# Unitarity-safe models of non-minimal inflation in supergravity

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## Abstract

We show that models of chaotic inflation based on the \(\phi ^p\) potential and a linear non-minimal coupling to gravity, \(f_{{\mathcal {R}}}=1+c_{{\mathcal {R}}}\phi \), can be done consistent with data in the context of Supergravity, retaining the perturbative unitarity up to the Planck scale, if we employ logarithmic Kähler potentials with prefactors \(-p(1+n)\) or \(-p(n+1)-1\), where \(-0.035\lesssim n\lesssim 0.007\) for \(p=2\) or \(-0.0145\lesssim n\lesssim 0.006\) for \(p=4\). Focusing, moreover, on a model employing a gauge non-singlet inflaton, we show that a solution to the \(\mu \) problem of MSSM and baryogenesis via non-thermal leptogenesis can be also accommodated.

## 1 Introduction

*non-minimal inflation*(nMI) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Between the numerus models, which may be proposed in this context,

*universal attractor models*(UAMs) [14] occupy a prominent position since they exhibit an attractor towards an inflationary phase excellently compatible with data [15] for \(c_{{\mathcal {R}}}\gg 1\) and \(\phi \le m_{\mathrm{P}}\) – where \(m_{\mathrm{P}}\) is the reduced Planck mass. UAMs consider a monomial potential of the type

*Einstein frame*(EF) where the inflationary potential, \(\widehat{V}_{\mathrm{attr}}\), takes the form

*Jordan frame*(JF) to EF.

*Ultraviolet*(UV) cut-off scale [16, 17, 18]

*vacuum expectation value*(v.e.v) \(\left\langle {\phi } \right\rangle \) [23, 24, 25, 26, 27], or introducing a sizable kinetic mixing in the inflaton sector which dominates over \(\tilde{f}_{{\mathcal {R}}}\) [28, 29, 30, 31, 32, 33, 34].

Here we propose a novel solution – applied only in the context of *Supergravity* (SUGRA) – to the aforementioned problem, by exclusively considering \(q=2\) in Eq. (2) – cf. Ref. [35]. In this case, the canonically normalized inflaton \({\widehat{\phi }}\) is related to the initial field \(\phi \) as \({\widehat{\phi }}\sim c_{{\mathcal {R}}}\phi \) at the vacuum of the theory, in sharp contrast to what we obtain for \(q>2\) where \({\widehat{\phi }}\simeq \phi \). As a consequence, the small-field series of the various terms of the action expressed in terms of \({\widehat{\phi }}\), does not contain \(c_{{\mathcal {R}}}\) in the numerators, preventing thereby the reduction of \(\Lambda _{\mathrm{UV}}\) below \(m_{\mathrm{P}}\) [7, 18]. The same conclusion may be drawn within the JF since no dangerous inflaton-inflaton-graviton interaction appears because \({\tilde{f}}_{{\mathcal {R}},\phi \phi }=0\) for \(q=2\) [7]. Note that the importance of a scalar field with a totally or partially linear non-minimal coupling to gravity in unitarizing Higgs inflation within non-SUSY settings is highlighted in Refs. [36, 37].

A permanently linear \(\tilde{f}_{{\mathcal {R}}}\) can be reconciled with an inflationary plateau, similar to that obtained in Eq. (3), in the context of SUGRA, by suitably selecting the employed Kähler potentials. Indeed, this kind of models is realized in SUGRA using logarithmic or semilogarithmic Kähler potentials [8, 9, 10, 11, 12] with the prefactor \((-N)\) of the logarithms being related to the exponent of the denominator in Eq. (3). Therefore, by conveniently adjusting *N* we can achieve, in principle, a flat enough EF potential for any *p* in Eq. (1) but taking exclusively \(q=2\) in Eq. (2). As we show in the following, this idea works for \(p\le 4\) in Eq. (1) supporting nMI compatible with the present data [15]. For \(p=4\) we also show that the inflaton may be identified with a gauge singlet or non-singlet field. In the latter case, models of non-minimal Higgs inflation are introduced, which may be embedded in a more complete extension of MSSM offering a solution to the \(\mu \) problem [38] and allowing for an explanation of *baryon asymmetry of the universe* (BAU) [39] via *non-thermal leptogenesis* (nTL) [40, 41]. The resulting models employ one parameter less than those used in Refs. [33, 34] whereas the gauge-symmetry-breaking scale is constrained to values well below the MSSM unification scale contrary to what happens in Refs. [27, 33, 34].

