# Starobinsky Inflation: From Non-SUSY to SUGRA Realizations

Authors: Constantinos Pallis; Nicolaos Toumbas
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

We review the realization of Starobinsky-type inflation within induced-gravitysupersymmetric (SUSY) and non-SUSY models. In both cases, inflation is in agreement with the current data and can be attained for sub-Planckian values of the inflation. The corresponding effective theories retain perturbative unitarity up to the Planck scale and the inflation mass is predicted to be 3·1013GeV. The supergravity embedding of these models is achieved by employing two gauge singlet chiral superfields, a superpotential that is uniquely determined by a continuous R and a discrete Zn symmetry and several (semi)logarithmic Kähler potentials that respect these symmetries. Checking various functional forms for the noninflation accompanying field in the Kähler potentials, we identify four cases which stabilize it without invoking higher order terms.


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## Body

## 1. Introduction

The idea that the universe underwent a period of exponential expansion, called inflation [1–3], has proven useful not only for solving the horizon and flatness problems of standard cosmology but also for providing an explanation for the scale invariant perturbations, which are responsible for generating the observed anisotropies in theCosmic Microwave Background (CMB). One of the first incarnations of inflation is due to Starobinsky. To date, this attractive scenario remains predictive, since it passes successfully all the observational tests [4, 5]. Starobinsky considered adding an R2 term, where R is the Ricci scalar, to the standard Einstein action in order to source inflation. Recall that gravity theories based on higher powers of R are equivalent to standard gravity theories with one additional scalar degree of freedom (see, e.g., [6, 7]). As a result, Starobinsky inflation is equivalent to inflation driven by a scalar field with a suitable potential, so it admits several interesting realizations [8–29].Following this route, we show in this work thatinduced-gravity inflation (IGI) [30–38] is effectively Starobinsky-like, reproducing the structure and the predictions of the original model. Within IGI, the inflation exhibits a strong coupling to R and the reduced Planck scale is dynamically generated through thevacuum expectation value (v.e.v.) of the inflation at the end of inflation. Therefore, the inflation acquires a Higgs-like behavior as in theories of induced gravity [36–42]. Apart from being compatible with data, the resulting theory respects perturbative unitarity up to the Planck scale [29–31]. Therefore, no concerns about the validity of the corresponding effective theory arise. This is to be contrasted with models ofnonminimal inflation (nMI) [43–54] based on a ϕn potential with negligible v.e.v. for the inflation ϕ. Although these models yield similar observational predictions with the Starobinsky model, they admit anultraviolet (UV) scale well below mP for n>2, leading to complications with naturalness [55–57].Nonetheless, IGI allows us to embed Starobinsky inflation withinN=1supergravity (SUGRA) in an elegant way. The embedding is achieved by incorporating two chiral superfields, a modulus-like field T and a matter-like field S appearing in the superpotential, W, as well as various Kähler potentials, K, consistent with an R and discrete Zn symmetries [29, 31, 58]; see also [20–22, 28, 32]. In some cases [20, 29, 31, 58], the employed K’s parameterize specific Kähler manifolds, which appear in no-scale models [59–61]. Moreover, this scheme ensures naturally a low enough reheating temperature, potentially consistent with the gravitino constraint [29, 62, 63] if connected with a version of theMinimal SUSY Standard Model (MSSM).An important issue in embedding IGI in SUGRA is the stabilization of the matter-like fieldS. Indeed, when K parameterizes the SU(2,1)/(SU(2)×U(1)) Kähler manifold [20, 21], the inflationary trajectory turns out to be unstablewith respect to the fluctuations of S. This difficulty can be overcome by adding a sufficiently large term kS|S|4, with kS>0 and |kS|~1, in the logarithmic function appearing in K, as suggested in [64] for models of nonminimal (chaotic) inflation [47–49] and applied in [50–54, 65–70]. This solution, however, deforms slightly the Kähler manifold [71]. More importantly, it violates the predictability of Starobinsky inflation, since mixed terms kST|S|2|T|2 with kST≥0.01, which cannot be ignored (without tuning), have an estimable impact [31, 72–74] on the dynamics and the observables. Moreover, this solution becomes complicated when more than two fields are considered, since all quartic terms allowed by symmetries have to be considered, and the analysis of the stabilization mechanism becomes tedious (see, e.g., [31, 72–74]). Alternatively, it was suggested to use a nilpotent superfield S [75] or a charged field under a gauged R symmetry [71].In this review, we revisit the issue of stabilizingS, disallowing terms of the form |S|2m, m>1, without caring much about the structure of the Kähler manifold. Namely, we investigate systematically several functions hi(|S|2) (with i=1,…,11) that appear in the choices for K, and we find four acceptable forms that lead to the stabilization of S during and after IGI. The output of this analysis is new, providing results that did not appear in the literature before. More specifically, we consider two principal classes of K’s, K3i, and K2i, distinguished by whether hi and T appear in the same logarithmic function. The resulting inflationary scenarios are almost indistinguishable. The case considered in [58] is included as one of the viable choices in K2i class. Contrary to [58], we impose here the same Zn symmetry on W and K. Consequently, the relevant expressions for the mass spectrum and the inflationary observables get simplified considerably compared to those displayed in [58]. As in the non-SUSY case, IGI may be realized using sub-Planckian values for the initial (noncanonically normalized) inflation field. The radiative corrections remain under control and perturbative unitarity is not violated up to mP [31, 58, 76], consistently with the consideration of SUGRA as an effective theory.Throughout this review we focus on the standardΛCDM cosmological model [4]. An alternative framework is provided by the running vacuum models [77–84] which turn out to yield a quality fit to observations, significantly better than that of ΛCDM. In this case, the acceleration of the universe, either during inflation or at late times, is not attributed to a scalar field but rather arises from the modification of the vacuum itself, which is dynamical. A SUGRA realization of Starobinsky inflation within this setting is obtained in [18].The plan of this paper is as follows. In Section2, we establish the realization of Starobinsky inflation as IGI in a non-SUSY framework. In Section 3 we introduce the formulation of IGI in SUGRA and revisit the issue of stabilizing the matter-like field S. The emerging inflationary models are analyzed in Section 4. Our conclusions are summarized in Section 5. Throughout, charge conjugation is denoted by a star (∗), the symbol ,z as subscript denotes derivation with respect to z, and we use units where the reduced Planck scale, mP=2.43·1018GeV, is set equal to unity.


## 2. Starobinsky Inflation from Induced Gravity

We begin our presentation demonstrating the connection betweenR2 inflation and IGI. We first review the formulation of nMI in Section 2.1 and then proceed to describe the inflationary analysis in Section 2.2. Armed with these prerequisites, we present R2 inflation as a type of nMI in Section 2.3 and exhibit its connection with IGI in Section 2.4.


### 2.1. Coupling Nonminimally the Inflation to Gravity

We consider an inflationϕ that is nonminimally coupled to the Ricci scalar R, via a coupling function fR(ϕ). We denote the inflation potential by VI(ϕ) and allow for a general kinetic function fK(ϕ)—in the cases of pure nMI [33–35, 45, 46] fK=1. TheJordan Frame (JF) action takes the form(1)S=∫d4x-g-12fRR+12fKgμν∂μϕ∂νϕ-VIϕ,where g is the determinant of the Friedmann-Robertson-Walker metric, gμν, with signature (+,-,-,-). We require 〈fR〉≃1 to ensure ordinary Einstein gravity at low energies.By performing a conformal transformation [45] to theEinstein frame (EF), we write the action(2)S=∫d4x-g^-12R^+12g^μν∂μϕ^∂νϕ^-V^Iϕ^,where a hat denotes an EF quantity. The EF metric is given by g^μν=fRgμν, and the canonically normalized field, ϕ^, and its potential, V^I, are defined as follows:(3)adϕ^dϕ=J=fKfR+32fR,ϕfR2,bV^I=VIfR2.For fR≫fK, the coupling function fR acquires a twofold role. On the one hand, it determines the relation between ϕ^ and ϕ. On the other hand, it controls the shape of V^I, thus affecting the observational predictions; see below. The analysis of nMI can be performed in the EF, using the standard slow-roll approximation. It is [33–35] completely equivalent with the analysis in the JF. We just have to keep track the relation between ϕ^ and ϕ.


### 2.2. Observational and Theoretical Constraints

A viable model of nMI must be compatible with a number of observational and theoretical requirements summarized in the following (cf. [85–88]).(1) The number of e-foldingsN^⋆ that the scale k⋆=0.05/Mpc experiences during inflation must be large enough for the resolution of the horizon and flatness problems of the standard hot Big Bang model; that is, [4, 45](4)N^⋆=∫ϕ^fϕ^⋆dϕ^V^IV^I,ϕ^=∫ϕfϕ⋆dϕJ2V^IV^I,ϕ≃61.7+ln⁡V^Iϕ⋆1/2V^Iϕf1/3+13ln⁡Trh+12ln⁡fRϕ⋆fRϕf1/3,where ϕ⋆[ϕ^⋆] is the value of ϕ[ϕ^] when k⋆ crosses the inflationary horizon. In deriving the formula above (cf. [65–67]), we take into account an equation-of-state with parameter wrh=0 [89], since V^I can be well approximated by a quadratic potential for low values of ϕ; see(20b), (32b), and (71b) below. Also Trh is the reheating temperature after nMI. We take a representative value Trh=4.1·10-10 throughout, which results in N^⋆≃53. The effective number of relativistic degrees of freedom at temperature Trh is taken grh=107.75 in accordance with the standard model spectrum. Lastly, ϕf[ϕ^f] is the value of ϕ[ϕ^] at the end of nMI, which in the slow-roll approximation can be obtained via the condition (5)max⁡ϵ^ϕf,η^ϕf=1,whereϵ^=12V^I,ϕ^V^I2=12J2V^I,ϕV^I2,η^=V^I,ϕ^ϕ^V^I=1J2V^I,ϕϕV^I-V^I,ϕV^IJ,ϕJ·Evidently nontrivial modifications of fR, and thus of J, may have a significant effect on the parameters above, modifying the inflationary observables.(2) The amplitudeAs of the power spectrum of the curvature perturbation generated by ϕ at k⋆ has to be consistent with the data [90]; that is,(6)As=123πV^Iϕ^⋆3/2V^I,ϕ^ϕ^⋆=Jϕ⋆23πV^Iϕ⋆3/2V^I,ϕϕ⋆≃4.627·10-5.As shown in Section 3.4, the remaining scalars in the SUGRA versions of nMI may be rendered heavy enough, so they do not contribute to As.(3) The remaining inflationary observables (the spectral indexns, its running αs, and the tensor-to-scalar ratio r) must be in agreement with the fitting of thePlanck,Baryon Acoustic Oscillations (BAO) and Bicep2/Keck Array data [4, 5] with the ΛCDM+r model; that is,(7)ans=0.968±0.009,br≤0.07,at the 95%confidence level (c.l.) with |αs|≪0.01. Although compatible with (7)(b), all data taken by the Bicep2/Keck Array CMB polarization experiments, up to the 2014 observational season (BK14) [5], seem to favor r’s of the order of 0.01, as the reported value is 0.028-0.025+0.026 at the 68% c.l.. These inflationary observables are estimated through the relations:(8)ans=1-6ϵ^⋆+2η^⋆,bαs=234η^⋆2-ns-12-2ξ^⋆,cr=16ϵ^⋆,where ξ^=V^I,ϕ^V^I,ϕ^ϕ^ϕ^/V^I2 and the variables with subscript ⋆ are evaluated at ϕ⋆.(4) The effective theory describing nMI remains valid up to a UV cutoff scaleΛUV, which has to be large enough to ensure the stability of our inflationary solutions; that is,(9)aV^Iϕ⋆1/4≤ΛUV,bϕ⋆≤ΛUV.As we show below, ΛUV≃1 for the models analyzed in this work, contrary to the cases of pure nMI with large fR, where ΛUV≪1. The determination of ΛUV is achieved expanding S in (2) about 〈ϕ〉. Although these expansions are not strictly valid [57] during inflation, we take ΛUV extracted this way to be the overall UV cutoff scale, since the reheating phase, realized via oscillations about 〈ϕ〉, is a necessary stage of the inflationary dynamics.


