Induced-Gravity Inflation in no-Scale Supergravity and Beyond

Supersymmetric versions of induced-gravity inflation are formulated within Supergravity (SUGRA) employing two gauge singlet chiral superfields. The proposed superpotential is uniquely determined by applying a continuous R and a discrete Z_n symmetry. We select two types of logarithmic Kahler potentials, one associated with a no-scale-type SU(2,1) / SU(2) x U(1)_R x Z_n Kahler manifold and one more generic. In both cases, imposing a lower bound on the parameter cR involved in the coupling between the inflaton and the Ricci scalar curvature - e.g. cR>76, 105, 310 for n=2,3 and 6 respectively -, inflation can be attained even for subplanckian values of the inflaton while the corresponding effective theory respects the perturbative unitarity. In the case of no-scale SUGRA we show that, for every n, the inflationary observables remain unchanged and in agreement with the current data while the inflaton mass is predicted to be 3x10^13 GeV. Beyond no-scale SUGRA the inflationary observables depend mildly on n and crucially on the coefficient involved in the fourth order term of the Kahler potential which mixes the inflaton with the accompanying non-inflaton field.


INTRODUCTION
The announcement of the recent PLANCK results [1,2] fuelled increasing interest in inflationary models implemented thanks to a strong enough non-minimal coupling between the inflaton field, φ, and the Ricci scalar curvature, R. Indeed, these models predict [2,3] a (scalar) spectral index n s , tantalizingly close to the value favored by observational data. The existing non-minimally coupled to Gravity inflationary models can be classified into two categories depending whether the non-minimal coupling to R is added into the conventional one, m 2 P R/2 -where m P = 2.44 · 10 18 GeV is the reduced Planck scale -or it replaces the latter. In the first case the vacuum expectation value (v.e.v) of the inflaton after inflation assumes sufficiently low values after inflation, such that a transition to Einstein gravity at low energy to be guarantied. In the second case, however, the term m 2 P R/2 is dynamically generated via the v.e.v of the inflaton; these models are, thus, named [4,5] Induced-Gravity (IG) inflationary models. Despite the fact that both models of non-Minimal Inflation (nMI) are quite similar during inflation and may be collectively classified into universal "attractor" models [6], they exhibit two crucial differences. Namely, in the second category, (i) the Einstein frame (EF) inflationary potential develops a singularity at φ = 0 and so, inflation is of Starobinsky-type [7] actually; (ii) The ultaviolet (UV) cut-off scale [8][9][10] of the theory, as it is recently realized [11,12], can be identified with m P and, thereby, concerns regarding the naturalness of inflation can be safely eluded. On the other hand, only some [10] of the remaining models of nMI can be characterized as unitarity safe.
In a recent paper [11] a supersymmetric (SUSY) version of IG inflation was, for first time, presented within no-scale [13][14][15] Supergravity (SUGRA). A Higgs-like modulus plays there the role of inflaton, in sharp contrast to Ref. [14] where the inflaton is matter-like. For this reason we call in Ref. [11] the inflationary model no-scale modular inflation. Although any connection with the no-scale SUSY breaking [13,16] is lost in that setting, we show that the model provides a robust cosmological scenario linking together non-thermal leptogenesis, neutrino physics and a resolution to the µ problem of the Minimal SUSY SM (MSSM). Namely, in Ref. [11], we employ a Kähler potential, K, corresponding to a SU (N, 1)/SU (N ) × U (1) R × Z 2 symmetric Kähler manifold. This symmetry fixes beautifully the form of K up to an holomorphic function Ω H which exclusively depends on the inflaton, φ, and its form Ω H ∼ φ 2 is fixed by imposing a Z 2 discrete symmetry which is also respected by the superpotential W . Moreover, the model possesses a continuous R symmetry, which reduces to the well-known R-parity 2 of MSSM. Thanks to the strong enough coupling between φ and R, inflation can be attained even for subplanckian values of φ, contrary to other SUSY realizations [15,17,18] of the Starobinsky-type inflation.
Most recently a more generic form of Ω H has been proposed [12] at the non-SUSY level. In particular, Ω H is specified as Ω H ∼ φ n and it was pointed out that the resulting IG inflationary models exhibit an attractor behavior since the inflationary observables and the mass of the inflaton at the vacuum are independent of the choice of n. It would be, thereby, interesting to investigate if this nice feature insists also in the SUSY realizations of these models. This aim gives us the opportunity to generalize our previous analysis [11] and investigate the inflationary predictions independently of the post-inflationary cosmological evolution. Namely, we here impose on Ω H a discrete Z n symmetry with n ≥ 2, and investigate its possible embedding in standard Poincaré SUGRA, without invoking the superconformal formulation -cf. Ref. [19]. We discriminate two possible embeddings, one based on a no-scale-type symmetry and one more generic, with the first of these being much more predictive. Namely, while the embedding of IG models in generic SUGRA gives adjustable results as regards the inflationary observables, -see also Ref. [20] -, no-scale SUGRA predicts independently of n results identical to those obtained in the non-SUSY case. Therefore, no-scale SUGRA consists a natural framework in which such models can be implemented.
Below, in Sec. 2, we describe the generic formulation of IG models within SUGRA. In Sec. 3 we present the basic ingredients of our IG inflationary models, derive the inflationary observables and confront them with observations. We also provide a detailed analysis of the UV behavior of these models in Sec. 4. Our conclusions are summarized in Sec. 5. Throughout the text, the subscript of type , χ denotes derivation with respect to (w.r.t) the field χ (e.g., ,χχ = ∂ 2 /∂χ 2 ) and charge conjugation is denoted by a star.

