Reconciling Induced-Gravity Inflation in Supergravity With The Planck 2013&BICEP2 Results

We generalize the embedding of induced-gravity inflation beyond the no-scale Supergravity presented in arXiv:1403.5486 employing two gauge singlet chiral superfields, a superpotential uniquely determined by applying a continuous R and a discrete Zn symmetries, and a logarithmic Kahler potential including all the allowed terms up to fourth order in powers of the various fields. We show that, increasing slightly the prefactor (-3) encountered in the adopted Kahler potential, an efficient enhancement of the resulting tensor-to-scalar ratio can be achieved rendering the predictions of the model consistent with the recent BICEP2 results, even with subplanckian excursions of the original inflaton field. The remaining inflationary observables can become compatible with the data by mildly tuning the coefficient involved in the fourth order term of the Kahler potential which mixes the inflaton with the accompanying non-inflaton field. The inflaton mass is predicted to be close to 10^14 GeV.


INTRODUCTION
Although compatible with the PLANCK (and WMAP) data [2], the models of induced-gravity (IG) inflation [3] formulated within standard Supergravity (SUGRA) yield [1] a low tensor-to-scalar ratio r ≃ 0.004 which fails to approach the recent BICEP2 results [4] -for other recent incarnations of IG inflation see Ref. [5,6]. More specifically, the BICEP2 collaboration has detected B-modes in the polarization of the cosmic microwave background radiation at large angular scales. If this observation is attributed to the primordial gravity waves predicted by inflation, it implies [4] r = 0.16 +0.06 −0.05 -after substraction of a dust foreground. Despite the fact that this result is subject to considerable uncertainties [7,8] and its interpretation as a detection of primordial gravitational waves is rather questionable, it motivates us to explore how IG inflation can also accommodate large r's -for similar recent attempts see Ref. [9][10][11]. In particular, taking into account both the PLANCK [2] and BICEP2 [4] data we find a simultaneously compatible region [12] 0.06 r 0.135 (1.1) at 95% confidence level (c.l.) which can be considered as the most exciting region where r values may be confined for models with low running, a s , of the spectral index, n s . In this paper we show that modifying modestly the implementation of IG inflation beyond the no-scale SUGRA [13] we can ensure a sizable augmentation of the resulting r's with respect to (w.r.t) those obtained in the the models presented in Ref. [1]. The key-ingredient of our generalization is the variation of the numerical prefactor encountered in the adopted Kähler potential. We show that increasing the conventional value (−3) of this prefactor by an amount of order 0.01, the inflationary potential acquires a moderate inclination accommodating, thereby, observable r's reconcilable with Eq. (1.1). At the same time, the remaining attractive features of these models [1,14] are kept almost intact. Most notably, the super-and Kähler potentials are fixed by an R and a discrete Z n symmetries, inflation is realized using subplanckian values of the initial (non-canonically normalized) inflaton field, the radiative corrections remain under control and the perturbative unitarity is respected up to the Planck scale, m P = 2.44 · 10 18 GeV [1,14,15].
Below we generalize in Sec. 2 the formulation of IG models within SUGRA. In Sec. 3 we present the basic ingredients of our IG inflationary models, derive the inflationary observables and test them against observations. We end-up with a brief analysis of the UV behavior of these models in Sec. 4 and the summary of our conclusions in Sec. 5. Throughout we follow closely the notation and the conventions adopted in Ref. [1], whose Sections, Equations, Tables and Figures are referred