### 2.3. From Nonminimal toR2 Inflation

R
2 inflation can be viewed as a type of nMI, if we employ an auxiliary field ϕ with the following input ingredients:(10)fK=0,fR=1+4cRϕ,V^I=ϕ2.Using the equation of motion for the auxiliary field, ϕ=cRR, we obtain the action of the original Starobinsky model (see, e.g., [71]):(11)S=∫d4x-g-12R+cR2R2.As we can see from (10), the model has only one free parameter (cR), enough to render it consistent with the observational data, ensuring at the same time perturbative unitarity up to the Planck scale. Using (10) and (3), we obtain the EF quantities:(12)aJ=26cRfR,bV^I=ϕ2fR2≃116cR2.For cR≫1, the plot of V^I versus ϕ is depicted in Figure 1(a). An inflationary era can be supported since V^I becomes flat enough. To examine further this possibility, we calculate the slow-roll parameters. Plugging (12) into (5) yields(13)ϵ^=112cR2ϕ2,η^=1-4cRϕ12cR2ϕ2·Notice that η<0 since V^I is slightly concave downwards, as shown in Figure 1(a). The value of ϕ at the end of nMI is determined via (5), giving(14)ϕf=max⁡123cR,16cR⟹ϕf=123cR·Figure 1
The inflationary potentialV^I as a function of ϕ for R2 inflation (a) and IGI with ϕ⋆=1 (b). Values corresponding to ϕ⋆ and ϕf are also indicated.
(a)
(b)Under the assumption thatϕf≪ϕ⋆, we can obtain a relation between N^⋆ and ϕ⋆ via (4) (15)N^⋆≃3cRϕ⋆.The precise value of cR can be determined enforcing (6). Recalling that N^⋆≃53, we get(16)As1/2≃N^⋆122πcR=4.627·10-5⟹cR≃2.3·104.The resulting value of cR is large enough so that (17)ϕ⋆≃N^⋆3cR≃8.3·10-4≪1consistently with (9)(b); see Figure 1(a). Impressively, the remaining observables turn out to be compatible with the observational data of (7). Indeed, inserting the above value of ϕ⋆ into (8) (N^⋆=53), we get(18a)ns≃N^⋆-3N^⋆-1N^⋆2≃1-2N^⋆-3N^⋆2≃0.961;(18b)αs≃-N^⋆-34N^⋆+32N^⋆4≃-2N^⋆2-152N^⋆3≃-7.6·10-4;(18c)r≃12N^⋆2≃4.2·10-3.Without the simplification of (15), we obtain numerically ns=0.964, αs=-6.7·10-4, and r=3.7·10-3. We see that ns turns out to be appreciably lower than unity thanks to the negative values of η; see (13). The mass of the inflation at the vacuum is(19)m^δϕ=V^I,ϕ^ϕ^1/2=V^I,ϕϕJ21/2=123cR≃1.25·10-5i.e.3·1013GeV.As we show below this value is salient future in all models of Starobinsky inflation.Furthermore, the model provides an elegant solution to the unitarity problem [55–57], which plagues models of nMI with fR~ϕn≫fK, n>2, and fK=1. This stems from the fact that ϕ^ and ϕ do not coincide at the vacuum, as (12)(a) implies ϕ^=〈J〉ϕ=23cRϕ. In fact, if we expand the second term in theright-hand side (r.h.s.) of (2) about 〈ϕ〉=0, we find(20a)J2ϕ˙2=1-223ϕ^+2ϕ^2-⋯ϕ^˙2.Similarly, expanding V^I in (12)(b), we obtain (20b)V^I=ϕ^224cR21-223ϕ^+2ϕ^2-⋯.Since the coefficients of the above series are of order unity, independent of cR, we infer that the model does not face any problem with perturbative unitarity up to the Planck scale.


### 2.4. Induced-Gravity Inflation

It would be certainly beneficial to realize the structure and the predictions ofR2 inflation in a framework that deviates minimally from Einstein gravity, at least in the present cosmological era. To this extent, we incorporate the idea of induced gravity, according to which mP is generated dynamically [41, 42] via the v.e.v. of a scalar field ϕ, driving a phase transition in the early universe. The simplest way to implement this scheme is to employ a double-well potential for ϕ; for scale invariant realizations of this idea, see [39, 40]. On the other hand, an inflationary stage requires a sufficiently flat potential, as in (10). This can be achieved at large field values if we introduce a quadratic fR [33–38]. More explicitly, IGI may be defined as nMI with the following input ingredients:(21)fK=1,fR=cRϕ2,VI=λϕ2-M224.Given that 〈ϕ〉=M, we recover Einstein gravity at the vacuum if(22)fRϕ=1⟹M=1cR.We see that in this model there is one additional free parameter, namely, λ appearing in the potential, as compared to R2 model.Equations (3) and (21) imply(23)aJ≃6ϕ,bV^I=λfϕ24cR4ϕ4≃λ4cR2withfϕ=1-cRϕ2.For cR≫1, the plot of V^I versus ϕ is shown in Figure 1(b). As in R2 model, V^I develops a plateau, so an inflationary stage can be realized. To check its robustness, we compute the slow-roll parameters. Equations (5) and (23) give(24)ϵ^=43fϕ2,η^=41+fϕ3fϕ2·IGI is terminated when ϕ=ϕf, determined by the condition(25)ϕf=max⁡1+2/3cR,53cR⟹ϕf=1+2/3cR·Under the assumption thatϕf≪ϕ⋆, (4) implies the following relation between N^⋆ and ϕ⋆:(26)N^⋆≃3cRϕ⋆24⟹ϕ⋆≃2N^⋆3cR≫ϕf.Imposing (9)(b) and setting N^⋆≃53, we derive a lower bound on cR:(27)ϕ⋆≤1⟹cR≥4N^⋆3≃71.Contrary to R2 inflation, cR does not control exclusively the normalization of (6), thanks to the presence of an extra factor of λ. This is constrained to scale with cR. Indeed, we have(28)As1/2≃λN^⋆62πcR=4.627·10-5⟹cR≃42969λforN^⋆≃53.If, in addition, we impose the perturbative bound λ≤3.5, we end-up with following ranges:(29)77≲cR≲8.5·104,2.8·10-6≲λ≲3.5,where the lower bounds on cT and λ correspond to ϕ⋆=1; see Figure 1(b). Within the allowed ranges, m^δϕ remains constant, by virtue of (28). The mass turns out to be(30)m^δϕ=λ3cR≃1.25·10-5,essentially equal to that estimated in (19). Moreover, using (26) and (8), we extract the remaining observables(31a)ns=4N^⋆-154N^⋆+13-4N^⋆2≃1-2N^⋆-92N^⋆2≃0.961;(31b)αs=-128N^⋆4N^⋆+93-4N^⋆4≃-2N^⋆2-212N^⋆3≃-7.7·10-4;(31c)r=1923-4N^⋆2≃12N^⋆2≃4.4·10-3.Without making the approximation of (26), we obtain numerically (ns,αs,r)=(0.964,-6.6·10-4,3.7·10-3). These results practically coincide with those of R2 inflation, given in (18a)–(18c), and they are in excellent agreement with the observational data presented in (7).As in the previous section, the model retains perturbative unitarity up tomP. To verify this, we first expand the second term in the r.h.s. of (1) about δϕ^=ϕ-M≃0, with J given by (23)(a). We find(32a)J2ϕ˙2=1-23δϕ^+12δϕ^2-⋯δϕ^˙2withδϕ^≃6cRδϕ.Expanding V^I given by (23)(b), we get(32b)V^I=λ26cR2δϕ^21-32δϕ^+2524δϕ^2-⋯.Therefore, ΛUV=1 as for R2 inflation. Practically identical results can be obtained if we replace the quadratic exponents in (21) with n≥3 as first pointed out in [30]. This generalization can be elegantly performed [31, 32] within SUGRA, as we review below.


## 2.1. Coupling Nonminimally the Inflation to Gravity

We consider an inflationϕ that is nonminimally coupled to the Ricci scalar R, via a coupling function fR(ϕ). We denote the inflation potential by VI(ϕ) and allow for a general kinetic function fK(ϕ)—in the cases of pure nMI [33–35, 45, 46] fK=1. TheJordan Frame (JF) action takes the form(1)S=∫d4x-g-12fRR+12fKgμν∂μϕ∂νϕ-VIϕ,where g is the determinant of the Friedmann-Robertson-Walker metric, gμν, with signature (+,-,-,-). We require 〈fR〉≃1 to ensure ordinary Einstein gravity at low energies.By performing a conformal transformation [45] to theEinstein frame (EF), we write the action(2)S=∫d4x-g^-12R^+12g^μν∂μϕ^∂νϕ^-V^Iϕ^,where a hat denotes an EF quantity. The EF metric is given by g^μν=fRgμν, and the canonically normalized field, ϕ^, and its potential, V^I, are defined as follows:(3)adϕ^dϕ=J=fKfR+32fR,ϕfR2,bV^I=VIfR2.For fR≫fK, the coupling function fR acquires a twofold role. On the one hand, it determines the relation between ϕ^ and ϕ. On the other hand, it controls the shape of V^I, thus affecting the observational predictions; see below. The analysis of nMI can be performed in the EF, using the standard slow-roll approximation. It is [33–35] completely equivalent with the analysis in the JF. We just have to keep track the relation between ϕ^ and ϕ.