EMBEDDING IG INFLATION IN SUGRA
In Sec. 2.1 we present the basic formulation of a theory which exhibits non-minimal coupling of scalar fields to R within SUGRA and in Sec. 2.2 we outline our strategy in constructing viable models of IG inflation. The general framework for the analysis of the emerged models is given in Sec. 2.3.

THE GENERAL SET-UP
Our starting point is the EF action for N gauge singlet scalar fields z α within SUGRA [21,22] which can be written as where summation is taken over the scalar fields z α , K αβ = K ,z α z * β with Kβ α K αγ = δβ γ , g is the determinant of the EF metric g µν , R is the EF Ricci scalar curvature, V is the EF F-term SUGRA scalar potential which can be extracted once the superpotential W and the Kähler potential K have been selected, by applying the standard formula Note that D-term contributions into V do not exist since we consider gauge singlet z α 's. By performing a conformal transformation and adopting a frame function Ω which is related to K as follows we arrive at the following action where g µν = − (3/Ω) g µν and V = Ω 2 V /9 are the JF metric and potential respectively, we use the shorthand notation Ω α = Ω ,z α and Ωᾱ = Ω ,z * ᾱ and A µ is the purely bosonic part of the on-shell value of an auxiliary field given by It is clear from Eq. (2.3) that S exhibits non-minimal couplings of the z α 's to R. However, Ω enters the kinetic terms of the z α 's too. In general, Ω can be written as [21] − where Ω K is a dimensionless real function while Ω H is a dimensionless, holomorphic function.
For Ω H > Ω K , Ω K expresses mainly the kinetic terms of the z α 's whereas Ω H represents the non-minimal coupling to gravity -note that Ω αβ is independent of Ω H since Ω H,z α z * β = 0.
To realize the idea of IG, we have to assume that Ω H depends on a Higgs-like modulus, z 1 := Φ whose the v.e.v generates the conventional term of the Einstein gravity at the SUSY vacuum, i.e.
where we take into account that the phase of Φ, argΦ is stabilized to zero; we thus get Ω H = Ω * H . In order to get canonical kinetic terms, we need [21] A µ = 0 and Ω Kαβ ≃ 0 or δ αβ . The first condition is attained when the dynamics of the z α 's is dominated only by the real moduli |z α |. The second condition is satisfied by the choice with sufficiently small coefficients k α and k αβ ≃ 1. Here we assume that the z α 's are charged under a global symmetry, so as mixed terms of the form z α z * β are disallowed. The inclusion of the fourth order term for the accompanying non-inflaton field, z 2 := S is obligatory in order to evade [21] a tachyonic instability occurring along this direction during IG inflation. As a consequence, all the allowed terms are to be considered in the analysis for consistency. Let us here note that such a consistency is not observed in the SUGRA incarnations of similar models [6,21]. On the other hand, if we assume that the emergent Kähler manifold associated with K can be identified with SU (N, 1)/SU (N ) × U (1) R × Z n -where the symmetries U (1) R and Z n are specified in Sec. 2.2 -and highly simplifies the realization of IG inflation. The option in Eq. (2.8) is inspired by the early models of soft SUSY breaking [13] and defines [15] no-scale SUGRA. We below show details of these two realizations of IG inflation.