GENERALIZING THE EMBEDDING OF THE IG INFLATION IN SUGRA
According to the scheme proposed in Ref. [1], the implementation of IG inflation in SUGRA requires at least two singlet superfields, i.e., z α = Φ, S, with Φ (α = 1) and S (α = 2) being the inflaton and a "stabilizer" field respectively. The superpotential W of the model has the form which is (i) invariant under the action of a global Z n discrete symmetry, i.e., where we take into account that the phase of Φ, argΦ, is stabilized to zero. If Ω H is the holomorphic part of the frame function Ω and dominates it, Eq. (2.4) assures a transition to the conventional Einstein gravity realizing, thereby, the idea of IG [3]. Our main point in this paper is that this construction remains possible for a broad class of relations between Ω and the Kähler potential K. Indeed, if we perform a conformal transformation defining the JF metric g µν through the relation where m is a dimensionless (small in our approach) parameter which quantifies the deviation from the standard set-up [16], the EF action -where V is the F-term SUGRA scalar potential given below -, is written in the JF as follows [16] with V = Ω 2 9(1+m) 2 V being the JF potential. If we specify the following relation between Ω and K, and employ the definition [16] of the purely bosonic part of the on-shell value of the auxiliary field we arrive at the following action where Ω K includes the kinetic terms for the z α 's and takes the form with sufficiently small coefficients k αβ i.e. k αβ ≪ c R . As a consequence, Ω H represents the nonminimal coupling to gravity and so Eq. (2.4) dynamically generates m P . As for m = 0, when the dynamics of the z α 's is dominated only by the real moduli |z α | or if z α = 0 for α = 1 [16], we can obtain A µ = 0 in Eq. (2.10). The only difference w.r.t the case with m = 0 is that now the scalar fields z α have not canonical kinetic terms in the JF due to the term proportional to Ω α Ωβ = δ αβ . This fact does not cause any problem, since the canonical normalization of the inflaton keeps its strong dependence on c R included in Ω H whereas the non-inflaton fields become heavy enough during inflation and so, they do not affect the dynamics -see Sec. 3.1. Note that our present set-up lies on beyond the no-scale SUGRA embedding of IG inflation since the framework of the no-scale SUGRA [13] is defined by supplementing Eq. (R.1-2.8) with the imposition m = 0. Indeed, only under this condition the cosmological constant term into the EF F-term SUGRA scalar potential -see below -vanishes. The resulting through Eq. (2.8) Kähler potential is Recall that the fourth order term for S is included to cure the problem of a tachyonic instability occurring along this direction [16] and the remaining terms of the same order are considered for consistency -the factors of 2 are added just for convenience.

THE INFLATIONARY SCENARIO
In this section we describe the inflationary potential of our model in Sec. 3.1. We then exhibit a number of constraints imposed (Sec. 3.2) and present our analytic and numerical results in Sec. 3.3 and 3.4 respectively.

THE INFLATIONARY POTENTIAL
The EF F-term (tree level) SUGRA scalar potential, V IG0 , of IG inflation is obtained from W and K in Eqs. (2.1) and (2.13) respectively by applying (for z α = Φ, S) the well-known formula where F α = W ,z α + K ,z α W/m 2 P and S is placed at the origin. Here we take into account that The functions f R and f SΦ , defined as follows -cf. Eq. (R.1-3.26): are computed along the inflationary track, i.e., for using the standard parametrization for Φ and S Besides the inflationary plateau which emerges for m = 0 and studied in Ref. [1], a chaotic-type potential (bounded from below) is generated for m < 0. More specifically, V IG0 can be cast in the following from -cf. Eq. (R.1-3.25a): ) coincides with f φφ and f Φ defined in Eq. (R.1-3.10). Confining ourselves to n = 2 -which, as we justify in Sec. 3.4 consists the most interesting choice -V IG0 takes the form whereas the corresponding EF Hubble parameter is The stability of the configuration in Eq. (3.2) can be checked verifying the validity of the conditions where m 2 χ α are the eigenvalues of the mass matrix with elements M 2 αβ = ∂ 2 V IG0 /∂ χ α ∂ χ β and hat denotes the EF canonically normalized fields defined by the kinetic terms in Eq. (2.6) as follows where the dot denotes derivation w.r.t the JF cosmic time and the hatted fields read where The spinors ψ Φ and ψ S associated with S and Φ are normalized similarly, i.e., Integrating the first equation in Eq. (3.8b) we can identify the EF field: where φ c is a constant of integration and we take into account Eqs. (2.1) and (2.4). Upon diagonalization of M 2 αβ , we construct the mass spectrum of the theory along the path of Eq. (3.2). Taking advantage of the fact that c R ≪ 1 and the limits k Φ → 0 and k SΦ → 0 we find the expressions of the relevant masses squared, arranged in Table 1, which approach rather well the quite lengthy, exact expressions taking into account in our numerical computation. In the limit m = 0 the expressions in Table R.1-1 are recovered. We have numerically verified 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. They enter a phase of oscillations about zero with reducing amplitude and so the x φ dependence in their normalization -see Eq. (3.8b) -does not affect their dynamics. As usually -cf. Ref. [1,14] -the lighter eignestate of M 2 αβ is m 2 s which here can become positive and heavy enough for k S 0.1 -see Sec. 3.4. Inserting, finally, the mass spectrum of the model in the well-known Coleman-Weinberg formula, we calculate the one-loop corrected V IG where Λ is a renormalization group (RG) mass scale. We determine it by requiring [17] ∆V (φ ⋆ ) = 0.
To reduce the possible [17] dependence of our results on the choice of Λ, we confine ourselves to λ and k S values which do not enhance these corrections -see Sec. 3.4.