## 2.2. Observational and Theoretical Constraints

A viable model of nMI must be compatible with a number of observational and theoretical requirements summarized in the following (cf. [85–88]).(1) The number of e-foldingsN^⋆ that the scale k⋆=0.05/Mpc experiences during inflation must be large enough for the resolution of the horizon and flatness problems of the standard hot Big Bang model; that is, [4, 45](4)N^⋆=∫ϕ^fϕ^⋆dϕ^V^IV^I,ϕ^=∫ϕfϕ⋆dϕJ2V^IV^I,ϕ≃61.7+ln⁡V^Iϕ⋆1/2V^Iϕf1/3+13ln⁡Trh+12ln⁡fRϕ⋆fRϕf1/3,where ϕ⋆[ϕ^⋆] is the value of ϕ[ϕ^] when k⋆ crosses the inflationary horizon. In deriving the formula above (cf. [65–67]), we take into account an equation-of-state with parameter wrh=0 [89], since V^I can be well approximated by a quadratic potential for low values of ϕ; see(20b), (32b), and (71b) below. Also Trh is the reheating temperature after nMI. We take a representative value Trh=4.1·10-10 throughout, which results in N^⋆≃53. The effective number of relativistic degrees of freedom at temperature Trh is taken grh=107.75 in accordance with the standard model spectrum. Lastly, ϕf[ϕ^f] is the value of ϕ[ϕ^] at the end of nMI, which in the slow-roll approximation can be obtained via the condition (5)max⁡ϵ^ϕf,η^ϕf=1,whereϵ^=12V^I,ϕ^V^I2=12J2V^I,ϕV^I2,η^=V^I,ϕ^ϕ^V^I=1J2V^I,ϕϕV^I-V^I,ϕV^IJ,ϕJ·Evidently nontrivial modifications of fR, and thus of J, may have a significant effect on the parameters above, modifying the inflationary observables.(2) The amplitudeAs of the power spectrum of the curvature perturbation generated by ϕ at k⋆ has to be consistent with the data [90]; that is,(6)As=123πV^Iϕ^⋆3/2V^I,ϕ^ϕ^⋆=Jϕ⋆23πV^Iϕ⋆3/2V^I,ϕϕ⋆≃4.627·10-5.As shown in Section 3.4, the remaining scalars in the SUGRA versions of nMI may be rendered heavy enough, so they do not contribute to As.(3) The remaining inflationary observables (the spectral indexns, its running αs, and the tensor-to-scalar ratio r) must be in agreement with the fitting of thePlanck,Baryon Acoustic Oscillations (BAO) and Bicep2/Keck Array data [4, 5] with the ΛCDM+r model; that is,(7)ans=0.968±0.009,br≤0.07,at the 95%confidence level (c.l.) with |αs|≪0.01. Although compatible with (7)(b), all data taken by the Bicep2/Keck Array CMB polarization experiments, up to the 2014 observational season (BK14) [5], seem to favor r’s of the order of 0.01, as the reported value is 0.028-0.025+0.026 at the 68% c.l.. These inflationary observables are estimated through the relations:(8)ans=1-6ϵ^⋆+2η^⋆,bαs=234η^⋆2-ns-12-2ξ^⋆,cr=16ϵ^⋆,where ξ^=V^I,ϕ^V^I,ϕ^ϕ^ϕ^/V^I2 and the variables with subscript ⋆ are evaluated at ϕ⋆.(4) The effective theory describing nMI remains valid up to a UV cutoff scaleΛUV, which has to be large enough to ensure the stability of our inflationary solutions; that is,(9)aV^Iϕ⋆1/4≤ΛUV,bϕ⋆≤ΛUV.As we show below, ΛUV≃1 for the models analyzed in this work, contrary to the cases of pure nMI with large fR, where ΛUV≪1. The determination of ΛUV is achieved expanding S in (2) about 〈ϕ〉. Although these expansions are not strictly valid [57] during inflation, we take ΛUV extracted this way to be the overall UV cutoff scale, since the reheating phase, realized via oscillations about 〈ϕ〉, is a necessary stage of the inflationary dynamics.


## 2.3. From Nonminimal toR2 Inflation

R
2 inflation can be viewed as a type of nMI, if we employ an auxiliary field ϕ with the following input ingredients:(10)fK=0,fR=1+4cRϕ,V^I=ϕ2.Using the equation of motion for the auxiliary field, ϕ=cRR, we obtain the action of the original Starobinsky model (see, e.g., [71]):(11)S=∫d4x-g-12R+cR2R2.As we can see from (10), the model has only one free parameter (cR), enough to render it consistent with the observational data, ensuring at the same time perturbative unitarity up to the Planck scale. Using (10) and (3), we obtain the EF quantities:(12)aJ=26cRfR,bV^I=ϕ2fR2≃116cR2.For cR≫1, the plot of V^I versus ϕ is depicted in Figure 1(a). An inflationary era can be supported since V^I becomes flat enough. To examine further this possibility, we calculate the slow-roll parameters. Plugging (12) into (5) yields(13)ϵ^=112cR2ϕ2,η^=1-4cRϕ12cR2ϕ2·Notice that η<0 since V^I is slightly concave downwards, as shown in Figure 1(a). The value of ϕ at the end of nMI is determined via (5), giving(14)ϕf=max⁡123cR,16cR⟹ϕf=123cR·Figure 1
The inflationary potentialV^I as a function of ϕ for R2 inflation (a) and IGI with ϕ⋆=1 (b). Values corresponding to ϕ⋆ and ϕf are also indicated.
(a)
(b)Under the assumption thatϕf≪ϕ⋆, we can obtain a relation between N^⋆ and ϕ⋆ via (4) (15)N^⋆≃3cRϕ⋆.The precise value of cR can be determined enforcing (6). Recalling that N^⋆≃53, we get(16)As1/2≃N^⋆122πcR=4.627·10-5⟹cR≃2.3·104.The resulting value of cR is large enough so that (17)ϕ⋆≃N^⋆3cR≃8.3·10-4≪1consistently with (9)(b); see Figure 1(a). Impressively, the remaining observables turn out to be compatible with the observational data of (7). Indeed, inserting the above value of ϕ⋆ into (8) (N^⋆=53), we get(18a)ns≃N^⋆-3N^⋆-1N^⋆2≃1-2N^⋆-3N^⋆2≃0.961;(18b)αs≃-N^⋆-34N^⋆+32N^⋆4≃-2N^⋆2-152N^⋆3≃-7.6·10-4;(18c)r≃12N^⋆2≃4.2·10-3.Without the simplification of (15), we obtain numerically ns=0.964, αs=-6.7·10-4, and r=3.7·10-3. We see that ns turns out to be appreciably lower than unity thanks to the negative values of η; see (13). The mass of the inflation at the vacuum is(19)m^δϕ=V^I,ϕ^ϕ^1/2=V^I,ϕϕJ21/2=123cR≃1.25·10-5i.e.3·1013GeV.As we show below this value is salient future in all models of Starobinsky inflation.Furthermore, the model provides an elegant solution to the unitarity problem [55–57], which plagues models of nMI with fR~ϕn≫fK, n>2, and fK=1. This stems from the fact that ϕ^ and ϕ do not coincide at the vacuum, as (12)(a) implies ϕ^=〈J〉ϕ=23cRϕ. In fact, if we expand the second term in theright-hand side (r.h.s.) of (2) about 〈ϕ〉=0, we find(20a)J2ϕ˙2=1-223ϕ^+2ϕ^2-⋯ϕ^˙2.Similarly, expanding V^I in (12)(b), we obtain (20b)V^I=ϕ^224cR21-223ϕ^+2ϕ^2-⋯.Since the coefficients of the above series are of order unity, independent of cR, we infer that the model does not face any problem with perturbative unitarity up to the Planck scale.


## 2.4. Induced-Gravity Inflation

It would be certainly beneficial to realize the structure and the predictions ofR2 inflation in a framework that deviates minimally from Einstein gravity, at least in the present cosmological era. To this extent, we incorporate the idea of induced gravity, according to which mP is generated dynamically [41, 42] via the v.e.v. of a scalar field ϕ, driving a phase transition in the early universe. The simplest way to implement this scheme is to employ a double-well potential for ϕ; for scale invariant realizations of this idea, see [39, 40]. On the other hand, an inflationary stage requires a sufficiently flat potential, as in (10). This can be achieved at large field values if we introduce a quadratic fR [33–38]. More explicitly, IGI may be defined as nMI with the following input ingredients:(21)fK=1,fR=cRϕ2,VI=λϕ2-M224.Given that 〈ϕ〉=M, we recover Einstein gravity at the vacuum if(22)fRϕ=1⟹M=1cR.We see that in this model there is one additional free parameter, namely, λ appearing in the potential, as compared to R2 model.Equations (3) and (21) imply(23)aJ≃6ϕ,bV^I=λfϕ24cR4ϕ4≃λ4cR2withfϕ=1-cRϕ2.For cR≫1, the plot of V^I versus ϕ is shown in Figure 1(b). As in R2 model, V^I develops a plateau, so an inflationary stage can be realized. To check its robustness, we compute the slow-roll parameters. Equations (5) and (23) give(24)ϵ^=43fϕ2,η^=41+fϕ3fϕ2·IGI is terminated when ϕ=ϕf, determined by the condition(25)ϕf=max⁡1+2/3cR,53cR⟹ϕf=1+2/3cR·Under the assumption thatϕf≪ϕ⋆, (4) implies the following relation between N^⋆ and ϕ⋆:(26)N^⋆≃3cRϕ⋆24⟹ϕ⋆≃2N^⋆3cR≫ϕf.Imposing (9)(b) and setting N^⋆≃53, we derive a lower bound on cR:(27)ϕ⋆≤1⟹cR≥4N^⋆3≃71.Contrary to R2 inflation, cR does not control exclusively the normalization of (6), thanks to the presence of an extra factor of λ. This is constrained to scale with cR. Indeed, we have(28)As1/2≃λN^⋆62πcR=4.627·10-5⟹cR≃42969λforN^⋆≃53.If, in addition, we impose the perturbative bound λ≤3.5, we end-up with following ranges:(29)77≲cR≲8.5·104,2.8·10-6≲λ≲3.5,where the lower bounds on cT and λ correspond to ϕ⋆=1; see Figure 1(b). Within the allowed ranges, m^δϕ remains constant, by virtue of (28). The mass turns out to be(30)m^δϕ=λ3cR≃1.25·10-5,essentially equal to that estimated in (19). Moreover, using (26) and (8), we extract the remaining observables(31a)ns=4N^⋆-154N^⋆+13-4N^⋆2≃1-2N^⋆-92N^⋆2≃0.961;(31b)αs=-128N^⋆4N^⋆+93-4N^⋆4≃-2N^⋆2-212N^⋆3≃-7.7·10-4;(31c)r=1923-4N^⋆2≃12N^⋆2≃4.4·10-3.Without making the approximation of (26), we obtain numerically (ns,αs,r)=(0.964,-6.6·10-4,3.7·10-3). These results practically coincide with those of R2 inflation, given in (18a)–(18c), and they are in excellent agreement with the observational data presented in (7).As in the previous section, the model retains perturbative unitarity up tomP. To verify this, we first expand the second term in the r.h.s. of (1) about δϕ^=ϕ-M≃0, with J given by (23)(a). We find(32a)J2ϕ˙2=1-23δϕ^+12δϕ^2-⋯δϕ^˙2withδϕ^≃6cRδϕ.Expanding V^I given by (23)(b), we get(32b)V^I=λ26cR2δϕ^21-32δϕ^+2524δϕ^2-⋯.Therefore, ΛUV=1 as for R2 inflation. Practically identical results can be obtained if we replace the quadratic exponents in (21) with n≥3 as first pointed out in [30]. This generalization can be elegantly performed [31, 32] within SUGRA, as we review below.


## 3. Induced-Gravity Inflation in SUGRA

In Section3.1, we present the general SUGRA setting, where IGI is embedded. Then, in Section 3.2, we examine a variety of Kähler potentials, which lead to the desired inflationary potential; see Section 3.3. We check the stability of the inflationary trajectory in Section 3.4.