MODELING IG INFLATION IN SUGRA
As we anticipated above, the realization of the idea of IG in SUGRA requires at least two singlet superfields, i.e., z α = S, Φ; Φ is a Higgs-like superfield whose the v.e.v generates m P and S is an accompanying superfield, whose the stabilization at the origin assists us to isolate the contribution of Φ into V , Eq. (2.1b). To see how this structure works, let us below specify the form of Ω H and W .
Inspired by Ref. [12], we here determine Ω H by postulating its invariance under the action of a global Z n discrete symmetry. Therefore it can be written as with k being a positive integer. Restricting ourselves to subplanckian values of Φ and assuming relatively low λ k 's, we can say that Z n uniquely determines the form of Ω H . Confining ourselves to a such situation we ignore henceforth the k-dependent terms in Eq. (2.9). On the other hand, W has to be selected so as to achieve the arrangement of Eq. (2.6). The simplest choice is that used in the models of F-term hybrid inflation [23]. As a consequence Ω H (Φ) has to be involved also in the superpotential W of our model which has the form and can be uniquely determined if we impose, besides Z n , a nonanomalous R symmetry Indeed, U (1) R symmetry ensures the linearity of W w.r.t S which is crucial for the success of our construction. To verify that W leads to the desired Ω H we minimize the SUSY limit, V SUSY , of V , obtained from the latter, when m P tends to infinity. This is where the complex scalar components of Φ and S are denoted by the same symbol. From Eq. (2.12a), we find that the SUSY vacuum lies at S = 0 and Ω H = 1/2, (2.12b) as required by Eq. (2.6). Let us emphasize that soft SUSY breaking effects explicitly break U (1) R to a discrete subgroup. Usually [11] combining the latter with the Z f 2 fermion parity, yields the wellknown R-parity of MSSM, which guarantees the stability of the lightest SUSY particle and therefore it provides a well-motivated CDM candidate.
The selected W and K by construction give also rise to a stage of IG inflation. Indeed, placing S at the origin, the only surviving term of V in Eq. (2.1b) is where the functions f R and f SΦ are computed along the inflationary track, i.e., , an inflationary plateau emerges since the resulting V IG0 in Eq. (2.13a) is almost constant. Therefore, Φ involved in the definition of Ω H , Eq. (2.9), arises naturally as an inflaton candidate. Note that the non-vanishing values of Φ during IG inflation break spontaneously the imposed Z n ; no domain walls are thus produced due to the spontaneous breaking of Z n at the SUSY vacuum, Eq. (2.12b).