THE INFLATIONARY REQUIREMENTS
Based on V IG in Eq. (3.10) we can proceed to the analysis of IG inflation in the EF [3], employing the standard slow-roll approximation [18]. We have just to convert the derivations and integrations w.r.t φ to the corresponding ones w.r.t φ keeping in mind the dependence of φ on φ, Eq. (3.8b). In particular, the observational requirements which are imposed on our inflationary scenario are outlined in the following. The number of e-foldings, N ⋆ , that the scale k ⋆ = 0.05/Mpc suffers during IG inflation has to be adequate to resolve the horizon and flatness problems of standard big bang, i.e., [2,19] at the end of IG inflation, which can be found from the condition are the well-known slow-roll parameters and T rh is the reheat temperature after IG inflation, which is taken T rh = 10 9 GeV throughout.

3.2.2
The amplitude A s of the power spectrum of the curvature perturbation generated by φ at k ⋆ has to be consistent with data [2] where the variables with subscript ⋆ are evaluated at φ = φ ⋆

3.2.4.
Since SUGRA is an effective theory below m P the existence of higher-order terms in W and K, Eqs. (2.1) and (2.13), appears to be unavoidable. Therefore the stability of our inflationary solutions can be assured if we entail where m P is the UV cutoff scale of the effective theory for the present models, as shown in Sec. 4.

ANALYTIC RESULTS
Plugging Eqs. (3.5) and (3.8b) into Eq. (3.12) and taking k Φ ≃ 0, we obtain the following approximate expressions for the slow-roll parameters Taking the limit of the expressions above for k SΦ ≃ 0 we can analytically solve the condition in Eq. (3.12) w.r.t x φ . The results are The end of IG inflation mostly occurs at φ f = φ 1f because this is mainly the maximal value of the two solutions above.
Since φ f ≪ φ ⋆ , we can estimate N ⋆ through Eq. (3.11) neglecting φ f . Our result is Ignoring the first term in the last equality and solving w.r.t x ⋆ we extract φ ⋆ as follows Although a radically different dependence of φ ⋆ on N ⋆ arises compared to the models of Ref.
[1] -cf. Eq. (R.1-3.17b) -φ ⋆ can again remain subplanckian for large c R 's. Indeed, As emphasized in Ref. [1], this achievement is crucial for the viability of our proposal, since it assures the stability of the inflationary computation against higher-order corrections from non-renormalizable terms in Ω H -see Eq. (2.1). Note that Ω H is totally defined in terms of Φ. In other words, our setting is independent of φ ⋆ which can be found by employing Eq.
From this expression we see that m < 0 and k SΦ < 0 assist us to reduce n s sizably lower than unity as required in Eq. (3.15a). Making use of Eqs. (3.19b), (3.17) and (3.14c) we arrive at From the last result we conclude that primarily |m| = 0 and secondary m < 0 help us to increase r.   To appreciate the validity of our analytic estimates, we test them against our numerical ones. The relevant results are displayed in Table 2. We use four sets of input parameters -see also Sec. 3.4 -and we present their response by applying the formulae of Sec. 3.3 (first four columns to the right of the leftmost one) or using the formulae of Sec. 3.2 with V IG given in Eq. (3.10) (next four columns). We see that the results are quite close to each other with an exception regarding φ ⋆ whose the numerical and analytic values appreciably differ. This fact can be attributed to the inaccuracy of Eq. (3.19b) whose the derivation is based on a number of efficient simplifications. Despite this deviation, the absence of φ f from Eq. (3.19a) assists us to evaluate rather accurately N ⋆ and the analytic values of φ ⋆ , r and n s are rather close to the numerical ones. As anticipated in Eq. (3.20), φ ⋆ is independent of c R (and k SΦ ). Finally, from the two last rows of Table 2 we see that the formulas of Table 1 are reliable enough. As can be deduced by the relevant expressions, m 2 s is a monotonically increasing function of x φ and so its minimal value is encountered for φ = φ f . On the contrary, the minimal m 2 θ is located at φ = φ ⋆ . It is clear that n s and r obtained in Table 2 are perfectly consistent with both the PLANCK and BICEP2 results -cf. Eqs. (1.1) and (3.14a,b). The most impressive point, however, is that these large r values are accommodated with subplanckian values of φ. As first stressed in Ref. [19], this fact does not contradict to the Lyth bound [20], since the latter bound is applied to the EF inflaton field, φ which remains transplanckian and close to the value shown in Eq. (3.20). Therefore, large r's do not necessarily [21] correlate with transplanckian excursions of φ within IG inflation. It is also notable that our set-up is clearly distinguishable with the so-called α-attractor models [11] where the absolute value of the coefficient of the logarithm in the Kähler potential is also involved in the definition of the superpotential, as dictated by the superconformal motivation of these models.