### 3.1. The General Set-Up

To realize IGI within SUGRA [29, 31, 32, 58], we must use two gauge singlet chiral superfields zα, with z1=T and z2=S being the inflation and a “stabilizer” superfield, respectively. Throughout this work, the complex scalar fields zα are denoted by the same superfield symbol. The EF effective action is written as follows [47–49]:(33a)S=∫d4x-g^-12R^+Kαβ-g^μν∂μzα∂νz∗β--V^,where Kαβ¯=K,zαz∗β¯ is the Kähler metric and Kαβ¯ its inverse (Kαβ¯Kβ¯γ=δγα). V^ is the EF F-term SUGRA potential, given in terms of the Kähler potential K and the superpotential W by the following expression:(33b)V^=eKKαβ¯DαWDβ¯∗W∗-3W2withDαW=W,zα+K,zαW.Conformally transforming to the JF withfR=-Ω/N, where N is a dimensionless positive parameter, S takes the form(34)S=∫d4x-gΩ2NR+34NΩ∂μΩ∂μΩ-1NΩKαβ¯∂μzα∂μz∗β¯-VwithV=Ω2N2V^.Note that N=3 reproduces the standard set-up [47–49]. Let us also relate Ω and K by(35)-ΩN=e-K/N⟹K=-Nln⁡-ΩN.Then taking into account the definition [47–49] of the purely bosonic part of the auxiliary field when on shell,(36)Aμ=iKα∂μzα-Kα¯∂μz∗α¯6,we arrive at the following action:(37a)S=∫d4x-gΩ2NR+Ωαβ-+3-NNΩαΩβ-Ω∂μzα∂μz∗β--27N3ΩAμAμ-V.By virtue of (35), Aμ takes the form(37b)Aμ=-iNΩα∂μzα-Ωα¯∂μz∗α¯6Ωwith Ωα=Ω,zα and Ωα¯=Ω,z∗α¯. As can be seen from (37a), -Ω/N introduces a nonminimal coupling of the scalar fields to gravity. Ordinary Einstein gravity is recovered at the vacuum when(38)-ΩN≃1.Starting with the JF action in (37a), we seek to realize IGI, postulating the invariance of Ω under the action of a global Zn discrete symmetry. With S stabilized at the origin, we write(39)-ΩN=ΩHT+ΩH∗T∗withΩHT=cTTn+∑k=2∞λkTkn,where k is a positive integer. If T≤1 during IGI and assuming that λk’s are relatively small, the contributions of the higher powers of T in the expression above are small, and these can be dropped. As we verify later, this can be achieved when the coefficient cT is large enough. Equivalently, we may rescale the inflation, setting T→T~=cT1/nT. Then the coefficients λk of the higher powers in the expression of Ω get suppressed by factors of cT-k. Thus, Zn and the requirement T≤1 determine the form of Ω, avoiding a severe tuning of the coefficients λk. Under these assumptions, K in (35) takes the form(40)K0=-Nln⁡fT+f∗T∗withfT≃cTTn,where S is assumed to be stabilized at the origin.Equations (35) and (38) require that T and S acquire the following v.e.v.s:(41)T≃12cT1/n,S=0.These v.e.v.s can be achieved, if we choose the following superpotential [31, 32]:(42)W=λSTn-12cT.Indeed the corresponding F-term SUSY potential, VSUSY, is found to be(43)VSUSY=λ2Tn-12cT2+λ2n2STn-12and is minimized by the field configuration in (21).As emphasized in [29, 31, 58], the forms of W and ΩH can be uniquely determined if we limit ourselves to integer values for n (with n>1) and T≤1 and impose two symmetries:(i)
AnR symmetry under which S and T have charges 1 and 0, respectively.(ii)
A discrete symmetryZn under which only T is charged.For simplicity we assume here that both W and ΩH respect the same Zn, contrary to the situation in [58]. This assumption simplifies significantly the formulae in Sections 3.3 and 3.4. Note, finally, that the selected Ω in (39) does not contribute in the kinetic term involving ΩTT∗ in (37a). We expect that our findings are essentially unaltered even if we include in the r.h.s. of (39) a term -(T-T∗)2/2N [32] or -|T|2/N [31] which yields ΩTT∗=1≪cT; the former choice, though, violates Zn symmetry above.


### 3.2. Proposed Kähler Potentials

It is obvious from the considerations above that the stabilization ofS at zero during and after IGI is of crucial importance for the viability of our scenario. This key issue can be addressed if we specify the dependence of the Kähler potential on S. We distinguish the following basic cases:(44)K3i=-n3ln⁡fT+f∗T∗+hiX,K2i=-n2ln⁡fT+f∗T∗+hiX,where the various choices hi, i=1,…,11, are specified in Table 1 and X is defined as follows:(45)X=-S2n3forK=K3iS2forK=K2i.As shown in Table 1 we consider exponential, logarithmic, trigonometric, and hyperbolic functions. Note that K31 and K21 parameterize SU(2,1)/SU(2)×U(1) and SU(1,1)/U(1)×U(1) Kähler manifolds, respectively, whereas K23 parameterizes the SU(1,1)/U(1)×SU(2)/U(1) Kähler manifold; see [58].Table 1
Definition of the varioushi(X)’s, hi′′(0)=d2hi(0)/dX2 and masses squared of the fluctuations of s and s- along the inflationary trajectory in (46) for K=K3i and K2i.
i
h
i
(
X
)
h
i
′
′
(
0
)
m
^
s
2
/
H
^
I
2
K
=
K
3
i
K
=
K
2
i
1
X
0
-
2
+
2
n
/
f
ϕ
2
3
⋅
2
n
-
1
/
f
ϕ
2
2
e
X
-
1
1
2
(
4
-
n
)
/
2
c
T
ϕ
n
-
2
+
2
n
/
f
ϕ
2
-
6
+
3
⋅
2
n
-
1
/
f
ϕ
2
3
ln
⁡
(
X
+
1
)
-
1
-
2
(
1
+
2
1
-
n
/
2
c
T
ϕ
n
)
6
(
1
+
2
n
-
1
/
f
ϕ
2
)
4
-
cos
⁡
(
arcsin
⁡
1
+
X
)
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
5
sin
⁡
(
arccos
⁡
1
+
X
)
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
6
tan
⁡
X
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
7
-
cot
⁡
(
arcsin
⁡
1
+
X
)
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
8
cosh
⁡
(
arcsin
⁡
1
+
X
)
-
2
2
2
(
5
-
n
)
/
2
c
T
ϕ
n
-
2
+
2
n
/
f
ϕ
2
3
⋅
2
n
-
1
/
f
ϕ
2
-
6
2
9
sinh
⁡
(
X
)
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
10
tanh
⁡
(
X
)
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
11
-
coth
⁡
(
arcsinh
⁡
1
+
X
)
+
2
-
2
2
2
n
/
f
ϕ
2
-
2
(
7
-
n
)
/
2
c
T
ϕ
n
-
2
3
⋅
2
n
-
1
/
f
ϕ
2
+
12
2To show that the proposedK’s are suitable for IGI, we have to verify that they reproduce V^I in (23)(b) when n=2, and they ensure the stability of S at zero. These requirements are checked in the following two sections.


### 3.3. Derivation of the Inflationary Potential

SubstitutingW of (42) and a choice for K in (44) (with the hi’s given in Table 1) into (33b), we obtain a potential suitable for IGI. The inflationary trajectory is defined by the constraints(46)S=T-T∗=0,ors=s¯=θ=0,where we have expanded T and S in real and imaginary parts as follows:(47)T=ϕ2eiθ,S=s+is¯2·Along the path of (46), V^ reads(48)V^I=V^θ=s=s¯=0=eKKSS∗W,S2.From (42) we get W,S=f-1/2. Also, (44) yields(49)eK=2f+hi0-n3forK=K3iehi02fn2forK=K2i,where we take into account that f(T)=f∗(T∗) along the path of (46). Moreover, KSS∗=1/KSS∗ can be obtained from the Kähler metric, which is given by(50)Kαβ¯=diagKTT∗,KSS∗=diagn3n22ϕ2,hi′02f+hi0forK=K3idiagn2n22ϕ2,hi′0forK=K2i,where a prime denotes a derivative with respect to X. Note that KTT∗ for K=K2i (and S≠0) does not involve the field S in its denominator, and so no geometric destabilization [91] can be activated, contrary to the K=K3i case. Inserting W,S and the results of (49) and (50) into (33b), we obtain(51)V^I=λ21-2f2cT2·2f+hi01-n3hi′0forK=K3iehi02fn2hi′0forK=K2i.Recall thatf~ϕn; see (40). Then V^I develops a plateau, with almost constant potential energy density, for cT≫1 and ϕ<1 (or cT=1 and ϕ≫1), if we impose the following conditions:(52)2n=nn3-1forK=K3inn2forK=K2i⟹n3=3forK=K3in2=2forK=K2i.This empirical criterion is very precise since the data on ns allows only tiny (of order 0.001) deviations [28]. Actually, the requirement cT≫1 and the synergy between the exponents in W and K’s assist us to tame the well-known η problem within SUGRA with a mild tuning. If we insert (52) into (51) and compare the result for n=2 with (23)(b) (replacing also λ2 with λ), we see that the two expressions coincide, if we set(53)hi0=0,hi′0=1.As we can easily verify the selected hi in Table 1 satisfy these conditions. Consequently, V^I in (51) and the corresponding Hubble parameter H^I take their final form:(54)aV^I=λ2fϕ24cT4ϕ2n,bH^I=V^I1/23=λfϕ23cT2ϕn,with fϕ=2n/2-1-cTϕn<0 reducing to that defined in (23)(b). Based on these expressions, we investigate in Section 4 the dynamics and predictions of IGI.