Below, we first – in Sect. 2 – describe the SUGRA set-up of our models and prove that these are unitarity-conserving in Sect. 3. Then, in Sect. 4, we analyze the inflationary dynamics and predictions. In Sect. 5 we concentrate on the case of nMI driven by a Higgs field and propose a possible post-inflationary completion. We conclude in Sect. 6. Unless otherwise stated, we use units where the reduced Planck scale \(m_{\mathrm{P}}= 2.4\times 10^{18}~{\mathrm{GeV}}\) is set to be unity.

## 2 Supergravity framework

In Sect. 2.1 we describe the generic formulation of our models within SUGRA, and then we apply it for a gauge singlet and non-singlet inflaton in Sects. 2.2 and 2.3 respectively.

### 2.1 General framework

We focus on the part of the EF action within SUGRA related to the complex scalars \(z^{\alpha }\) – denoted by the same superfield symbol – which has the form [8, 9, 10, 11]

*z*denotes derivation

*with respect to*(w.r.t) the field

*z*. Also, \(\widehat{V}\) is the EF SUGRA potential which can be found once we select a superpotential

*W*in Eq. (24) and a Kähler potential

*K*via the formula

*g*. The remaining terms in the

*right-hand side*(r.h.s) of the equation above describes contribution from the F terms. The contribution from the D terms vanishes for a gauge singlet inflaton and can be eliminated during nMI for a gauge non-singlet inflaton, by identifying it with the radial part of a conjugate pair of Higgs superfields – see Sect. 2.3. In both our scenaria, we employ a “stabilizer” field

*S*placed at the origin during nMI. Thanks to this arrangement, the term \(3{\vert W\vert ^2}\) in \(\widehat{V}\) vanishes, avoiding thereby a possible runaway problem, and the derivation of \(\widehat{V}\) is facilitated since the non-vanishing terms arise from those proportional to \(W_{,S}\) and \(W^*_{,S^*}\) – see Sects. 2.2.2 and 2.3.2 below.

*N*may be considered in general as a free parameter with interesting consequences not only on the inflationary observables [7, 25, 26, 29, 32, 42, 43] but also on the consistency of the effective theory, as we show below.

### 2.2 Gauge-singlet inflaton

Below, in Sect. 2.2.1, we specify the necessary ingredients (super- and Kähler potentials) which allow us to implement our scenario with a gauge-singlet inflaton. Then, in Sect. 2.2.2, we outline the derivation of the inflationary potential.

#### 2.2.1 Set-up

Fields | Eigenstates | Masses squared | |||||
---|---|---|---|---|---|---|---|

\(K=K_1\) | \(K=K_2\) | \(K=K_{3}\) | \(K=K_4\) | \(K=K_{5}\) | |||

4 Real | \({\widehat{\theta }}\) | \({\widehat{m}}_{\theta }^2\) | \(6(1-1/N){\widehat{H}}_{\mathrm{CI}}^2\) | \(6{\widehat{H}}_{\mathrm{CI}}^2\) | |||

Scalars | \({\widehat{s}}, \widehat{\bar{s}}\) | \( {\widehat{m}}_{s}^2\) | \(6c_{{\mathcal {R}}}\phi {\widehat{H}}_{\mathrm{CI}}^2/N\) | \(6{\widehat{H}}_{\mathrm{CI}}^2/N_S\) | |||

2 Weyl spinors | \(\widehat{\psi }_\pm \) | \(\widehat{m}^2_{\psi \pm }\) | \(3p\left( 1-nc_{{\mathcal {R}}}\phi ^2\right) ^2{\widehat{H}}_{\mathrm{CI}}^2/Nc_{{\mathcal {R}}}^2\phi ^2\) |

*S*(\({\alpha }=2)\) being the inflaton and a “stabilizer” field respectively. More specifically, we adopt the superpotential

*R*symmetry under which

*S*and \(\Phi \) have charges 1 and 0; (ii) a global

*U*(1) symmetry with assigned charges \(-1\) and 2 /

*p*for

*S*and \(\Phi \). To obtain a linear non-minimal coupling of \(\Phi \) to gravity, though, we have to violate the latter symmetry as regards \(\Phi \). Indeed, we propose the following set of Kähler potentials

*S*during and after nMI. Their possible forms are given in Refs. [33, 34]. Just for definiteness, we adopt here only their logarithmic form, i.e.,

*K*’s contain up to quadratic terms of the various fields. Also \(F_{{\mathcal {R}}}\) (and \(F_{{\mathcal {R}}}^*\)) is exclusively included in the logarithmic part of the

*K*’s whereas \(F_{-}\) may or may not accompany it in the argument of the logarithm. Note finally that, although quadratic nMI is analyzed in Refs. [6, 7, 14] too, the present set of

*K*’s is examined for first time.