NUMERICAL RESULTS
As shown in Eqs. (2.1), (2.13) and (3.11), this inflationary scenario depends on the parameters: Besides the free parameters employed in Sec. R.1-3.3.3, we here have m which is constrained to negative values in order to ensure the boundedness from below of V IG0 -see Eq. (3.5). Using the reasoning of Sec. R.1-3.3.3, we set k Φ = 0.5 and T rh = 10 9 GeV. On the other hand, m 2 s becomes positive with k S 's lower than those used in Sec. R.1-3.3.3 since positive contributions from m < 0 arise here -see in Table 1. Moreover, due to the relatively large λ's encountered in our scheme, if k S takes a value of order unity m 2 s grows more efficiently than in the cases with m = 0, rendering thereby the radiative corrections in Eq. (3.10) sizeable for very large c R 's. To avoid such a certainly unpleasant dependence of the model predictions on the radiative corrections we tune somehow k S to lower values than those used in Sec. R.1-3.3.3. E.g. we set k S = 0.1 throughout. For the same reason we confine ourselves to the lowest possible n, n = 2. Eqs. (3.11), (3.13) and (3.16) assist us to restrict λ (or c R ≥ 1) and φ ⋆ . By adjusting m and k SΦ we can achieve not only n s 's in the range of Eq. (3.15a) but also r's in the optimistic region of Eq. (1.1).
The structure of V IG as a function of φ for m < 0 (and n = 2) is visualized in Fig. 1, where we depict V IG versus φ for φ ⋆ = m P and the selected values of λ, k SΦ and m, shown in the label. These choices require that c R 's are (1.7, 5.6, 26) · 10 3 and result to n s = 0.96 and r = 0.053, 0.096, 0.16 for increasing |m|'s -light gray, black and gray line correspondingly. It would be instructive to compare Fig. 1 with Fig. R.1-1, where V IG for m = 0 is displayed -the fact that we employ a vanishing k SΦ in Fig. R.1-1 does not invalidate the comparison since the impact of k SΦ on the form of V IG is almost invisible. We remark that in Fig. 1 (i) The values of V IG0 for φ = φ ⋆ are one order of magnitude larger than those encountered in Another difference of the present set-up regarding those of Ref. [1] is that for m = 0 we obtain constantly η ⋆ < 0 whereas we here obtain η ⋆ > 0 for n s > 0.97 and η ⋆ < 0 for lower n s values. Confronting the models under consideration with the constraints of Eqs. From Fig. 2-(a) we remark that c R remains almost proportional to λ but the dependence on k SΦ is stronger than that shown in Fig. R.1-2-(a1) and (a2). Also as |m| increases, the allowed areas become smaller favoring larger c R 's and λ's. From Fig. 2-(b) we notice that the allowed k SΦ 's get concentrated around zero as |m| increases and so the relevant tuning increases. Finally from Fig. 2-(c) and (d) we conclude that decreasing m below zero, r and a s increase w.r.t their standard values -cf. Eq. (R. 1-3.22) and discussion below Eq. (R.1-3.32c). As a consequence, r for m = −0.04 and −0.05 approaches the range of Eq. (1.1) -which explains (conservatively) the recent BICEP2 results -being at the same time compatible with the PLANCK (and WMAP) measurements. For m = −0.0625, r reaches its (almost) maximal possible value in our set-up which lie close to the BICEP2 central r value -see Sec. 1 above Eq. (1.1). On the other hand, a s remains sufficiently low; it is thus consistent with the fitting of data with the standard ΛCDM model -see Eq. (R.1-3.6b). Namely, |a s | never exceeds 4 · 10 −3 and it is mostly positive. It is clear, therefore, that it is much smaller than its best-fit value of roughly −0.02 which may help [4,22] to relieve the tension between the BICEP2 and the PLANCK data as regards the bounds on r. Furthermore, the resulting r and a s depend only on the input m and k SΦ (or n s ) and are independent on λ (or c R ). This feature can be verified by our analytical estimate for r in Eq. respectively. Consequently, our model can fit both PLANCK and BICEP2 results employing just one more parameter (m) than those employed in Ref. [1]. It is worth noticing that a decrease of k SΦ below zero is imperative in order to achieve a simultaneous fulfillment of Eq. (3.15a) and (1.1). Indeed, selecting k SΦ = 0 the increase of the prefactor (−3) in K generates an enhancement of r which is accompanied by an increase of n s beyond the range of Eq. (3.15a). Increasing, finally, n above 2 the required λ and c R values become larger and so the allowed regions are considerably shrunk; we thus do not pursue further our investigation.