FRAMEWORK OF INFLATIONARY ANALYSIS
To consolidate the validity of the inflationary proposal we have to check the stability of the inflationary direction θ = s =s = 0, (2.14) w.r.t the fluctuations of the various fields, which are expanded in real and imaginary parts as follows To this end we examine the validity of the extremum and minimum conditions, i.e., Here m 2 χ α are the eigenvalues of the mass matrix with elements and hat denotes the EF canonically normalized fields. The kinetic terms of the various scalars in Eq. (2.1a) can be brought into the following form where the dot denotes derivation w.r.t the JF cosmic time and the hatted fields are defined as follows Note, in passing, that the spinors ψ Φ and ψ S associated with the superfields S and Φ are normalized similarly, i.e., 16b), we can construct the scalar mass spectrum of the theory along the direction in Eq. (2.14) -see Sec. 3.2.1 and 3.3.1. Besides the stability requirement in Eq. (2.16a), from the derived spectrum we can numerically verify that the various masses remain greater than H IG during the last 50 e-foldings of inflation, and so any inflationary perturbations of the fields other than the inflaton are safely eliminated. Due to the large effective masses that θ, s and s in Eq. (2.16b) acquire during inflation, they enter a phase of oscillations about zero with reducing amplitude. As a consequence, the φ dependence in their normalization -see Eq. (2.17b) -does not affect their dynamics. Moreover, we can observe that the fermionic (4) and bosonic (4) degrees of freedom are equal -here we take into account that φ is not perturbed. Employing the well-known Coleman-Weinberg formula [24], we find that the one-loop corrected inflationary potential is where Λ is a renormalization group mass scale, m θ and m s = ms are defined in Eq. (2.16a) and m ψ ± are the mass eigenvalues which correspond to eigenstates ψ ± ≃ ( ψ S ± ψ Φ )/ √ 2. As we numerically verify, the one-loop corrections have no impact on our results, since the slope of the inflationary path is generated at the classical level and the various masses are proportional to the weak coupling λ.

3 THE INFLATIONARY SCENARIA
In this section we outline the salient features and the predictions of our inflationary scenaria in Secs. 3.2 and 3.3 respectively, testing them against a number of criteria introduced in Sec. 3.1.

INFLATIONARY OBSERVABLES -CONSTRAINTS
A successful inflationary scenario has to be compatible with a number of observational requirements which are outlined in the following.
has to be at least enough to resolve the horizon and flatness problems of standard big bang, i.e., [2] where we assumed that IG inflation is followed in turn by a decaying-inflaton, radiation and matter domination, T rh is the reheat temperature after IG inflation, at the end of IG inflation, which can be found, in the slow-roll approximation and for the considered in this paper models, from the condition where the slow-roll parameters can be calculated as follows: The amplitude A s of the power spectrum of the curvature perturbation generated by φ at the pivot scale k * must to be consistent with data [2] where we assume that no other contributions to the observed curvature perturbation exists.
3.1.3. The (scalar) spectral index, n s , its running, a s , and the scalar-to-tensor ratio r -estimated through the relations: 2 η ǫ and the variables with subscript * are evaluated at φ = φ * -must be in agreement with the fitting of the data [2] with ΛCDM model, i.e., at 95% confidence level (c.l.) 3.1.4. To avoid corrections from quantum gravity and any destabilize of our inflationary scenario due to higher order non-renomralizable terms -see Eq. (2.9) -, we impose two additional theoretical constraints on our models -keeping in mind that As we show in Sec. 4, the UV cutoff of our model is m P and so no concerns regarding the validity of the effective theory arise.