### 3.4. Stability of the Inflationary Trajectory

We proceed to check the stability of the direction in (46) with respect to the fluctuations of the various fields. To this end, we have to examine the validity of the extremum and minimum conditions; that is,(55)a∂V^I∂z¯^αs=s¯=θ=0=0,bm^z¯α2>0withz¯α=θ,s,s¯.Here m^z¯α2 are the eigenvalues of the mass matrix with elements(56)M^αβ2=∂2V^I∂z¯^α∂z¯^βs=s¯=θ=0withz¯α=θ,s,s¯and a hat denotes the EF canonically normalized field. The canonically normalized fields can be determined if we bring the kinetic terms of the various scalars in (33a) into the following form:(57a)Kαβ-z˙αz˙∗β-=12ϕ^˙2+θ^˙2+12s^˙2+s-^˙2,where a dot denotes a derivative with respect to the JF cosmic time. Then the hatted fields are defined as follows:(57b)dϕ^dϕ=J=KTT∗,θ^=Jθϕ,s^,s¯^=KSS∗s,s¯,where by virtue of (52) and (53), the Kähler metric of (50) reads(57c)Kαβ¯=diagKTT∗,KSS∗=3n22ϕ2,2n/2-1cTϕnforK=K3in2ϕ2,1forK=K2i.Note that the spinor componentsψT and ψS of the superfields T and S are normalized in a similar way; that is, ψ^T=KTT∗ψΦ and ψ^S=KSS∗ψS. In practice, we have to make sure that all the m^z¯α2’s are not only positive, but also greater than H^I2 during the last 50–60 e-foldings of IGI. This guarantees that the observed curvature perturbation is generated solely by ϕ, as assumed in (6). Nonetheless, two-field inflationary models which interpolate between the Starobinsky and the quadratic model have been analyzed in [92–95]. Due to the large effective masses that the scalars acquire during IGI, they enter a phase of damped oscillations about zero. As a consequence, ϕ dependence in their normalization (see (57b)) does not affect their dynamics.We can readily verify that (55)(a) is satisfied for all the three z¯α’s. Regarding (55)(b), we diagonalize M^αβ2 (56) and we obtain the scalar mass spectrum along the trajectory of (46). Our results are listed in Table 1 together with the masses squared m^ψ±2 of the chiral fermion eigenstates ψ^±=(ψ^T±ψ^S)/2. From these results, we deduce the following:(i)
For both classes ofK’s in (44), (55)(b) is satisfied for the fluctuations of θ^; that is, m^θ2>0, since fϕ<0. Moreover, m^θ2≫H^I2 because cT≫1.(ii)
WhenK=K3i and hi′′(0)=0, we obtain m^s2<0. This occurs for i=1,4,…,7,9 and 10, as shown in Table 1. For i=1, our result reproduces those of similar models [31, 47–52, 68–70]. The stability problem can be cured if we include in K3i a higher order term of the form kS|S|4 with kS~1, or assuming that S2=0 [75]. However, a probably simpler solution arises if we take into account the results accumulated in Table 2. It is clear that the condition m^s2>H^I2 can be satisfied when hi′′(0)>0 with |hi′′(0)|≥1. From Table 1, we see that this is the case for i=2 and 8.(iii)
WhenK=K2i and hi′′(0)=0, we obtain m^s2>0, but m^s2<H^I2. Therefore, S may seed inflationary perturbations, leading possibly to large non-Gaussianities in the CMB, contrary to observations. From the results listed in Table 2, we see that the condition m^s2≫H^I2 requires hi′′(0)<0 with |hi′′(0)|≥1. This occurs for i=3 and 11. The former case was examined in [58].Table 2
Mass-squared spectrum forK=K3i and K2i along the inflationary trajectory in (46).
Fields
Eigenstates
Masses squared
K=K3i
K=K2i
1 real scalar
θ^
m^θ2/H^I2
4
(
2
n
-
2
-
c
T
ϕ
n
f
ϕ
)
/
f
ϕ
2
6
(
2
n
-
2
-
c
T
ϕ
n
f
ϕ
)
/
f
ϕ
2
1 complex
s
^
,
s
-
^
m^s2/H^I2
2
n
/
f
ϕ
2
-
2
3
⋅
2
n
-
1
/
f
ϕ
2
scalar
+
2
2
-
n
/
2
c
T
ϕ
n
h
i
′
′
(
0
)
-
6
h
i
′
′
(
0
)
2 Weyl spinors
ψ
^
±
m^ψ±2/H^I2
2n/fϕ2
6⋅2n-3/fϕ2To highlight further the stabilization ofS during and after IGI we present in Figure 2m^s2/H^I2 as a function of ϕ for the various acceptable K’s identified above. In particular, we fix n=2 and ϕ⋆=1, setting K=K32 or K=K38 in Figure 2(a) and K=K23 or K=K2,11 in Figure 2(b). The parameters of the models (λ and cT) corresponding to these choices are listed in third and fifth rows of Table 3. Evidently m^s2/H^I2 remain larger than unity for ϕf≤ϕ≤ϕ⋆, where ϕ⋆ and ϕf are also depicted. However, in Figure 2(b)  m^s2/H^I2 exhibits a constant behavior and increases sharply as ϕ decreases below 0.2. On the contrary, m^s2/H^I2 in Figure 2(a) is an increasing function of ϕ for ϕ≳0.2, with a clear minimum at ϕ≃0.2. For ϕ≲0.2, m^s2/H^I2 increases drastically as in Figure 2(b) too.Table 3
Input and output parameters of the models which are compatible with (4) for N^⋆=53.2, (6), and (7).
Kähler potentialK
Parameters
Input
Output
n
c
T
ϕ
⋆
λ
(
1
0
-
3
)
ϕ
f
n
s
α
s
(
1
0
-
4
)
r
(
1
0
-
3
)
K
3
i
1
1
54.5
0.022
1.5
0.964
-6.3
3.6
K
2
i
1
1
80
0.028
1.7
0.964
-6.6
2.5
K
3
i
2
77
1
1.7
0.17
0.964
-6.7
3.7
K
3
i
3
109
1
2.4
0.3
0.964
-6.5
3.7
K
2
i
2
113
1
2
0.15
0.964
-6.7
2.5
K
2
i
3
159
1
3
0.3
0.964
-6.7
2.6Figure 2
The ratiom^s2/H^I2 as a function of ϕ for n=2 and ϕ⋆=1. We set (a) K=K32 or K=K38 and (b) K=K23 or K=K2,11. The values corresponding to ϕ⋆ and ϕf are also depicted.
(a)
(b)Employing the well-known Coleman-Weinberg formula [96], we find from the derived mass spectrum (see Table 1) the one-loop radiative corrections, ΔV^I, to V^I, depending on renormalization group mass scale Λ. It can be verified that our results are insensitive to ΔV^I, provided that Λ is determined by requiring ΔV^I(ϕ⋆)=0 or ΔV^I(ϕf)=0. A possible dependence of the results on the choice of Λ is totally avoided [31] thanks to the smallness of ΔV^I, for Λ≃(1-1.8)·10-5; see Section 4.2 too. These conclusions hold even for ϕ>1. Therefore, our results can be accurately reproduced by using exclusively V^I in (54)(a).


## 3.1. The General Set-Up

To realize IGI within SUGRA [29, 31, 32, 58], we must use two gauge singlet chiral superfields zα, with z1=T and z2=S being the inflation and a “stabilizer” superfield, respectively. Throughout this work, the complex scalar fields zα are denoted by the same superfield symbol. The EF effective action is written as follows [47–49]:(33a)S=∫d4x-g^-12R^+Kαβ-g^μν∂μzα∂νz∗β--V^,where Kαβ¯=K,zαz∗β¯ is the Kähler metric and Kαβ¯ its inverse (Kαβ¯Kβ¯γ=δγα). V^ is the EF F-term SUGRA potential, given in terms of the Kähler potential K and the superpotential W by the following expression:(33b)V^=eKKαβ¯DαWDβ¯∗W∗-3W2withDαW=W,zα+K,zαW.Conformally transforming to the JF withfR=-Ω/N, where N is a dimensionless positive parameter, S takes the form(34)S=∫d4x-gΩ2NR+34NΩ∂μΩ∂μΩ-1NΩKαβ¯∂μzα∂μz∗β¯-VwithV=Ω2N2V^.Note that N=3 reproduces the standard set-up [47–49]. Let us also relate Ω and K by(35)-ΩN=e-K/N⟹K=-Nln⁡-ΩN.Then taking into account the definition [47–49] of the purely bosonic part of the auxiliary field when on shell,(36)Aμ=iKα∂μzα-Kα¯∂μz∗α¯6,we arrive at the following action:(37a)S=∫d4x-gΩ2NR+Ωαβ-+3-NNΩαΩβ-Ω∂μzα∂μz∗β--27N3ΩAμAμ-V.By virtue of (35), Aμ takes the form(37b)Aμ=-iNΩα∂μzα-Ωα¯∂μz∗α¯6Ωwith Ωα=Ω,zα and Ωα¯=Ω,z∗α¯. As can be seen from (37a), -Ω/N introduces a nonminimal coupling of the scalar fields to gravity. Ordinary Einstein gravity is recovered at the vacuum when(38)-ΩN≃1.Starting with the JF action in (37a), we seek to realize IGI, postulating the invariance of Ω under the action of a global Zn discrete symmetry. With S stabilized at the origin, we write(39)-ΩN=ΩHT+ΩH∗T∗withΩHT=cTTn+∑k=2∞λkTkn,where k is a positive integer. If T≤1 during IGI and assuming that λk’s are relatively small, the contributions of the higher powers of T in the expression above are small, and these can be dropped. As we verify later, this can be achieved when the coefficient cT is large enough. Equivalently, we may rescale the inflation, setting T→T~=cT1/nT. Then the coefficients λk of the higher powers in the expression of Ω get suppressed by factors of cT-k. Thus, Zn and the requirement T≤1 determine the form of Ω, avoiding a severe tuning of the coefficients λk. Under these assumptions, K in (35) takes the form(40)K0=-Nln⁡fT+f∗T∗withfT≃cTTn,where S is assumed to be stabilized at the origin.Equations (35) and (38) require that T and S acquire the following v.e.v.s:(41)T≃12cT1/n,S=0.These v.e.v.s can be achieved, if we choose the following superpotential [31, 32]:(42)W=λSTn-12cT.Indeed the corresponding F-term SUSY potential, VSUSY, is found to be(43)VSUSY=λ2Tn-12cT2+λ2n2STn-12and is minimized by the field configuration in (21).As emphasized in [29, 31, 58], the forms of W and ΩH can be uniquely determined if we limit ourselves to integer values for n (with n>1) and T≤1 and impose two symmetries:(i)
AnR symmetry under which S and T have charges 1 and 0, respectively.(ii)
A discrete symmetryZn under which only T is charged.For simplicity we assume here that both W and ΩH respect the same Zn, contrary to the situation in [58]. This assumption simplifies significantly the formulae in Sections 3.3 and 3.4. Note, finally, that the selected Ω in (39) does not contribute in the kinetic term involving ΩTT∗ in (37a). We expect that our findings are essentially unaltered even if we include in the r.h.s. of (39) a term -(T-T∗)2/2N [32] or -|T|2/N [31] which yields ΩTT∗=1≪cT; the former choice, though, violates Zn symmetry above.