THE EFFECTIVE CUT-OFF SCALE
The realization of IG inflation with m < 0 retains the perturbative unitarity up to m P as the models described in Ref. [1] do -cf. Ref. [14,15]. Focusing first on the JF computation, we remark that the argument goes as analyzed in Sec. R.1-4.1 with F K taking the form Moving on to the EF, recall -see Eqs. (3.8b) and (3.9) -that the canonically normalized inflaton, acquires mass which is calculated to be We remark that m δφ turns out to be largely independent of n as in Eq. (R.1-4.5). However, due to the modified λ − c R relation -see Eq. (3.21) -its numerical value increases slightly w.r.t its value in the models of Ref. [1]. E.g., taking φ ⋆ = 0.6m P and m = −(0.04−0.625) we get 6.9 m δφ /10 13 GeV 9.2 for n s in the range of Eq. (R.1-3.6a).
To check the limit of the validity of the effective theory, we expand J 2φ2 involved in Eq. (2.6) about φ in terms of δφ in Eq. (4.2) and we arrive at the following result (1 + m)m 2 P − · · · · (4.5) Hence, we can conclude from Eqs. (4.4) and (4.5) that in this case also Λ UV = m P , in agreement with our analysis in the JF.

CONCLUSIONS
Prompted by the recent excitement -see e.g. Ref. [9][10][11] -in the are(n)a of inflationary model building, we carried out a confrontation of IG inflation, formulated beyond the no-scale SUGRA, with the PLANCK [2] and BICEP2 results [4] -regardless of the ongoing debate on the ultimate validity of the latter [7,8]. As in our original paper, Ref. [1], the inflationary models are tied to a superpotential, which realizes easily the idea of IG, and a logarithmic Kähler potential, which includes all the allowed terms up to the fourth order in powers of the various fields -see Eq. (2.13). The models are totally defined imposing two global symmetries -a continuous R and a discrete Z n symmetryin conjunction with the requirement that the original inflaton takes subplanckian values. Extending our work in Ref. [1] we allow for deviations from the prefactor (−3) multiplying the logarithm of the Kähler potential-see Eq. (2.13). We parameterized these deviation by a factor (1 + m). Fixing n = 2, confining m to the range −(4 − 6.25)% and adjusting λ, c R and (−k SΦ ) in the ranges 0.09 − 3.5, (1.7 − 64.5) · 10 3 and 0.019 − 0.93 correspondingly, we achieved inflationary solutions that are simultaneously PLANCK-and BICEP2-friendly, i.e. we obtained n s ≃ 0.96 and 0.05 r 0.16 with negligible small a s . A mild tuning of k S to values of order 0.1 is adequate such that the oneloop radiative corrections remain subdominant. Moreover, the corresponding effective theory remains trustable up to m P , as in the other cases analyzed in Ref. [1]. In closing, we could say that incarnations of IG inflation beyond the no-scale SUGRA provide us with the adequate flexibility needed to obtain larger r's without disturbing the remaining attractive features of this inflationary model.