the bisectional curvature along the S = 0 plane: notation of [43] by mi2j = eG Gij 1+ V jm3=2j2 where G G G , physical scalar elds, this simpli es to GiGj +(Gi +GiG ) G ( G j +G Gj ) Rij G G ; (2.18) = (S; T i) and i = 1; : : : ; N . Under the assumption (1.4) for the mi2j = eG Gij 1 + + Gi G G j RijSSG G S S ; (2.19) Rbisec = RT T SSG G T T SS = GT T (VT T (FS2 + V ) VT VT ) : (FS2 + V )2 During in ation at V jFSj2, it is proportional to slow roll parameters In contrast, at the minimum of the potential, Rbisec jin G T T VT T V VT VT V 2 = It therefore sets the scale for the sum of masses of both T -components, and stability requires a positive value for the bisectional curvature. 2.3 Stability analysis For a model with a single in aton super eld model, we nd that the supersymmetric scalar partner of in aton (the so-called sin aton) is always stabilized at its origin as shown below. The general formula for the non-holomorphic masses of the scalar elds is given in the V jm3=2j2 { 6 { (2.13) (2.14) (2.15) (2.16) (2.17) Tracing this formula yields the average mass: ma2ve = N G 1 ij mi2j = 1 N eG N 1 + V jm3=2j2 In particular, for N = 1, this expression can be reduced to + Gi Gj G G ij Gij RijSSG G S S : (2.20) ma2ve = V + m23=2 + m23=2(jGT T j2(GT T )2 G T T RT T SSGSGS) = V + m23=2 + m23=2jGT T j2(GT T )2 + Rbisec(jFSj2 + V ) ; (2.21) HJEP07(21)5 emphasizing the importance of the bisectional curvature. During in ation, the in aton mass is very small, and the rst 3 terms are positive. The last term is given by the linear combination of slow-roll parameters (2.16). Using the experimental values of ns and r it comes out negative, but is always smaller than the rst two positive contributions thanks to the slow-roll suppression. Thus, during in ation, we have shown that the sin aton direction in a single super eld model is always stable, or equivalently our assumption T = T is satis ed automatically. Apart from the in ationary era, we discuss the minimum of our model. The nilpotent super eld is well de ned only if GS 6= 0 and GSS 6= 0. Due to the absence of a propagating scalar in S, the stability requirement is equivalent to the condition that the propagating scalars have stable vacua at GS 6= 0 and also GSS 6= 0. Then, we need to require positive masses for the scalar elds at the minimum. The general minimization condition of the scalar potential is Vi = GiV + eG (riG G G + Gi) = 0: Since V = 0 at the minimum, we obtain the condition riG G G + Gi = 0. For Gi = 0, the condition is equivalent to riGS = 0. Then, the mass matrix at the minimum is simply given by mi2j = eG [Gij + Gij GjkGkj + RijSS(GSS)2]: Assuming Gij GjkGkj = O(1) and RijSS = 0, the averaged mass becomes m2ave = O(m23=2): Therefore, to disentangle the scalar mass and the SUSY breaking scale, we need to introduce large Gij G jk Gkj or RijSS(GSS)2. Moreover, the scale of the averaged mass does not tell us the mass of each scalar and their positivity, and therefore, we need to discuss the stability at the minimum for each case. With our choice of GSS in the single-modulus N = 1 case, the averaged mass becomes m2ave = eG (1 + GT T G T T GT T ) + G T T VT T : The last term comes from the bisectional curvature and it is not necessarily related to the SUSY breaking scale. Thus, with a proper choice of V , the SUSY breaking and the mass of the in ation sector can be disentangled. (2.22) (2.23) (2.24) (2.25) { 7 { Our main goal here is to give example of geometric models of in ation which are de ned by a geometry of the D3-brane in the CY bulk geometry. For this purpose it is natural to use logarithmic Kahler potentials for the moduli elds T i of the kind ln(T i + T ). However, i once we use nilpotent super eld geometry as a tool in model building, we nd that the shift symmetric Kahler potentials for the moduli elds i are also particularly e cient. We will start therefore with the model of polynomial in ation with the Kahler potential 1 2 ( )2. of 3 main parameters: the amplitude of the perturbations As, the spectral index ns and the tensor to scalar ratio r. According to [46{48], one can properly describe any set of these parameters in the context of the 3-parameter polynomial in ationary models with the potential One could try to implement the models with such potentials in supergravity [47, 48], using the general approach developed in [ 32, 49, 50 ], but the resulting potentials can reproduce the potential (3.1) only approximately, see a discussion of this issue in [51]. Meanwhile, as we will see now, the potential (3.1) can be easily obtained in the context of the new geometric approach discussed in our paper. We will consider the Kahler function (3.1) (3.2) (3.3) (3.4) HJEP07(21)5 W02 jFSj2 + V ( ; ) : Here the part of the potential vanishing at the minimum is { 8 { m2 4 V ( ; ) = ( + ) One can show that the potential of these elds is stable at = 0, and the in aton elds 2 = p1 ( +i ). has the desirable potential m2 2 2 V( ) = 1 a (1 + b a ) + ; where 3jW0j2 is the vacuum energy/cosmological constant at the minimum of the potential, and the gravitino mass at the minimum is equal to m3=2 = W0. The potential for = 0 is shown in gure 1. As we already mentioned, in ation-related Planck data [44, 45] consist of three main parameters, As, ns and r. The value of As can be easily tuned by a proper choice of M . The parameters a and b are responsible for ns and r. For example, for a = 0:12 and b = 0:29, the perturbations generated at the moment corresponding to N = 58 e-folding HJEP07(21)5 ϕ a + a2b 2 for a = 0:12 and b = 0:30 (upper curve), b = 0:29 (middle), and b = 0:28 (lower curve). The potential is shown in units of m2, with in Planck units. For b = 0:29 (the middle curve), at the moment corresponding to N = 58 e-folding from the end of in ation one has ns = 0:965 and r = 0:012, perfectly matching the Planck data. from the end of in ation have ns = 0:965 and r = 0:012, perfectly matching the Planck data [44, 45]. Thus we found the desirable polynomial potential, and much more: we have full exibility to describe arbitrary cosmological constant and SUSY breaking in this simple model. Finally, in ation in this model may begin close to the Planck density, which easily solves the problem of initial conditions for in ation, as explained in [ 52, 53 ]. 3.2 T-models Moving on to a hyperbolic instead of a at geometry for the scalar manifold, the Kahler function in disk variables can be written as log (1 ZZ)2 (1 Z2)(1 Z ) 2 + S + S + W02 jFSj2 + m2ZZ SS: (3.5) Note that this employs a Kahler frame that has a manifest in aton shift symmetry [38]. One can check that GZ = 0 and GS = 1, i.e. the theory has all required properties. The canonical in aton ' is de ned by relation Z = tanh p' . The in aton potential is VjZ=Z = + m2 tanh2 p { 9 { where 3W02. The axion mass along the in aton trajectory for m2 = 2(m2 + 2W02) m2 2 2 3 cosh2 p ' 6 cosh p ' 6 As expected from the observation in section 2.3, the mass of the axion is positive during in ation: m2 = 2(m2 + 2W02) > 2V 6H2 for ' 6 . This means that the eld is strongly stabilized and its perturbations are not generated during in ation. Moreover, the ' 6 + ; 1 3 p 6 = 0 is 4 : (3.6) (3.7) = 1. The height of the potential here and in other gures is in units m2 and the values of the elds are in Planck mass units. stability condition is satis ed along the full in aton trajectory for all . In particular, the masses of the elds ' and at the minimum of the potential at ' = = 0 are given by gure 2, which shows the potential V ('; ) in the limit m2 for the particular case If we use a more general function log (1 ZZ)2 (1 Z2)(1 2 Z ) W02 jFSj2 + f (ZZ) and the potential is Then, the axion mass becomes VjZ=Z = FS2 3W02 + f tanh2 p ' 6 m2 = 4W02 + 2f + cosh q 32 sech4 p where the prime denotes the derivative with respect to the argument tanh2 p' . The last term becomes O(p )H2 whereas the second term is 6H2 and is much larger than the last 6 term. Therefore, the axion mass is positive as we expected from the general discussion in section 2.3. The minimum is = 0 and the mass of the in aton and axion at the minimum are m2 = f 0(0) 3 ; m2 = 4W02 + f 0(0) 3 ; where we have used f (0) = 0, which is our general assumption on V . One can check that this coincides with the general formula (2.25) from the previous section. A simple case using half-plane variables is log (T + T )2 4T T V jT =T = One can check that GT = 0 and GSjmin = 1 6= 0, i.e. the theory has all required properties. The axion mass squared during in ation is It is positive de nite during and after in ation. Note that at the minimum ' = 0, the axion mass squared becomes 4W02 = 4m23=2. If we take a more general function log (T + T )2 4T T and the potential is In this case, the mass of the axion is given by VjZ=Z = jFSj2 3W02 + M 2f 2eq 32 ' : m2a = 4W02 + 2f 2eq 32 ' = 4W02 + 6H2; which