## 3.2. Proposed Kähler Potentials

It is obvious from the considerations above that the stabilization ofS at zero during and after IGI is of crucial importance for the viability of our scenario. This key issue can be addressed if we specify the dependence of the Kähler potential on S. We distinguish the following basic cases:(44)K3i=-n3ln⁡fT+f∗T∗+hiX,K2i=-n2ln⁡fT+f∗T∗+hiX,where the various choices hi, i=1,…,11, are specified in Table 1 and X is defined as follows:(45)X=-S2n3forK=K3iS2forK=K2i.As shown in Table 1 we consider exponential, logarithmic, trigonometric, and hyperbolic functions. Note that K31 and K21 parameterize SU(2,1)/SU(2)×U(1) and SU(1,1)/U(1)×U(1) Kähler manifolds, respectively, whereas K23 parameterizes the SU(1,1)/U(1)×SU(2)/U(1) Kähler manifold; see [58].Table 1
Definition of the varioushi(X)’s, hi′′(0)=d2hi(0)/dX2 and masses squared of the fluctuations of s and s- along the inflationary trajectory in (46) for K=K3i and K2i.
i
h
i
(
X
)
h
i
′
′
(
0
)
m
^
s
2
/
H
^
I
2
K
=
K
3
i
K
=
K
2
i
1
X
0
-
2
+
2
n
/
f
ϕ
2
3
⋅
2
n
-
1
/
f
ϕ
2
2
e
X
-
1
1
2
(
4
-
n
)
/
2
c
T
ϕ
n
-
2
+
2
n
/
f
ϕ
2
-
6
+
3
⋅
2
n
-
1
/
f
ϕ
2
3
ln
⁡
(
X
+
1
)
-
1
-
2
(
1
+
2
1
-
n
/
2
c
T
ϕ
n
)
6
(
1
+
2
n
-
1
/
f
ϕ
2
)
4
-
cos
⁡
(
arcsin
⁡
1
+
X
)
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
5
sin
⁡
(
arccos
⁡
1
+
X
)
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
6
tan
⁡
X
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
7
-
cot
⁡
(
arcsin
⁡
1
+
X
)
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
8
cosh
⁡
(
arcsin
⁡
1
+
X
)
-
2
2
2
(
5
-
n
)
/
2
c
T
ϕ
n
-
2
+
2
n
/
f
ϕ
2
3
⋅
2
n
-
1
/
f
ϕ
2
-
6
2
9
sinh
⁡
(
X
)
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
10
tanh
⁡
(
X
)
0
-
2
(
1
-
2
n
-
1
/
f
ϕ
2
)
3
⋅
2
n
-
1
/
f
ϕ
2
11
-
coth
⁡
(
arcsinh
⁡
1
+
X
)
+
2
-
2
2
2
n
/
f
ϕ
2
-
2
(
7
-
n
)
/
2
c
T
ϕ
n
-
2
3
⋅
2
n
-
1
/
f
ϕ
2
+
12
2To show that the proposedK’s are suitable for IGI, we have to verify that they reproduce V^I in (23)(b) when n=2, and they ensure the stability of S at zero. These requirements are checked in the following two sections.


## 3.3. Derivation of the Inflationary Potential

SubstitutingW of (42) and a choice for K in (44) (with the hi’s given in Table 1) into (33b), we obtain a potential suitable for IGI. The inflationary trajectory is defined by the constraints(46)S=T-T∗=0,ors=s¯=θ=0,where we have expanded T and S in real and imaginary parts as follows:(47)T=ϕ2eiθ,S=s+is¯2·Along the path of (46), V^ reads(48)V^I=V^θ=s=s¯=0=eKKSS∗W,S2.From (42) we get W,S=f-1/2. Also, (44) yields(49)eK=2f+hi0-n3forK=K3iehi02fn2forK=K2i,where we take into account that f(T)=f∗(T∗) along the path of (46). Moreover, KSS∗=1/KSS∗ can be obtained from the Kähler metric, which is given by(50)Kαβ¯=diagKTT∗,KSS∗=diagn3n22ϕ2,hi′02f+hi0forK=K3idiagn2n22ϕ2,hi′0forK=K2i,where a prime denotes a derivative with respect to X. Note that KTT∗ for K=K2i (and S≠0) does not involve the field S in its denominator, and so no geometric destabilization [91] can be activated, contrary to the K=K3i case. Inserting W,S and the results of (49) and (50) into (33b), we obtain(51)V^I=λ21-2f2cT2·2f+hi01-n3hi′0forK=K3iehi02fn2hi′0forK=K2i.Recall thatf~ϕn; see (40). Then V^I develops a plateau, with almost constant potential energy density, for cT≫1 and ϕ<1 (or cT=1 and ϕ≫1), if we impose the following conditions:(52)2n=nn3-1forK=K3inn2forK=K2i⟹n3=3forK=K3in2=2forK=K2i.This empirical criterion is very precise since the data on ns allows only tiny (of order 0.001) deviations [28]. Actually, the requirement cT≫1 and the synergy between the exponents in W and K’s assist us to tame the well-known η problem within SUGRA with a mild tuning. If we insert (52) into (51) and compare the result for n=2 with (23)(b) (replacing also λ2 with λ), we see that the two expressions coincide, if we set(53)hi0=0,hi′0=1.As we can easily verify the selected hi in Table 1 satisfy these conditions. Consequently, V^I in (51) and the corresponding Hubble parameter H^I take their final form:(54)aV^I=λ2fϕ24cT4ϕ2n,bH^I=V^I1/23=λfϕ23cT2ϕn,with fϕ=2n/2-1-cTϕn<0 reducing to that defined in (23)(b). Based on these expressions, we investigate in Section 4 the dynamics and predictions of IGI.


## 3.4. Stability of the Inflationary Trajectory

We proceed to check the stability of the direction in (46) with respect to the fluctuations of the various fields. To this end, we have to examine the validity of the extremum and minimum conditions; that is,(55)a∂V^I∂z¯^αs=s¯=θ=0=0,bm^z¯α2>0withz¯α=θ,s,s¯.Here m^z¯α2 are the eigenvalues of the mass matrix with elements(56)M^αβ2=∂2V^I∂z¯^α∂z¯^βs=s¯=θ=0withz¯α=θ,s,s¯and a hat denotes the EF canonically normalized field. The canonically normalized fields can be determined if we bring the kinetic terms of the various scalars in (33a) into the following form:(57a)Kαβ-z˙αz˙∗β-=12ϕ^˙2+θ^˙2+12s^˙2+s-^˙2,where a dot denotes a derivative with respect to the JF cosmic time. Then the hatted fields are defined as follows:(57b)dϕ^dϕ=J=KTT∗,θ^=Jθϕ,s^,s¯^=KSS∗s,s¯,where by virtue of (52) and (53), the Kähler metric of (50) reads(57c)Kαβ¯=diagKTT∗,KSS∗=3n22ϕ2,2n/2-1cTϕnforK=K3in2ϕ2,1forK=K2i.Note that the spinor componentsψT and ψS of the superfields T and S are normalized in a similar way; that is, ψ^T=KTT∗ψΦ and ψ^S=KSS∗ψS. In practice, we have to make sure that all the m^z¯α2’s are not only positive, but also greater than H^I2 during the last 50–60 e-foldings of IGI. This guarantees that the observed curvature perturbation is generated solely by ϕ, as assumed in (6). Nonetheless, two-field inflationary models which interpolate between the Starobinsky and the quadratic model have been analyzed in [92–95]. Due to the large effective masses that the scalars acquire during IGI, they enter a phase of damped oscillations about zero. As a consequence, ϕ dependence in their normalization (see (57b)) does not affect their dynamics.We can readily verify that (55)(a) is satisfied for all the three z¯α’s. Regarding (55)(b), we diagonalize M^αβ2 (56) and we obtain the scalar mass spectrum along the trajectory of (46). Our results are listed in Table 1 together with the masses squared m^ψ±2 of the chiral fermion eigenstates ψ^±=(ψ^T±ψ^S)/2. From these results, we deduce the following:(i)
For both classes ofK’s in (44), (55)(b) is satisfied for the fluctuations of θ^; that is, m^θ2>0, since fϕ<0. Moreover, m^θ2≫H^I2 because cT≫1.(ii)
WhenK=K3i and hi′′(0)=0, we obtain m^s2<0. This occurs for i=1,4,…,7,9 and 10, as shown in Table 1. For i=1, our result reproduces those of similar models [31, 47–52, 68–70]. The stability problem can be cured if we include in K3i a higher order term of the form kS|S|4 with kS~1, or assuming that S2=0 [75]. However, a probably simpler solution arises if we take into account the results accumulated in Table 2. It is clear that the condition m^s2>H^I2 can be satisfied when hi′′(0)>0 with |hi′′(0)|≥1. From Table 1, we see that this is the case for i=2 and 8.(iii)
WhenK=K2i and hi′′(0)=0, we obtain m^s2>0, but m^s2<H^I2. Therefore, S may seed inflationary perturbations, leading possibly to large non-Gaussianities in the CMB, contrary to observations. From the results listed in Table 2, we see that the condition m^s2≫H^I2 requires hi′′(0)<0 with |hi′′(0)|≥1. This occurs for i=3 and 11. The former case was examined in [58].Table 2
Mass-squared spectrum forK=K3i and K2i along the inflationary trajectory in (46).
Fields
Eigenstates
Masses squared
K=K3i
K=K2i
1 real scalar
θ^
m^θ2/H^I2
4
(
2
n
-
2
-
c
T
ϕ
n
f
ϕ
)
/
f
ϕ
2
6
(
2
n
-
2
-
c
T
ϕ
n
f
ϕ
)
/
f
ϕ
2
1 complex
s
^
,
s
-
^
m^s2/H^I2
2
n
/
f
ϕ
2
-
2
3
⋅
2
n
-
1
/
f
ϕ
2
scalar
+
2
2
-
n
/
2
c
T
ϕ
n
h
i
′
′
(
0
)
-
6
h
i
′
′
(
0
)
2 Weyl spinors
ψ
^
±
m^ψ±2/H^I2
2n/fϕ2
6⋅2n-3/fϕ2To highlight further the stabilization ofS during and after IGI we present in Figure 2m^s2/H^I2 as a function of ϕ for the various acceptable K’s identified above. In particular, we fix n=2 and ϕ⋆=1, setting K=K32 or K=K38 in Figure 2(a) and K=K23 or K=K2,11 in Figure 2(b). The parameters of the models (λ and cT) corresponding to these choices are listed in third and fifth rows of Table 3. Evidently m^s2/H^I2 remain larger than unity for ϕf≤ϕ≤ϕ⋆, where ϕ⋆ and ϕf are also depicted. However, in Figure 2(b)  m^s2/H^I2 exhibits a constant behavior and increases sharply as ϕ decreases below 0.2. On the contrary, m^s2/H^I2 in Figure 2(a) is an increasing function of ϕ for ϕ≳0.2, with a clear minimum at ϕ≃0.2. For ϕ≲0.2, m^s2/H^I2 increases drastically as in Figure 2(b) too.Table 3
Input and output parameters of the models which are compatible with (4) for N^⋆=53.2, (6), and (7).
Kähler potentialK
Parameters
Input
Output
n
c
T
ϕ
⋆
λ
(
1
0
-
3
)
ϕ
f
n
s
α
s
(
1
0
-
4
)
r
(
1
0
-
3
)
K
3
i
1
1
54.5
0.022
1.5
0.964
-6.3
3.6
K
2
i
1
1
80
0.028
1.7
0.964
-6.6
2.5
K
3
i
2
77
1
1.7
0.17
0.964
-6.7
3.7
K
3
i
3
109
1
2.4
0.3
0.964
-6.5
3.7
K
2
i
2
113
1
2
0.15
0.964
-6.7
2.5
K
2
i
3
159
1
3
0.3
0.964
-6.7
2.6Figure 2
The ratiom^s2/H^I2 as a function of ϕ for n=2 and ϕ⋆=1. We set (a) K=K32 or K=K38 and (b) K=K23 or K=K2,11. The values corresponding to ϕ⋆ and ϕf are also depicted.
(a)
(b)Employing the well-known Coleman-Weinberg formula [96], we find from the derived mass spectrum (see Table 1) the one-loop radiative corrections, ΔV^I, to V^I, depending on renormalization group mass scale Λ. It can be verified that our results are insensitive to ΔV^I, provided that Λ is determined by requiring ΔV^I(ϕ⋆)=0 or ΔV^I(ϕf)=0. A possible dependence of the results on the choice of Λ is totally avoided [31] thanks to the smallness of ΔV^I, for Λ≃(1-1.8)·10-5; see Section 4.2 too. These conclusions hold even for ϕ>1. Therefore, our results can be accurately reproduced by using exclusively V^I in (54)(a).