#### 2.2.2 Inflationary potential

*S*according to the parametrization

*K*’s in Eqs. (9a)–(9e), reads

*K*’s. Indeed, \(K_{{\alpha }{{\bar{\beta }}}}\) along the configuration in Eq. (12) takes the form

*p*, it yields integer

*N*in Eqs. (9a)–(9e), i.e., \(N=p+1\) for \(K=K_1\) and \(K_2\) or \(N=p\) for \(K=K_3-K_5\). Although integer

*N*’s are more friendly to string theory – and give observationally acceptable results as shown in Sect. 4.2 –, non-integer

*N*’s are also acceptable [25, 26, 29, 32, 33, 34, 42, 43] and assist us to cover the whole allowed domain of the observables. More specifically, for \(n<0\), \({\widehat{V}}_{\mathrm{CI}}\) remains an increasing function of \(\phi \), whereas for \(n>0\), it develops a local maximum \({\widehat{V}}_{\mathrm{CI}}(\phi _{\mathrm{max}})\) where

Fields | Eigenstates | Masses squared | |||||
---|---|---|---|---|---|---|---|

\(K=K_1\) | \(K=K_2\) | \(K=K_{3}\) | \(K=K_4\) | \(K=K_{5}\) | |||

4 Real | \({\widehat{\theta }}_{+}\) | \({\widehat{m}}_{\theta +}^2\) | \(3(1-1/N){\widehat{H}}_{\mathrm{HI}}^2\) | \(3{\widehat{H}}_{\mathrm{HI}}^2\) | |||

Scalars | \({\widehat{\theta }}_\Phi \) | \({\widehat{m}}_{ \theta _\Phi }^2\) | \(M^2_{BL}\) | \(M^2_{BL}+6{\widehat{H}}_{\mathrm{HI}}^2\) | \(M^2_{BL}\) | \(M^2_{BL}+6(1+1/N_S){\widehat{H}}_{\mathrm{HI}}^2\) | |

\({\widehat{s}}, \widehat{\bar{s}}\) | \( {\widehat{m}}_{s}^2\) | \(6{\widehat{H}}_{\mathrm{HI}}^2c_{{\mathcal {R}}}\phi /N\) | \(6{\widehat{H}}_{\mathrm{HI}}^2/N_S\) | ||||

1 Gauge boson | \(A_{BL}\) | \( M_{BL}^2\) | \(g^2\phi ^2/f_{{\mathcal {R}}}\) | \(g^2\phi ^2\) | \(g^2\phi ^2/f_{{\mathcal {R}}}\) | \(g^2\phi ^2\) | |

4 Weyl | \(\widehat{\psi }_\pm \) | \(\widehat{m}^2_{\psi \pm }\) | \(3\left( c_{{\mathcal {R}}}(N-5)\phi -4\right) ^2{\widehat{H}}_{\mathrm{HI}}^2/Nc_{{\mathcal {R}}}^2\phi ^2\) | \(3\left( c_{{\mathcal {R}}}(N-4)\phi -4\right) ^2{\widehat{H}}_{\mathrm{HI}}^2/Nc_{{\mathcal {R}}}^2\phi ^2\) | |||

Spinors | \(\uplambda _{BL}, {\widehat{\psi }}_{\Phi -}\) | \(M_{BL}^2\) | \(g^2\phi ^2/f_{{\mathcal {R}}}\) | \(g^2\phi ^2\) | \(g^2\phi ^2/f_{{\mathcal {R}}}\) | \(g^2\phi ^2\) |

*s*) arranged in Table 1, which approach rather well the quite lengthy, exact expressions taken into account in our numerical computation. We infer that \(\widehat{m}^2_{z^{\alpha }}\gg {\widehat{H}}_{\mathrm{CI}}^2={\widehat{V}}_{\mathrm{CI}}/3\) for \(1<N<6\) and \(K=K_1, K_2\) or for \(0<N_S<6\) and \(K=K_{3} - K_5\). Therefore \(m^2_{z^{\alpha }}\) are not only positive but also heavy enough during nMI. In Table 1 we display the masses \(\widehat{m}^2_{\psi \pm }\) of the spinors \(\widehat{\psi }_\pm = (\widehat{\psi }_{S}\pm \widehat{\psi }_{\Phi })/\sqrt{2}\) too. We define \(\widehat{\psi }_{S}=\sqrt{K_{SS^*}}\psi _{S}\) and \(\widehat{\psi }_{\Phi }=\sqrt{K_{\Phi \Phi ^*}}\psi _{\Phi }\) where \(\psi _\Phi \) and \(\psi _S\) are the Weyl spinors associated with

*S*and \(\Phi \) respectively.