NO-SCALE SUGRA
According to our analysis in Sec. 2.2, IG inflation in the context of no-scale SUGRA can be achieved adopting a Kähler potential which depends at least on two gauge singlet superfields -the inflaton Φ and an accompanying one S -and has the form since f SΦ = 1 and f R = 2c R x n φ /2 n/2 where we introduce the dimensionless quantities Obviously V IG0 in Eq. (3.9) develops a plateau with almost constant potential energy density corresponding to the Hubble parameter Along the configuration of Eq. (2.14) K αβ defined in Eq. (2.17a) takes the form where the explicit form of Ω H in Eq. (2.9) is taken into account. Integrating the first equation in Eq. (3.12) we can identify the EF field: where we take into account Eqs. (2.9) and (2.12b). Also φ c is a constant of integration.
Following the general analysis in Sec. 2.3 we derive the mass spectrum along the configuration of Eq. (2.14). Our results are arranged in Table 1. We see there that k S 1 assists us to achieve m 2 s > 0in accordance with Ref. [15,17,18]. Inserting the extracted masses in Eq. (2.18) we can proceed to the numerical analysis of IG inflation in the EF [4], employing the standard slow-roll approximation [25] -see Sec. 3.2.3. For the sake of the presentation, however, we first -see Sec. 3.2.2 -present analytic results based on Eq. (3.11), which are quite close to the numerical ones.

ANALYTICAL RESULTS
The duration of the slow-roll IG inflation is controlled by the slow-roll parameters which, according to their definition in Eq. (3.3b), are calculated to be (3.14) The termination of IG inflation is triggered by the violation of the ǫ criterion at φ = φ f given by since the violation of the η criterion occurs at φ =φ f such that In the EF, φ f remains independent of c R and n, since substituting Eq. (3.15a) into Eq. (3.13) we obtain Given that φ f ≪ φ * , we can find a relation between φ * and N * as follows Obviously, IG inflation consistent with Eq. (3.7b) can be achieved if for N * ≃ 52. Therefore, we need relatively large c R 's which increase with n. On the other hand, φ * remains transplanckian, since plugging Eq. (3.17a) into Eq. (3.13) we find which gives φ * = 5.3m P for φ c = 0. Despite this fact, our construction remains stable under possible corrections from non-renormalizable terms in Ω H since these are expressed in terms of initial field Φ, and can be harmless for |Φ| ≤ m P .

4) we find A s as follows
for N * ≃ 52. Therefore, enforcing Eq. (3.4) we obtain a relation between λ and c R which turns out to be independent of n. Replacing φ * by Eq. (3.17a) into Eq. (3.5) we estimate, finally, the inflationary observable through the relations: for N * ≃ 52. These outputs are fully consistent with the observational data, Eq. (3.6).

NUMERICAL RESULTS
The inflationary scenario under consideration depends on the parameters: λ, c R , k S and T rh .
Our results are essentially independent of k S 's, provided that we choose them so as m 2 s > 0 for every allowed λ and c R -see Table 1. We therefore set k S = 1 throughout our calculation. We also choose Λ ≃ 10 13 GeV so as the one-loop corrections in Eq. (2.18) vanish at the SUSY vacuum, Eqs. (2.12b) and (2.6). Finally we choose T rh = 10 9 GeV as suggested by reliable post-inflationary scenariasee Ref. [11]. Upon substitution of V IG from Eqs. The variation of V IG as a function of φ for two different values of n can be easily inferred from Fig. 1, where we depict V IG versus φ for φ * = m P and n = 2 or n = 6. The imposition φ * = m P  Table 1 remain well above H IG both during and after IG inflation for the selected k S . E.g., for n = 3 and c R = 495 (corresponding to λ = 0.01) we obtain  6a). Therefore, the inclusion of the variant exponent n ≥ 2, compared to the initial model of Ref. [11], does not affect the successful predictions on the inflationary observables.

BEYOND NO-SCALE SUGRA
If we lift the assumption of no-scale SUGRA in Eq. (2.8), Ω takes its more general form, obtained by inserting Eqs. (2.7) and (2.9) into Eq. (2.5); the resulting through Eq. (2.2) Kähler potential is (3.23) where the factors of 2 are added just for convenience. The description of the inflationary potential, our analytical and numerical results are exhibited below in Secs. 3.3.1, 3.3.2 and 3.3.3 correspondingly.