is positive de nite and consistent with our general argument in section 2.3. = 1=3 creates the in aton potential with = 2=3. Here we considered an example with M = 6 m. 3.4 3.4.1 Two-disk merger models Here we consider the model with two half-planes T1;2 and 3 i = 1 for i = 1; 2. As the previous work [54] where the merger of di erent attractors was discussed (albeit in disk coordinates), we dynamically realize the in ationary trajectory where two half-plane moduli directions merge during last 50-60 e-foldings. Instead of the use of the superpotential for stabilization [54], we use the geometry, 1 4 V = (T1 + T 1 + T2 + T 2) M 2(T1 + T 1 T2 T 2)2: (3.21) 2 2 The last term in (3.21) leads to the merger of in ationary trajectories of T i as shown in gure 4. We represent eld Ti as Ti = e 2 i), where i are canonical, and i are canonical in the small i limit. The in aton direction on merger trajectory is ' = p1 ( 1 + 2) and the orthogonal direction is ' = p1 ( 1 + 2) the potential of the canonically normalized in aton eld ' is 2). During in ation with p 2 i (1 + p e ')2. As is the case of the previous work [54], the direction 2) acquires a light or tachyonic mass for su ciently large : the mass of is stable for last N e-foldings is M 2 > m22N . As explained in [54], this simply means that the exponentially at and long dS plateau in the upper right corner of gure 4 is slightly curved, and the elds tend to move towards its boundaries. Then they slide along these boundaries towards the point where these boundaries merge and the diagonal deep gorge is formed, as shown at the center of gure 4. After that, all elds become stable along the in ationary trajectory with 1 = and the in aton potential coincides with the E-model potential (3.14). The eld value of ' at the last N e-folding is given by 'N = log(4N ). The condition that the merger trajectory 2 = p1 ' 2 At the minimum, GS = 1, the metric is GSS = jFSj2 W02 13 , and the SUSY breaking is realized with m3=2 = W0. Thus, this model generalizes the E-model disk merger described in [54], but now one can have arbitrary values of the cosmological constant and the gravitino mass. 3.4.2 The disk merger model is also possible for T-models. We consider the following system, In models with multiple elds, the stability of the axionic direction is not guaranteed by the discussion in section 3.1 and we need to discuss the stability of the trajectory for each case. For the current model, the axionic directions are stabilized with masses m12 = m22 = 4W02 + 2m2(cosh2 ' + cosh ' (cosh ' + 1)2 1) : G = log W02 gSS = 1 W02 jFSj2 + 1 X2 log The scalar potential is 2 + S + S + gSSSS; (Z1 + Z1) (Z2 + Z2) 2 : + m2 2 (jZ1j2 + jZ2j2) + and the last term gives the dynamical constraint 1 = 2 where we have de ned canonical elds as Zi = tanh ip+2i i . During in ation with 1 = 2 = p12 ' the potential is Turning to stability, the mass eigenvalues of axionic directions on in ationary trajectory 1 = 2 = p1 ' are given by 2 (3.23) (3.24) HJEP07(21)5 (3.25) (3.26) (3.27) (3.28) (3.29) = 1=3 creates the in aton T-model potential with In this gure we show the potential with M = 10 m. The masses are positive. At the minimum, mi2 = m22 + 4W02. Instead, the mass of the direction is M 2 > m22N . As in the E-model discussion, for the very large values of the in aton eld, such that is tachyonic. In order for this instability to take place outside of the observable window of N e-folds, one has again has to impose the condition As in the previous section, at the minimum, GS = 1, the metric is GSS = jFSj2 the SUSY breaking is realized with m3=2 = W0. Thus, this model generalizes the T-model disk merger described in [54], but now one can have arbitrary values of the cosmological W02 13 , and m2 = m2 + 2M 2 cosh ' m2 2 cosh 4 ' : 2 (3.30) constant and the gravitino mass. 3.4.3 Cascade in ation The two-disk merger (the fusion of two di erent attractors) is not the only interesting feature of the two-disk model studied above. Figure 5 shows only the lower part of the potential, which is su cient to illustrate the e ect of the disk merger. However, the upper part of the potential tells us an equally interesting story. To explain it, we will show the potential including its upper part, for a toy model with m = M , see gure 6. One can easily recognize the minimum of the potential, near which one may have in ation with = 2=3 for the models with M m. However, another important part of the potential is the existence of 4 di erent dS plateaus. The lower ones have the height m2, one can see them also in gure 5. The upper ones have the height m2 + M 2. They exist even in the absence of the disk interactions, for M = 0, in which case the height of each plateau is equal to m2. The existence of these plateaus follows from the general expression for the potential of the elds 1 and 2 in that model: + m2 2 tanh2 p1 + tanh2 p2 2 2 + tanh p 1 2 tanh p 2 2 2 : (3.31) = 1=3 for m = M . 