### 2.3 Gauge non-singlet inflaton

Following the strategy of the previous section, we show below, in Sect. 2.3.1, how we can establish a model of unitarity-conserving nMI driven by a gauge non-singlet inflaton and then, in Sect. 2.3.2, we outline the derivation of the corresponding inflationary potential.

#### 2.3.1 Set-up

*S*, which is a gauge-singlet chiral superfield. Here, we take for simplicity the group \(U(1)_{B-L}\) where

*B*and

*L*denote the baryon and lepton number respectively. We base our construction on the superpotential [48]

*M*are parameters which can be made positive by field redefinitions. \(W_{\mathrm{HI}}\) is the most general renormalizable superpotential consistent with a continuous R symmetry [48] under which

*S*and \(W_{\mathrm{HI}}\) are equally charged whereas \({\bar{\Phi }}\Phi \) is uncharged. To obtain nMI – which is actually promoted to Higgs inflation – with a linear non-minimal coupling to gravity we combine \(W_{\mathrm{HI}}\) with one of the Kähler potentials in Eqs. (9a)–(9e) where the functions \(F_{{\mathcal {R}}}\) and \(F_{-}\) are now defined as

*K*’s respect the symmetries of \(W_{\mathrm{HI}}\). As in the case of Sect. 2.2.1, \(F_{-}\) ensures that the kinetic terms of \(\bar{\Phi }\) and \(\Phi \) do not enter the expression of \(f_{{\mathcal {R}}}\) along the inflationary trough – cf. Refs. [27, 29, 30, 31, 32].

Comparing the resulting *K*’s with the ones used in Refs. [33, 34] we may notice that here \(F_{-}\) is not accompanied by an independent variable \(c_-\) and the real function \(F_+\) is here replaced by the combination of the holomorphic function \(F_{{\mathcal {R}}}\) and its anti-holomorphic. Therefore, the present models are more economical since they include one parameter less. On the other hand, the presence of unity in the argument of the logarithms distinguishes clearly the present models from those in Ref. [27] where the absence of unity enforces us to invoke a large inflaton v.e.v.

#### 2.3.2 Inflationary potential

*S*in Eq. (13) and expressing \(\Phi \) and \({\bar{\Phi }}\) as follows

*n*in Eq. (20) we end up with

*K*’s in Eqs. (9a)–(9e) with \(F_{{\mathcal {R}}}\) and \(F_{-}\) given in Eq. (25), \(K_{{\alpha }{{\bar{\beta }}}}\) along the configuration in Eq. (27) takes the form

To check the stability of inflationary direction in Eq. (27) w.r.t the fluctuations of the non-inflaton fields, we derive the mass-squared spectrum of the various scalars defined in Eqs. (33a) and (33b). Taking the limit \(c_{{\mathcal {R}}}\gg 1\), we find the approximate expressions listed in Table 2 which are rather accurate at the horizon crossing of the pivot scale. As in case of Table 1, we again deduce that \(\widehat{m}^2_{z^{\alpha }}\gg {\widehat{H}}_{\mathrm{HI}}^2={\widehat{V}}_{\mathrm{HI}}/3\) for \(1<N<6\) and \(K=K_1, K_2\) or for \(0<N_S<6\) and \(K=K_{3} - K_5\). In Table 2 we also display the masses \(M_{BL}\) of the gauge boson \(A_{BL}\) and the corresponding fermions. The non-vanishing of \(M_{BL}\) signals the fact that \(U(1)_{B-L}\) is broken during nMI and so no cosmic string are produced at its end. Finally, the unspecified eigenstates \(\widehat{\psi }_\pm \) are defined as \(\widehat{\psi }_\pm =(\widehat{\psi }_{\Phi +}\pm \widehat{\psi }_{S})/\sqrt{2}\) – cf. Table 1.

## 3 Effective cut-off scale

The motivation of our proposal originates from the fact that \(f_{{\mathcal {R}}}\) in Eq. (16) assures that the perturbative unitarity is retained up to \(m_{\mathrm{P}}\) although that the attainment of nMI for \(\phi \le m_{\mathrm{P}}\) requires large \(c_{{\mathcal {R}}}\)’s – as expected from the UAMs [5, 6, 7] and verified in Sect. 4 below. To show that this achievement is valid, we extract below the UV cut-off scale, \(\Lambda _{\mathrm{UV}}\), expanding the action in the JF – see Sect. 3.1 – and in the EF – see Sect. 3.2. Throughout this section we find it convenient to restore \(m_{\mathrm{P}}\) in the formulas. We concentrate also on the versions of our model with gauge singlet inflaton. However, this analysis covers also the case of a gauge non-singlet inflaton for \(M\ll m_{\mathrm{p}}\), \(p=4\) and \(\lambda ^2\) replaced by \(\lambda ^2/4\).