THE INFLATIONARY POTENTIAL
The tree-level scalar potential in this case has its general form in Eq. (2.13a) where f R and f SΦ are calculated by employing their definitions in Eq. (2.13b) as follows Taking into account the form of f R above, V IG0 can be cast as follows Moreover, the EF canonically normalized inflaton, φ, is found via Eq. (2.17b) with J 2 given by Consequently, J turns out to be close to that obtained in Sec. 3.2.1. Following the standard procedure of Sec. 2.3 we construct the mass spectrum of the theory along the path of Eq. (2.14). The precise expressions of the relevant masses squared, taken into account in our numerical computation, are rather lengthy due to the numerous contributions to V IG0 , Eq. (3.25a). Our findings, though, can be considerably simplified, if we perform an expansion for small x φ 'sretaining f Φ intact -, consistently with our restriction, Eq. (3.7). If we keep the lowest order terms, the masses squared for the scalars reduce to those displayed in Table 1, whereas the mass squared of the chiral fermions shown in Table 1 has to be multiplied by the factor 1 + k SΦ c R x 2+n φ /2 n/2−1 n. (3.27) As in the case of Sec. 3.2, employing the mass spectrum along the direction of Eq. (2.14), we can calculate V IG in Eq. (2.18) to further analyze the model.

ANALYTICAL RESULTS
Upon substitution of Eq. (3.25b) into Eq. (3.3b), we can extract the slow-roll parameters which determine the strength of the inflationary stage. Performing expansions about x φ ≃ 0, as above, we can extract approximate expressions which assist us to interpret the numerical results presented in Sec. 3.3.3. Namely, we find As it may be numerically verified, φ * and φ f do not decline a lot from their values in Eqs. (3.17a) and (3.15a), which can be served for our estimations below. In particular, replacing V IG0 from Eq. (3.25b) in Eq. (3.4) we obtain (3.29) Comparing this expression with the one obtained in the case of no-scale SUGRA, Eq. (3.19), we remark that λ acquires a mild dependence on both k SΦ and n. Inserting, Eq. (3.17a) into Eq. (3.5) we can similarly provide an expression for n s . This is Therefore, a clear dependence of n s on n and k SΦ arises, with the second one being much more efficient. On the other hand, a s and r remain pretty close to those obtained in the absence of k SΦ -see Sec. 3.2.2.

NUMERICAL RESULTS
This inflationary scenario depends on the following parameters: λ, c R , k S , k SΦ , k Φ and T rh .
As in the case of Sec. 3.2.3 our results are independent of k S , provided that m 2 s > 0 -see in Table 1. The same is also valid for k Φ since the contribution from the second term in f R , Eq. (3.24), is overshadowed by the strong enough first term including c R ≫ 1. We therefore set k S = 1 and k Φ = 0.5. We also choose T rh = 10 9 GeV. Besides these values, in our numerical code, we use as input parameters c R , k SΦ and φ * . For every chosen c R ≥ 1, we restrict λ and φ * so that the conditions Eqs. (3.1), (3.4) and (3.7) are satisfied. By adjusting k SΦ we can achieve n s 's in the range of Eq. (3.6). Our results are displayed in Fig. 3-(a 1 ) and (a 2 ) [Fig. 3-(b 1 ) and (b 2 )], where we delineate the hatched regions allowed by Eqs. (3.1), (3.4), (3.6) and (3.7) in the λ − c R [λ − k SΦ ] plane. We take n = 2 in Fig. 3-(a 1 ) and (b 1 ) and n = 3 in Fig. 3-(a 2 ) and (b 2 ). The conventions adopted for the various lines are also shown. In particular, the dashed [dot-dashed] lines correspond to n s = 0.975 [n s = 0.946], whereas the solid (thick) lines are obtained by fixing n s = 0.96 -see Eq. (3.6). Along the thin line, which provides the lower bound for the regions presented in Fig. 3, the constraint of Eq. (3.7b) is saturated. At the other end, the perturbative bound on λ bounds the various regions.
From Fig. 3-(a 1 ) and (a 2 ) we see that c R remains almost proportional to λ and for constant λ, c R increases as n s decreases. From Fig. 3-(b 1 ) we remark that k SΦ is confined close to zero for n s = 0.96  (3.7) in the λ − c R plane (a 1 , a 2 ) and λ − k SΦ plane (b 1 , b 2 ) for k S = 1, k Φ = 0.5 and n = 2 (a 1 , b 1 ) or n = 3 (a 2 , b 2 ). The conventions adopted for the various lines are also shown. and λ < 0.16 or φ * > 0.1m P -see Eq. (3.17a). Therefore, a degree of tuning (of the order of 10 −2 ) is needed in order to reproduce the experimental data of Eq. (3.6a). On the other hand, for λ > 0.16 (or φ * < 0.1m P ), k SΦ takes quite natural (of order one) negative values -consistently with Eq. (3.30). This feature, however, does not insist for n = 3 -see Fig. 3-(b 2 ) -, where the allowed (hatched) region is considerably shrunk and so, k SΦ remains constantly below unity for any λ. As we explicitly verified, for n = 6 the results turn out to be even more concentrated about k SΦ ≃ 0. Therefore, we can conclude that this embedding of IG inflation in SUGRA favors low n values.