2 In ation may begin at the upper plateau, with 1 1 and 2 1, or with 1 1 and 2 1. Then the eld falls down to one of the lower plateaus, from which it moves towards the narrow gorge along 1 = 2 in the potential shown in gure 5, and eventually falls to the minimum at 1 = 2 = 0. One may call this multi-stage process a cascade in ation. For M 2 > 60m2 , all observational consequences of this regime are determined by the last stage of the process, described by the T-model potential with = 2=3 (3.31). However, the cascade regime is very interesting from the point of view of the theory of initial conditions for in ation. Indeed, suppose that the parameter M describing the disk interactions takes the simplest value M = O(1) in Planck units. Then the height of the upper potential will be Planckian, which allows to solve the problem of initial conditions for in ation in the simplest possible way, as described in [ 52, 53 ]. The Planck-size universe can be born with the scalar elds 1 and 2 at an in nite plateau with V = m2 + M 2 = O(1). According to [ 52, 53 ], the probability of this process is not expected to be exponentially suppressed. Once this happens, the cascade in ation begins, with observational predictions determined by the last stage of the process, matching the latest observational data. A more general solution to the problem of initial conditions for in ation, which applies to all models discussed in our paper, can be found in [55, 56]. We hope to return to a more detailed discussion of the cascade in ation in a separate publication. 3.5 Seven-disk merger model Finally, we brie y discuss the possible merger of several disks. Consider for instance, corresponding to seven disks with i = 1=3. The scalar potential is V = + m2 7 X i jZij2 + M 2 72 X and the last term gives the dynamical constraint i = j where we have de ned canonical elds as Zi = tanh ip+2i i . During in ation at i = j = p'7 , the scalar potential reads + m2 tanh2 p14 ; ' 1 7 r 2 7 ! 4 7 in terms of the canonically normalized in aton eld. The axionic directions are stabilized at their origin, and their masses are given by are stabilized at their origin with the mass i) i i+1 4W02 and is still positive. The rst two constant part dominate the mass and the remaining negative part is suppressed during in ation. At the minimum, the mass of the axions becomes m2i = 17 m2 + For real directions f ig, the following canonical mass eigenbasis is useful, ' = 7). The in aton is ' and moduli i (3.35) (3.36) (3.37) (3.38) m2 i = 1 7 2m2 + 4M 2 As the two disk models, the mass of the moduli i becomes small, and when 4M 2 < q 27 ', they becomes tachyonic. At the minimum ' = 0, the in aton and moduli mass are given by m2 = m2; 1 7 m2 i = m2 + M 2: Note that SUSY breaking takes place at the minimum; GS = 1 and Here again we see the advantage of using the new geometric class of models comparative to the earlier version of the seven-disk model in ref. [54] where we only studied an in ationary q GSGSSGS = p 3W0. stage. In the seven-disk models we expect a cascade in ation with a rich structure due to the multiplicity of di erent in ationary plateaus. The di erent possibilities arise from the possible sign choices for the seven moduli. For instance, one can either take four positive and three negative, in which case 12 out of the 21 mass terms contribute. Similarly, one can have ve and two, with ten mass terms etc. From this logic it follows that the potential at the dS plateaus may take 4 di erent values: V = + m2 + 16n72M2 , where n can be 0, 6, 10, or 12. 