Although the expansions about \(\left\langle {\phi } \right\rangle =0\), presented below, are not valid [19] during nMI, we consider \(\Lambda _{\mathrm{UV}}\) extracted this way as the overall cut-off scale of the theory for two reasons: (i) the reheating phase – realized via oscillations about \(\left\langle {\phi } \right\rangle \) – is an unavoidable stage of the inflationary dynamics; (ii) the result is within the range of validity of the low-energy theory and so this can be perturbatively extrapolated up to \(\Lambda _{\mathrm{UV}}\).

### 3.1 Jordan frame computation

### 3.2 Einstein frame computation

## 4 Inflation analysis

In Sects. 4.1 and 4.2 below we examine semi-analytically and numerically respectively, if \({\widehat{V}}_{\mathrm{CI}}\) in Eq. (21) may be consistent with a number of observational constraints. The analysis can be easily adapted to the case of \({\widehat{V}}_{\mathrm{HI}}\) in Eq. (29) performing the replacements mentioned in Sect. 3.

### 4.1 Semi-analytic results

*J*for \(K=K_1, K_3\) deviates slightly from that for \(K=K_2,K_4,K_5\) – see Eq. (19) – we have a discrimination as regards the expressions of \({\widehat{\epsilon }}\) and \({\widehat{\eta }}\) in these two cases. Indeed, our results are

*K*’s above we can numerically verify that Eq. (40a) is saturated for \(\phi =\phi _{\mathrm{f}}\), which is found from the condition

- For \(n=0\) and any
*K*in Eqs. (9a)–(9e), we obtainNote that for \(n\ne 0\) the formulas below for \({\widehat{N}_\star }\) cannot be reduced to the previous one.$$\begin{aligned} {\widehat{N}_\star }=\frac{Nc_{{\mathcal {R}}}}{2p}\phi _\star ~~\Rightarrow ~~\phi _\star \simeq \frac{2p{\widehat{N}_\star }}{Nc_{{\mathcal {R}}}}. \end{aligned}$$(45a) - For \(n\ne 0\) and \(K=K_1, K_3\), we obtainwhere \(e_n=e^{-2pn(n+1){\widehat{N}_\star }/N}\) and \(f_{n\star }=f_n(\phi _\star )\).$$\begin{aligned} {\widehat{N}_\star }=-\frac{N\ln f_{n\star }}{2n(1+n)p}~~\Rightarrow ~~\phi _\star \simeq \frac{1-e_n}{nc_{{\mathcal {R}}}}, \end{aligned}$$(45b)
- For \(n<0\) and \(K=K_2, K_4, K_5\), we obtainwhere \(W_k\) is the Lambert or product logarithmic function [51] with \(k=0\) and$$\begin{aligned} {\widehat{N}_\star }=-\frac{N\ln f_{n\star }}{2n(1+n)p}-\frac{\phi _\star ^2}{2np}~~\Rightarrow ~~\phi _\star \simeq \sqrt{\frac{N}{2(n+1)}}W_0(y), \end{aligned}$$(45c)$$\begin{aligned} y=2(1+n)e_n^2/Nn^2c_{{\mathcal {R}}}^2. \end{aligned}$$(45d)
- For \(n>0\) and \(K=K_2, K_4, K_5\), we obtainHere we are not able to solve the equation above w.r.t \(\phi _\star \). As a consequence, it is not doable to find an analytical expression for \(\phi _\star \) and the inflationary observables – see below. Therefore, in this portion of parameter space, our last resort is the numerical computation, whose the results are presented in Sect. 4.2.$$\begin{aligned} {\widehat{N}_\star }=-\frac{2+n^2c_{{\mathcal {R}}}^2N}{2nc_{{\mathcal {R}}}^2n^3(1+n)p}\ln f_{n\star }-\frac{3+f_{n\star }}{n^2pc_{{\mathcal {R}}}}\phi _\star . \end{aligned}$$(45e)

*J*given below Eq. (40b), we find

*K*’s in Eqs. (8) and (9a)–(9e), since these terms are exclusively expressed as functions of the initial field \(\Phi \) and remain harmless for \(\phi =\sqrt{2}|\Phi |\le 1\) – cf. Ref. [14].