## 4. Analysis of SUGRA Inflation

Keeping in mind that forK=K3i [K=K2i] the values i=2 and 8 [i=3 and 11] lead to the stabilization of S during and after IGI, we proceed with the computation of the inflationary observables for the SUGRA models considered above. Since the precise choice of the index i does not influence our outputs, here we do not specify henceforth the allowed i values. We first present, in Section 4.1, analytic results which are in good agreement with our numerical results displayed in Section 4.2. Finally we investigate the UV behavior of the models in Section 4.3.


### 4.1. Analytical Results

The duration of the IGI is controlled by the slow-roll parameters, which are calculated to be(58)ϵ^,η^=2n3fϕ2,21+n/22n/2-cTϕn3fϕ2forK=K3i2n-2fϕ2,2n/22n/2-cTϕnfϕ2forK=K2i.The end of inflation is triggered by the violation of ϵ^ condition when ϕ=ϕf given by(59a)ϵ^ϕf=1⟹ϕf≃2·1+2/32cT1/nforK=K3i1+22cT1/nforK=K2i.The violation of η^ condition occurs when ϕ=ϕ~f<ϕf:(59b)η^ϕ~f=1⟹ϕ~f≃2·56cT1/nforK=K3i32cT1/nforK=K2i.Givenϕf, we can compute N^⋆ via (4):(60)N^⋆=κ221-n/2cTϕ⋆n-ϕfn-nln⁡ϕ⋆ϕfwithκ=32forK=K3i1forK=K2i.Ignoring the logarithmic term and taking into account that ϕf≪ϕ⋆, we obtain a relation between ϕ⋆ and N^⋆:(61a)ϕ⋆≃2n/2N^⋆κcTn.Obviously, IGI, consistent with (9)(b), can be achieved if(61b)ϕ⋆≤1⟹cT≥2n/2N^⋆κ.Therefore, we need relatively large cT’s, which increase with n. On the other hand, ϕ^⋆ remains trans-Planckian, since solving the first relation in (57b) with respect to ϕ and inserting (61a), we find(62)ϕ^⋆≃ϕ^c+κln⁡2N^⋆κ≃5.2forK=K3i4.6forK=K2i,where the integration constant ϕ^c=0 and, as in the previous cases, we set N^⋆≃53. Despite this fact, our construction remains stable under possible corrections from higher order terms in fK, since when these are expressed in terms of initial field T, they can be seen to be harmless for |T|≤1.Upon substitution of (54) and (61a) into (6), we find(63)As1/2≃λ3-4N^⋆2962πcTN^⋆forK=K3iλ1-2N^⋆2163πcTN^⋆forK=K2i.Enforcing (6), we obtain a relation between λ and cT, which turns out to be independent of n. Indeed we have(64)λ≃6π2AscTN^⋆⟹cT≃42969λforK=K3i4π3AscTN^⋆⟹cT≃52627λforK=K2i.Finally, substituting the value of ϕ⋆ given in (61a) into (8), we estimate the inflationary observables. For K=K3i the results are given in (31a)–(31c). For K=K2i we obtain the relations:(65a)ns=4N^⋆N^⋆-3-31-2N^⋆2≃1-2N^⋆-3N^⋆2≃0.961;(65b)αs≃16N^⋆3+2N^⋆2N^⋆-14≃-2N^⋆2-7N^⋆3≃-0.00075;(65c)r≃321-2N^⋆2≃8N^⋆2+8N^⋆3≃0.0028.These outputs are consistent with our results in [58] for m=n and n11=n2=2 (in the notation of that reference).


### 4.2. Numerical Results

The analytical results presented above can be verified numerically. The inflationary scenario depends on the following parameters (see (42) and (44)): (66)n,cTandλ.Note that the stabilization of S with one of K32, K34, K23, and K2,11 does not require any additional parameter. Recall that we use Trh=4.1·10-9 throughout and N^⋆ is computed self-consistently for any n via (4). Our result is N^⋆≃53.2. For given n, the parameters above together with ϕ⋆ can be determined by imposing the observational constraints in (4) and (6). In our code we find ϕ⋆ numerically, without the simplifying assumptions used for deriving (61a). Inserting it into (8), we extract the predictions of the models.The variation ofV^I as a function of ϕ for two different values of n can be easily inferred from Figure 3. In particular, we plot V^I versus ϕ for ϕ⋆=1, n=2, or n=6, setting K=K3i in Figure 3(a) and K=K2i in Figure 3(b). Imposing ϕ⋆=1 for n=2 amounts to (λ,cT)=(0.0017,77) for K=K3i and (λ,cT)=(0.0017,113) for K=K2i. Also, ϕ⋆=1 for n=6 is obtained for (λ,cT)=(0.0068,310) for K=K3i and (λ,cT)=(0.0082,459) for K=K2i. In accordance with our findings in (61b), we conclude that increasing n (i) requires larger cT’s and, therefore, lower V^I’s to obtain ϕ≤1; (ii) larger ϕf and 〈ϕ〉 are obtained; see Section 4.3. Combining (59a) and (64) with (54)(a), we can conclude that V^I(ϕf) is independent of cT and to a considerable degree of n.Figure 3
The inflationary potentialV^I as a function of ϕ for ϕ⋆=1 and n=2 or n=6. We set (a) K=K3i and (b) K=K2i. The values corresponding to ϕ⋆ and ϕf are also depicted.
(a)
(b)Our numerical findings forn=1,2, and 3 and K=K3i or K=K2i are presented in Table 2. In the two first rows, we present results associated with Ceccoti-like models [97], which are defined by cT=n=1 and cannot be made consistent with the imposed Zn symmetry or with (9). We see that, selecting ϕ⋆≫1, we attain solutions that satisfy all the remaining constraints in Section 2.2. For the other cases, we choose a cT value so that ϕ⋆=1. Therefore, the presented cT is the minimal one, in agreement with (61b).In all cases shown in Table2, the model’s predictions for ns, αs, and r are independent of the input parameters. This is due to the attractor behavior [30–32] that these models exhibit, provided that cT is large enough. Moreover, these outputs are in good agreement with the analytical findings of (31a)–(31c) for K=K3i or (65a)–(65c) for K=K2i. On the other hand, the presented cT, λ, ϕ⋆, and ϕf values depend on n for every selected K. The resulting ns≃0.964 is close to its observationally central value; r is of the order of 0.001, and |αs| is negligible. Although the values of r lie one order of magnitude below the central value of the present combined Bicep2/Keck Array andPlanck results [5], these are perfectly consistent with the 95% c.l. margin in (7). The values of r for K=K3i or K=K2i distinguish the two cases. The difference is small, at the level of 10-3. However, it is possibly reachable by the next-generation experiment (e.g., the CMBPol experiment [98]) is expected to achieve a precision for r of the order of 10-3 or even 0.5·10-3. Finally, the renormalization scale Λ of the Coleman-Weinberg formula, found by imposing ΔV^I(ϕ⋆)=0, takes the values 7.8·10-5, 9.3·10-51.3·10-5, and 2.1·10-5 for K32, K38, K23, and K2,11, respectively. As a consequence, Λ depends explicitly on the specific choice of i used for K3i or K2i.The overall allowed parameter space of the model forn=2,3 and 6 is correspondingly(67a)77,105,310≲cT≲1.6·105,1.7,2.4,6.8·10-3≲λ≲3.54forK=K3i;(67b)113,159,453≲cT≲1.93·105,2,2.9,8.2·10-3≲λ≲3.54forK=K2i,where the parameters are bounded from above as in (29). Letting λ or cT vary within its allowed region above, we obtain the values of ns, αs, and r listed in Table 3 for K=K3i and K2i independently of n. Therefore, the inclusion of the variant exponent n>2, compared to the non-SUSY model in Section 2.4 does not affect the successful predictions of model.


### 4.3. UV Behavior

Following the approach described in Section2.2, we can verify that the SUGRA realizations of IGI retain perturbative unitarity up to mP. To this end, we analyze the small-field behavior of the theory, expanding S in (1) about(68)ϕ=2n-2/2ncT-1/n,which is confined in the ranges (0.0026–0.1), (0.021–0.24), and (0.17–0.48) for the margins of the parameters in (67a) and (67b).The expansion ofS is performed in terms of δϕ^ which is found to be(69)δϕ^=JδϕwithJ≃κnϕ=22-n/2nκncT1/n,where κ is defined in (60). Note, in passing, that the mass of δϕ^ at the SUSY vacuum in (41) is given by(70)m^δϕ=V^I,ϕ^ϕ^1/2≃λ2κcT≃26AsπN^⋆≃1.25·10-5,precisely equal to that found in (19) and (30). We observe that m^δϕ is essentially independent of n and κ, thanks to the relation between λ and cT in (64).Expanding the second term in the r.h.s. of (33a) about 〈ϕ〉 with J given by the first relation in (57b), we obtain(71a)J2ϕ˙2=1-2nκδϕ^+3n2κδϕ^2-4n3κ-3/2δϕ^3+⋯δϕ^˙2.On the other hand, V^I in (54)(a) can be expanded about 〈ϕ〉 as follows:(71b)V^I≃λ2ϕ^24κcT21-n+1κnδϕ^+1+n11+7n12κn2δϕ^2-⋯.Since the expansions above are cT independent, we infer that ΛUV=1 as in the other versions of Starobinsky-like inflation. The expansions above for K=K3i and n=2 reduce to those in (32a) and (32b). Moreover, these are compatible with the ones presented in [31] for K=K3i and those in [58] for K=K2i and n11=2. Our overall conclusion is that our models do not face any problem with perturbative unitarity up to mP.