4 THE EFFECTIVE CUT-OFF SCALE
An outstanding trademark of IG inflation is that it is unitarity-safe, despite the fact that its implementation with subplanckian φ's -see Eq. (3.17b) -requires relatively large c R 's. To show this we below extract the UV cut-off scale, Λ UV , of the effective theory first in the JF -Sec. 4.1 -and then in the EF -see Sec. 4.2. Although the expansions about φ presented below are not valid [9] during IG inflation, we consider the extracted this way Λ UV as the overall cut-off scale of the theory, since reheating is an unavoidable stage of the inflationary dynamics [10].

JORDAN FRAME COMPUTATION
The possible problematic process in the JF, which causes [8] concerns about the unitarity-violation, is the δφ − δφ scattering process via s-channel graviton, h µν , exchange -δφ represents an excitation of φ about φ , see below. The relevant vertex is c R δφ 2 ✷h/m P -with h = h µ µ -can be derived from the first term in the right-hand side of Eq. (2.3) expanding the JF metric g µν about the flat spacetime metric η µν and the inflaton φ abound its v.e.v as follows: Retaining only the terms with two derivatives of the excitations, the part of the lagrangian corresponding to the two first terms in the right-hand side of Eq. (2.3) takes the form where δ R = 1/2 [δ R = 2 2/n n(n − 1)/8] for n = 2 [n > 2] and the functions F EH and F R read (4.2c) The JF canonically normalized fieldsh µν and δφ are defined by the relations The interaction originating from the last term in the right-hand side of Eq. (4.2a) gives rise to a scattering amplitude which is written in terms of the center-of-mass energy E as follows (up to irrelevant numerical prefactors) since Ω H = 1/2 ≪ m 2 P Ω H,φ 2 ≃ 2 2/n n 2 c 2/n R /8 and Λ UV is identified as the UV cut-off scale in the JF, since A remains within the validity of the perturbation theory provided that E < Λ UV . Obviously, the argument above can be equally well applied to both implementations of IG inflation in SUGRA -see Sec. 3.2 and 3.3 -since the extra terms included in Eq. (3.23) -compared to Eq. (3.8) -are small enough and do not generate any problem with the perturbative unitarity.