*n*, in sharp contrast to UAMs where the same condition implies a relation between \(\lambda \) and \(c_{{\mathcal {R}}}\) [6, 12, 13]. Given that \(c_{{\mathcal {R}}}\) assumes large values, we expect that \(\lambda \) increases with

*p*and rapidly (for \(p\ge 5\) as we find numerically) violates the perturbative bound \(\lambda \le 2\sqrt{\pi }\simeq 3.5\). In particular, our results can be cast as following:

- For \(n\ne 0\) and \(K=K_1, K_3\), we obtainwhere \(f_{{\mathcal {R}}\star }=f_{{\mathcal {R}}}(\phi _\star )\). Taking into account that \(n\ll 1\) and \(f_{{\mathcal {R}}\star }\) is almost proportional to \(c_{{\mathcal {R}}}\) for large \(c_{{\mathcal {R}}}\), we can easily convince ourselves that the output above implies that \(\lambda /c_{{\mathcal {R}}}^{p/2}\) remains constant for fixed$$\begin{aligned} \lambda =2^{\frac{3}{2}+\frac{p}{4}}p\pi \sqrt{3A_{\mathrm{s}}}(1-nf_{{\mathcal {R}}\star }) f_{{\mathcal {R}}\star }^{\frac{np}{2}-1}c_{{\mathcal {R}}}^{{p}/{2}}/{N^{1/2}}, \end{aligned}$$(48b)
*n*. - For \(n<0\) and \(K=K_2, K_4, K_5\), we obtainfrom which we can again verify that the approximate proportionality of \(\lambda \) on \(c_{{\mathcal {R}}}^{p/2}\) holds.$$\begin{aligned} \lambda =2^{\frac{3}{2}+\frac{p}{4}}p\pi \sqrt{3A_{\mathrm{s}}}\frac{(1-nf_{{\mathcal {R}}\star }) f_{{\mathcal {R}}\star }^{\frac{np}{2}-1}c_{{\mathcal {R}}}^{1+\frac{p}{2}}}{(2f_{{\mathcal {R}}\star }^2+Nc_{{\mathcal {R}}}^2)^{1/2}}, \end{aligned}$$(48c)

*r*– are found from the relations [49, 50]

- For \(K=K_1\) and \(K_3\) we end up with a unified result$$\begin{aligned} n_{\mathrm{s}}\simeq & {} 1 - \frac{4p}{Nf_{{\mathcal {R}}\star }} -\frac{2p^2}{N}\left( n-\frac{1}{f_{{\mathcal {R}}\star }}\right) ^2 \nonumber \\\simeq & {} 1-\frac{2}{{\widehat{N}_\star }}+\frac{np}{{\widehat{N}_\star }}-\frac{2np}{N}; \end{aligned}$$(50a)$$\begin{aligned} r\simeq & {} \frac{16p^2}{N}\left( n-\frac{1}{f_{{\mathcal {R}}\star }}\right) ^2\simeq \frac{4N}{{\widehat{N}_\star }^2}-\frac{8np}{{\widehat{N}_\star }}; \end{aligned}$$(50b)For \(n=0\) the above results are also valid for \(K=K_2, K_4\) and \(K_5\) and yield observables identical with those obtained within UAMs [6, 8, 9, 10, 11, 12].$$\begin{aligned} a_{\mathrm{s}}\simeq & {} 8p^2(nf_{{\mathcal {R}}\star }-1)\frac{1+(1+n)p+f_{{\mathcal {R}}\star }(1-np)}{N^2f_{{\mathcal {R}}\star }^3}\nonumber \\\simeq & {} -\frac{2}{{\widehat{N}_\star }^2}+n\frac{1+p}{{\widehat{N}_\star }^2}. \end{aligned}$$(50c)
- For \(n<0\) and \(K=K_2, K_4\) and \(K_5\) we arrive at the following results$$\begin{aligned}&n_{\mathrm{s}}\simeq 1 + \frac{2pc_{{\mathcal {R}}}^2}{f_{{\mathcal {R}}\star }^2(2f_{{\mathcal {R}}\star }^2 +Nc_{{\mathcal {R}}}^2)^2}\Big (4 f_{{\mathcal {R}}\star }^3 (nf_{{\mathcal {R}}\star }-2) \nonumber \\&\quad -2N c_{{\mathcal {R}}}^2f_{{\mathcal {R}}\star }- (nf_{{\mathcal {R}}\star }-1)^2 (2f_{{\mathcal {R}}\star }^2 +Nc_{{\mathcal {R}}}^2) p\Big ); \end{aligned}$$(51a)with negligibly small \(a_{\mathrm{s}}\), as we find out numerically. Contrary to our previous results, here a \(c_{{\mathcal {R}}}\) dependence arises which complicates somehow the investigation of these models – see Sect. 4.2.$$\begin{aligned}&r\simeq \frac{16p^2}{N+2f_{{\mathcal {R}}\star }^2/c_{{\mathcal {R}}}^2}\left( n-\frac{1}{f_{{\mathcal {R}}\star }}\right) ^2 \end{aligned}$$(51b)