## 4.1. Analytical Results

The duration of the IGI is controlled by the slow-roll parameters, which are calculated to be(58)ϵ^,η^=2n3fϕ2,21+n/22n/2-cTϕn3fϕ2forK=K3i2n-2fϕ2,2n/22n/2-cTϕnfϕ2forK=K2i.The end of inflation is triggered by the violation of ϵ^ condition when ϕ=ϕf given by(59a)ϵ^ϕf=1⟹ϕf≃2·1+2/32cT1/nforK=K3i1+22cT1/nforK=K2i.The violation of η^ condition occurs when ϕ=ϕ~f<ϕf:(59b)η^ϕ~f=1⟹ϕ~f≃2·56cT1/nforK=K3i32cT1/nforK=K2i.Givenϕf, we can compute N^⋆ via (4):(60)N^⋆=κ221-n/2cTϕ⋆n-ϕfn-nln⁡ϕ⋆ϕfwithκ=32forK=K3i1forK=K2i.Ignoring the logarithmic term and taking into account that ϕf≪ϕ⋆, we obtain a relation between ϕ⋆ and N^⋆:(61a)ϕ⋆≃2n/2N^⋆κcTn.Obviously, IGI, consistent with (9)(b), can be achieved if(61b)ϕ⋆≤1⟹cT≥2n/2N^⋆κ.Therefore, we need relatively large cT’s, which increase with n. On the other hand, ϕ^⋆ remains trans-Planckian, since solving the first relation in (57b) with respect to ϕ and inserting (61a), we find(62)ϕ^⋆≃ϕ^c+κln⁡2N^⋆κ≃5.2forK=K3i4.6forK=K2i,where the integration constant ϕ^c=0 and, as in the previous cases, we set N^⋆≃53. Despite this fact, our construction remains stable under possible corrections from higher order terms in fK, since when these are expressed in terms of initial field T, they can be seen to be harmless for |T|≤1.Upon substitution of (54) and (61a) into (6), we find(63)As1/2≃λ3-4N^⋆2962πcTN^⋆forK=K3iλ1-2N^⋆2163πcTN^⋆forK=K2i.Enforcing (6), we obtain a relation between λ and cT, which turns out to be independent of n. Indeed we have(64)λ≃6π2AscTN^⋆⟹cT≃42969λforK=K3i4π3AscTN^⋆⟹cT≃52627λforK=K2i.Finally, substituting the value of ϕ⋆ given in (61a) into (8), we estimate the inflationary observables. For K=K3i the results are given in (31a)–(31c). For K=K2i we obtain the relations:(65a)ns=4N^⋆N^⋆-3-31-2N^⋆2≃1-2N^⋆-3N^⋆2≃0.961;(65b)αs≃16N^⋆3+2N^⋆2N^⋆-14≃-2N^⋆2-7N^⋆3≃-0.00075;(65c)r≃321-2N^⋆2≃8N^⋆2+8N^⋆3≃0.0028.These outputs are consistent with our results in [58] for m=n and n11=n2=2 (in the notation of that reference).


## 4.2. Numerical Results

The analytical results presented above can be verified numerically. The inflationary scenario depends on the following parameters (see (42) and (44)): (66)n,cTandλ.Note that the stabilization of S with one of K32, K34, K23, and K2,11 does not require any additional parameter. Recall that we use Trh=4.1·10-9 throughout and N^⋆ is computed self-consistently for any n via (4). Our result is N^⋆≃53.2. For given n, the parameters above together with ϕ⋆ can be determined by imposing the observational constraints in (4) and (6). In our code we find ϕ⋆ numerically, without the simplifying assumptions used for deriving (61a). Inserting it into (8), we extract the predictions of the models.The variation ofV^I as a function of ϕ for two different values of n can be easily inferred from Figure 3. In particular, we plot V^I versus ϕ for ϕ⋆=1, n=2, or n=6, setting K=K3i in Figure 3(a) and K=K2i in Figure 3(b). Imposing ϕ⋆=1 for n=2 amounts to (λ,cT)=(0.0017,77) for K=K3i and (λ,cT)=(0.0017,113) for K=K2i. Also, ϕ⋆=1 for n=6 is obtained for (λ,cT)=(0.0068,310) for K=K3i and (λ,cT)=(0.0082,459) for K=K2i. In accordance with our findings in (61b), we conclude that increasing n (i) requires larger cT’s and, therefore, lower V^I’s to obtain ϕ≤1; (ii) larger ϕf and 〈ϕ〉 are obtained; see Section 4.3. Combining (59a) and (64) with (54)(a), we can conclude that V^I(ϕf) is independent of cT and to a considerable degree of n.Figure 3
The inflationary potentialV^I as a function of ϕ for ϕ⋆=1 and n=2 or n=6. We set (a) K=K3i and (b) K=K2i. The values corresponding to ϕ⋆ and ϕf are also depicted.
(a)
(b)Our numerical findings forn=1,2, and 3 and K=K3i or K=K2i are presented in Table 2. In the two first rows, we present results associated with Ceccoti-like models [97], which are defined by cT=n=1 and cannot be made consistent with the imposed Zn symmetry or with (9). We see that, selecting ϕ⋆≫1, we attain solutions that satisfy all the remaining constraints in Section 2.2. For the other cases, we choose a cT value so that ϕ⋆=1. Therefore, the presented cT is the minimal one, in agreement with (61b).In all cases shown in Table2, the model’s predictions for ns, αs, and r are independent of the input parameters. This is due to the attractor behavior [30–32] that these models exhibit, provided that cT is large enough. Moreover, these outputs are in good agreement with the analytical findings of (31a)–(31c) for K=K3i or (65a)–(65c) for K=K2i. On the other hand, the presented cT, λ, ϕ⋆, and ϕf values depend on n for every selected K. The resulting ns≃0.964 is close to its observationally central value; r is of the order of 0.001, and |αs| is negligible. Although the values of r lie one order of magnitude below the central value of the present combined Bicep2/Keck Array andPlanck results [5], these are perfectly consistent with the 95% c.l. margin in (7). The values of r for K=K3i or K=K2i distinguish the two cases. The difference is small, at the level of 10-3. However, it is possibly reachable by the next-generation experiment (e.g., the CMBPol experiment [98]) is expected to achieve a precision for r of the order of 10-3 or even 0.5·10-3. Finally, the renormalization scale Λ of the Coleman-Weinberg formula, found by imposing ΔV^I(ϕ⋆)=0, takes the values 7.8·10-5, 9.3·10-51.3·10-5, and 2.1·10-5 for K32, K38, K23, and K2,11, respectively. As a consequence, Λ depends explicitly on the specific choice of i used for K3i or K2i.The overall allowed parameter space of the model forn=2,3 and 6 is correspondingly(67a)77,105,310≲cT≲1.6·105,1.7,2.4,6.8·10-3≲λ≲3.54forK=K3i;(67b)113,159,453≲cT≲1.93·105,2,2.9,8.2·10-3≲λ≲3.54forK=K2i,where the parameters are bounded from above as in (29). Letting λ or cT vary within its allowed region above, we obtain the values of ns, αs, and r listed in Table 3 for K=K3i and K2i independently of n. Therefore, the inclusion of the variant exponent n>2, compared to the non-SUSY model in Section 2.4 does not affect the successful predictions of model.


## 4.3. UV Behavior

Following the approach described in Section2.2, we can verify that the SUGRA realizations of IGI retain perturbative unitarity up to mP. To this end, we analyze the small-field behavior of the theory, expanding S in (1) about(68)ϕ=2n-2/2ncT-1/n,which is confined in the ranges (0.0026–0.1), (0.021–0.24), and (0.17–0.48) for the margins of the parameters in (67a) and (67b).The expansion ofS is performed in terms of δϕ^ which is found to be(69)δϕ^=JδϕwithJ≃κnϕ=22-n/2nκncT1/n,where κ is defined in (60). Note, in passing, that the mass of δϕ^ at the SUSY vacuum in (41) is given by(70)m^δϕ=V^I,ϕ^ϕ^1/2≃λ2κcT≃26AsπN^⋆≃1.25·10-5,precisely equal to that found in (19) and (30). We observe that m^δϕ is essentially independent of n and κ, thanks to the relation between λ and cT in (64).Expanding the second term in the r.h.s. of (33a) about 〈ϕ〉 with J given by the first relation in (57b), we obtain(71a)J2ϕ˙2=1-2nκδϕ^+3n2κδϕ^2-4n3κ-3/2δϕ^3+⋯δϕ^˙2.On the other hand, V^I in (54)(a) can be expanded about 〈ϕ〉 as follows:(71b)V^I≃λ2ϕ^24κcT21-n+1κnδϕ^+1+n11+7n12κn2δϕ^2-⋯.Since the expansions above are cT independent, we infer that ΛUV=1 as in the other versions of Starobinsky-like inflation. The expansions above for K=K3i and n=2 reduce to those in (32a) and (32b). Moreover, these are compatible with the ones presented in [31] for K=K3i and those in [58] for K=K2i and n11=2. Our overall conclusion is that our models do not face any problem with perturbative unitarity up to mP.


## 5. Conclusions and Perspectives

In this review we revisited the realization of induced-gravity inflation (IGI) in both a nonsupersymmetric and supergravity (SUGRA) framework. In both cases the inflationary predictions exhibit an attractor behavior towards those of Starobinsky model. Namely, we obtained a spectral indexns≃(0.960–0.965) with negligible running αs and a tensor-to-scalar ratio 0.001≲r≲0.005. The mass of the inflation turns out be close to 3·1013GeV. It is gratifying that IGI can be achieved for sub-Planckian values of the initial (noncanonically normalized) inflation, and the corresponding effective theories are trustable up to Planck scale, although a parameter has to take relatively high values. Moreover, the one-loop radiative corrections can be kept under control.In the SUGRA context this type of inflation can be incarnated using two chiral superfields,T and S, the superpotential in (42), which realizes easily the idea of induced gravity, and several (semi)logarithmic Kähler potentials K3i or K2i; see (44). The models are pretty much constrained upon imposing two global symmetries, a continuous R and a discrete Zn symmetry, in conjunction with the requirement that the original inflation, T, takes sub-Planckian values. We paid special attention to the issue of S stabilization during IGI and worked out its dependence on the functional form of the selected K’s with respect to |S|2. More specifically, we tested the functions hi(|S|2), which appear in K3i or K2i; see Table 1. We singled out h2(|S|2) and h8(|S|2) for K=K3i or h3(|S|2) and h11(|S|2) for K=K2i, which ensure that S is heavy enough, and so well stabilized during and after inflation. This analysis provides us with new results that do not appear elsewhere in the literature. Therefore, Starobinsky inflation realized within this SUGRA set-up preserves its original predictive power, since no mixing between |T|2 and |S|2 is needed for consistency in the considered K’s (cf. [31, 72, 73]).It is worth emphasizing that theS-stabilization mechanisms proposed in this paper can be also employed in other models of ordinary [47–49] or kinetically modified [65–67] nonminimal chaotic (and Higgs) inflation driven by a gauge singlet [47–49, 53, 54, 65–67] or nonsinglet [50–52, 68–70] inflation, without causing any essential alteration to their predictions. The necessary modifications involve replacing the |S|2 part of K with h2(|S|2), or h8(|S|2) if we have a purely logarithmic Kähler potential. Otherwise, the |S|2 part can be replaced by h3(|S|2) or h11(|S|2). Obviously, the last case can be employed for logarithmic or polynomial K’s with regard to the inflation terms.Let us, finally, remark that a complete inflationary scenario should specify a transition to the radiation dominated era. This transition could be facilitated in our setting [29, 62, 63] via the process of perturbative reheating, according to which the inflation after inflation experiences an oscillatory phase about the vacuum, given by (22) for the non-SUSY case or (41) for the SUGRA case. During this phase, the inflation can safely decay, provided that it couples to light degrees of freedom in the Lagrangian of the full theory. This process is independent of the inflationary observables and the stabilization mechanism of the noninflation field. It depends only on the inflation mass and the strength of the relevant couplings. This scheme may also explain the origin of the observed baryon asymmetry through nonthermal leptogenesis, consistently with the data from the neutrino oscillations [29]. It would be nice to obtain a complete and predictable transition to the radiation dominated era. An alternative graceful exit can be achieved in the running vacuum models, as described in the fourth paper of [77–84].



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