EINSTEIN FRAME COMPUTATION
Alternatively, Λ UV can be determined in EF, following the systematic approach of Ref. [10]. Note, in passing, that the EF (canonically normalized) inflaton, acquires mass which is given by Making use of Eq. (3.19) we find m δφ = 3·10 13 GeV for the case of no-scale SUGRA independently of the value of n -in accordance with the findings in Ref. [12]. Beyond no-scale SUGRA, replacing λ in Eq. (4.5) from Eq. (3.29), we find that m δφ inherits from λ a mild dependence on both n and k SΦ . E.g., for φ * = 0.5m P , n = 2 − 6 and n s in the range of Eq. The fact that δφ does not coincide with δφ -contrary to the standard Higgs inflation [8,9] -ensures that the IG models are valid up to m P . To show it, we write the EF action S in Eq. (2.1a) along the path of Eq. (2.14) as follows

CONCLUSIONS
In this work we showed that a wide class of IG inflationary models can be naturally embedded in standard SUGRA. Namely, we considered a superpotential which realize easily the IG idea and can be uniquely determined by imposing two global symmetries -a continuous R and a discrete Z n symmetry -in conjunction with the requirement that inflation has to occur for subplanckian values of the inflaton. On the other hand, we adopted two forms of Kähler potentials, one corresponding to the Kähler manifold SU (2, 1)/SU (2)×U (1) R ×Z n , inspired by no-scale SUGRA, and one more generic. In both cases, the tachyonic instability, occurring along the direction of the accompanying non-inflaton field, can be remedied by considering terms up to the fourth order in the Kähler potential. Thanks to the underlying symmetries the inflaton, φ appears predominantly as φ n in both the super-and Kähler potentials.
In the case of no-scale SUGRA, the inflaton is not mixed with the accompanying non-inflaton field in Kähler potential. As a consequence, the model predicts results identical to the non-SUSY case independently of the exponent n. In particular, we found n s ≃ 0.963, a s ≃ −0.00068 and r ≃ 0.0038, which are in excellent agreement with the current data, and m δφ = 3 · 10 13 GeV. Beyond no-scale SUGRA, all the possible terms up to the forth order in powers of the various fields are included in the Kähler potential. In this case, we can achieve n s precisely equal to its central observationally favored value, mildly tuning the coefficient k SΦ . Furthermore, a weak dependance of the results on n arises with the lower n's being more favored, since the required tuning on k SΦ is softer. In both cases a n-dependent lower bound on c R assists us to obtain inflation for subplanckian values of the inflaton, stabilizing thereby our proposal against possible corrections from higher order terms in Ω H . Furthermore we showed that the one-loop radiative corrections remain subdominant during inflation and the corresponding effective theory is trustable up to m P . Indeed, these models possess a built-in solution into long-standing naturalness problem [8,10] which plagued similar models. As demonstrated both in the EF and the JF, this solution relies on the dynamical generation of m P at the vacuum of the theory.
As a bottom line we could say that although no-scale SUGRA has been initially coined as a solution to the problem of SUSY breaking [13,16] ensuring a vanishing cosmological constant, it is by now recognized -see also [11,15,18] -that it provides a flexible framework for inflationary model building.
In fact, no-scale SUGRA is tailor-made for IG inflation since the predictive power of this inflationary model in more generic SUGRA incarnations is lost.

NOTE ADDED
When this work was under completion, results from BICEP2 experiment [26] were released according to which r = 0.16 +0.06 −0.05 . If we combine this result with Eq. (3.6c) we find a simultaneously compatible region 0.06 r 0.11 (at 95% c.l.) which, obviously, is not fulfilled by the models presented here, since the predicted r lies one order of magnitude lower -see Eq. (3.22) and comments below Eq. (3.31c). Although it is still premature to exclude any inflationary model with r lower than the above limit since the current data cannot definitively rule out other sources of gravitational waves -see e.g. Ref.
[27] -, we would like to emphasize that our conclusion related to the predictive power of no-scale SUGRA, thanks to simplifications of Eq. (2.8), remains valid. Indeed, no-scale SUGRA can assist us to supersymmetrize models with more exotic frame functions -see e.g. Ref. [6,28] which could ensure consistency with the aforementioned r values. These constructions do not remain generically intact beyond no-scale SUGRA.