### 4.2 Numerical results

*n*numerically. In particular, we confront the quantities in Eq. (44) with the observational requirements [39]

*p*and

*n*. The outputs, encoded as lines in the \(n_{\mathrm{s}}-r_{0.002}\) plane, are compared against the observational data [39, 52] in Fig. 1 for \(K=K_1\) and \(K_3\) (left plot) or \(K=K_2, K_4\) and \(K_5\) (right plot) – here \(r_{0.002}=16{\widehat{\epsilon }}(\phi _{0.002})\) where \(\phi _{0.002}\) is the value of \(\phi \) when the scale \(k=0.002/\mathrm{Mpc}\), which undergoes \(\widehat{N}_{0.002}=({\widehat{N}_\star }+3.22)\) e-foldings during nMI, crosses the horizon of nMI. We draw dashed [solid] lines for \(p=2\) [\(p=4\)] and show the variation of

*n*along each line. We take into account the data from

*Planck*and

*Baryon Acoustic Oscillations*(BAO) and the BK14 data taken by the Bicep2/

*Keck Array*CMB polarization experiments up to and including the 2014 observing season. Fitting the data above [15, 52] with \(\Lambda \)CDM\(+r\) model we obtain the marginalized joint \(68\%\) [\(95\%\)] regions depicted by the dark [light] shaded contours in Fig. 1. Approximately we get

*confidence level*(c.l.) with \(|a_{\mathrm{s}}|\ll 0.01\).

*n*which, though, remains close to zero signalizing an amount of tuning. In accordance with Eqs. (50a) and (50b), we find the allowed ranges

*n*varies in its allowed ranges above, we obtain

*r*is a little more enhanced. Contrary to Starobinsky inflation, though, \({\widehat{V}}_{\mathrm{CI}}\) is well defined at the origin as happens within the original UAMs and those in Refs. [32, 33, 34]. Obviously, the requirement – mentioned in Sect. 3 – \({\widehat{V}}_{\mathrm{CI}}^{1/4}\ll 1\) dictated from the validity of the effective theory is readily fulfilled.

*n*may deviate more appreciably from zero (mainly for \(p=2\)) and the maximal possible

*r*is somewhat larger. Moreover, these predictions depend harder on \(c_{{\mathcal {R}}}\) for \(|n|>0.01\), as expected from Eqs. (51a) and (51b). Therefore, in that regime, we could say that these models are less predictive than those based on \(K=K_1\) and \(K_3\). Our results below are presented for \(c_{{\mathcal {R}}}\) such that \(\phi _\star \simeq 1\). Namely, for \(p=2\), we find

*r*values are testable by the forthcoming experiments [53, 54, 55, 56], which are expected to measure

*r*with an accuracy of \(10^{-3}\). The tuning, finally, required for the attainment of hilltop nMI for \(n>0\) is very low, since \(\phi _\mathrm{max}\gg \phi _\star \).

*n*, the amplitudes of \(\lambda \) and \(c_{{\mathcal {R}}}\) can be bounded. This fact is illustrated in Fig. 3 where we display the allowed values \(c_{{\mathcal {R}}}\) versus \(\sqrt{\lambda }\) for \(p=4\) and \(K=K_{1}\) (gray lines) or \(K=K_5\) (light gray lines). We take \(n=0\) (solid lines) and \(n=-0.004\) (dashed lines). As anticipated in Eq. (46) for any

*n*there is a lower bound on \(c_{{\mathcal {R}}}\), above which \(\phi _\star \le 1\) stabilizing thereby the results against corrections from higher order terms – e.g., \(({\bar{\Phi }}\Phi )^l\) with \(l>1\) in Eq. (24). The perturbative bound \(\lambda =3.5\) limits the various lines at the other end. We observe that the ranges of the allowed lines are much more limited compared to other models – cf. Refs. [12, 30, 31] – and displaced to higher \(\lambda \) values as seen, also, by Eqs. (48a) and (48b). We find that for \(p=5\), \(\lambda \) corresponding to lowest possible \(c_{{\mathcal {R}}}\) violates the perturbative bound and so, our proposal can not be applied for \(p>4\).