4 Discussion It has been realized during the last few years that both the construction of de Sitter vacua in string theory as well as building in ationary models is facilitated by the concept of an upliting D3 brane. The positive energy contribution sourced by a D3 brane in e ective supergravity models is represented by a nilpotent multiplet S. Supersymmetry is spontaneously broken during in ation as well as at the exit from in ation, at the minimum of the potential, and never restored in the class of models we described here: D3 brane induced geometric in ationary models. The e ective supergravity of these models is described by the geometry of the CY moduli, G0(T i; T i) and by the geometry of the nilpotent super eld GSS(T i; T i)SS. In our models it is given by the expression G(T i; T i; S; S) = G0(T i; T i) + S + S + GSS(T i; T i)SS: (4.1) Subject to speci c assumptions about the geometry of T i moduli (1.4), satis ed by simple examples like a shift symmetric canonical geometry (3.2) or a disk geometry (3.13), one nds a simple relation between the in ationary potential and geometry of the D3 brane in the background of the T i elds: G SS(T i; T i) = V(T i; T i) + 3jm3=2j2 : This relation leads to a model building procedure of the following kind. Once the desired potential V(T i; T i) is determined, one can use the relation (4.2) to produce the geometry GSS(T i; T i). The remaining problem for each choice of in ationary model is to check that all non-in aton directions are stabilized. We have found that such a procedure leads to rather simple models with desirable properties. In particular, in models with one modulus T one nds that axions are stable during in ation. At the minimum, the masses of the in aton and axion also tend to be positive for the appropriate choices of the potentials where there is an exit from in ation in a nice agreement with the positivity of the S- eld metric GSS(T i; T i). at the minimum of the potential. These desirable stability properties of the potential are Our examples illustrate the main result of the paper: we build desirable cosmological models with in ationary potentials V(T i; T i) which are in agreement with the data, and we `read from the sky' the geometry of the D3 brane in CY bulk supporting these models as shown in eq. (4.2). The geometric nature of all these models manifests itself in the fact that the bisectional curvature is always present and is de ned by the slow-roll parameters as shown in section 2.2. At the exit from in ation at the minimum this curvature gives a positive contribution to the masses. We nd that this geometric formulation of e ective supergravity in ationary models inspired by string theory is the most powerful tool for model building. Their rst advantage is that they are easily associated with string theory due to fundamental role of the uplifting D3 brane, interacting with other moduli. The second advantage is that for speci c choices of Kahler geometries of the moduli elds T i, the only input comes from the nilpotent eld geometry, GSS(T i; T i), related to the potential. In previously existing models with generic superpotential W = Sf (T i) + g(T i), the main input is via two holomorphic functions f (T i) and g(T i), which should satisfy additional constraints. This made the model building more involved than in the approach developed in this paper. The third advantage is the fact that, by construction, the nilpotency condition FS 6= 0 is satis ed everywhere, including the minimum of the potential. The mere existence of the uplifting D3 brane interacting with the bulk geometry means that supersymmetry is nonlinearly realized and always spontaneously broken. In conclusion, the new cosmological models, D3 induced geometric models, de ned by a geometric Kahler function in eq. (4.1), lead to simple dynamical cosmological models of the in ationary evolution of the space-time, based on the geometry of the scalar manifold. The dynamics of these models is the consequence of their geometry. Acknowledgments We are grateful to E. Bergshoe , K. Dasgupta, S. Ferrara, D. Freedman, S. Kachru, E. McDonough, M. Scalisi, F. Quevedo, A. Uranga, A. Van Proeyen, A. Westphal, and T. Wrase for stimulating discussions and collaborations on related work. The work of RK, AL and YY is supported by SITP and by the US National Science Foundation grant PHY-1316699. The work of AL is also supported by the Templeton foundation grant \In ation, the Multiverse, and Holography". DR is grateful to SITP for the hospitality when this work was initiated. Open Access. 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Renata Kallosh, Andrei Linde, Diederik Roest, Yusuke Yamada. \( \overline{D3} \) induced geometric inflation, Journal of High Energy Physics, 2017, 1-22, DOI: 10.1007/JHEP07(2017)057