Fields | Eigenstates | Masses squared | ||
---|---|---|---|---|

\(K=K_1,K_2\) | \(K=K_3,K_4,K_5\) | |||

10 Real | \(\widehat{h}_{\pm },\widehat{{\bar{h}}}_{\pm }\) | \( {\widehat{m}}_{h\pm }^2\) | \(3{\widehat{H}}_{\mathrm{HI}}^2\left( 1+c_{{\mathcal {R}}}\phi /{N}\pm {4\lambda _\mu f_{{\mathcal {R}}}/\lambda \phi ^2}\right) \) | \(3{\widehat{H}}_{\mathrm{HI}}^2\left( 1+1/{N_X}\pm {4\lambda _\mu }/{\lambda \phi ^2}\right) \) |

Scalars | \(\widehat{{\tilde{\nu }}}^c_{i}, \widehat{\bar{{\tilde{\nu }}}}^c_{i}\) | \({\widehat{m}}_{i{\tilde{\nu }}^c}^2\) | \(3{\widehat{H}}_{\mathrm{HI}}^2c_{{\mathcal {R}}}\left( \phi /N+16\lambda ^2_{iN^c}/{\lambda ^2\phi }\right) \) | \(3{\widehat{H}}_{\mathrm{HI}}^2\left( 1+1/{N_X}+16\lambda ^2_{iN^c }/\lambda ^2\phi ^2\right) \) |

3 Weyl spinors | \({{\widehat{N}}_i^c}\) | \( {\widehat{m}}_{{iN^c}}^2\) | \(48{\widehat{H}}_{\mathrm{HI}}^2\lambda ^2_{iN^c }/\lambda ^2\phi ^2\) |

## 5 A post-inflationary completion

In a couple of recent papers [27, 33, 34] we attempt to connect the high-scale inflationary scenario based on \(W=W_{\mathrm{HI}}\) in Eq. (24) with the low energy physics, taking into account constraints from the observed BAU, neutrino data and MSSM phenomenology. It would be, therefore, interesting to check if this scheme can be applied also in the case of our present set-up where \(W_{\mathrm{HI}}\) in Eq. (24) cooperates with the *K*’s in Eqs. (9a) – (9e) where \(F_{{\mathcal {R}}}\) and \(F_{-}\) given in Eq. (25). The necessary extra ingredients for such a scenario are described in Sect. 5.1. Next, we show how we can correlate nMI with the generation of the \(\mu \) term of MSSM – see Sect. 5.2 – and the generation of BAU via nTL – see Sect. 5.3. Hereafter, we restore units, i.e., we take \(m_{\mathrm{P}}=2.433\cdot 10^{18}~{\mathrm{GeV}}\).

### 5.1 Relevant set-up

*i*-th generation \(SU(2)_{\mathrm{L}}\) doublet left-handed lepton superfields, and \({H_u}\) [\({H_d}\)] is the \(SU(2)_{\mathrm{L}}\) doublet Higgs superfield which couples to the up [down] quark superfields. Also, we assume that the superfields \(N_j^c\) have been rotated in the family space so that the coupling constants \(\lambda _i\) are real and positive. This is the so-called [24, 33, 34] \(N^c_i\) basis, where the \(N^c_i\) masses, \(M_{iN^c}\), are diagonal, real and positive.

*S*expressed by the functions \(F_{lS}\) with \(l=1,2,3\) in Eqs. (11a) – (11c) – see Refs. [33, 34]. Therefore, \(N_S\) may be renamed \(N_X\) henceforth. The inflationary trajectory in Eq. (12) has to be supplemented by the conditions

*S*in Eq. (13). The relevant masses squared are listed in Table 3 for \(K=K_1 -- K_5\), with hatted fields being defined as \({\widehat{s}}\) and \(\widehat{\bar{s}}\) in Eq. (23). Also we set

### 5.2 Solution to the \(\mu \) problem of MSSM

Supplementing, in Sect. 5.2.2, with soft SUSY breaking terms the SUSY limit of the SUGRA potential – found in Sect. 5.2.1 – we can show that our model assists us to understand the origin of \(\mu \) term of MSSM, consistently with the low-energy phenomenology – see Sect. 5.2.3.

#### 5.2.1 SUSY potential

*K*’s in Eqs. (9a)–(9e) with \(F_{{\mathcal {R}}}\) and \(F_{-}\) defined in Eq. (25) allows us to expand them for \(m_{\mathrm{P}}\rightarrow \infty \) up to quadratic terms obtaining \({\widetilde{K}}\). Focusing on the \(S-{\bar{\Phi }}-\Phi \) system we obtain

*S*, we can easily verify that their v.e.vs lie along the direction in Eq. (59).

CMSSM region | Parameters | |||||||
---|---|---|---|---|---|---|---|---|

\(|A_0|\) \(({\mathrm{TeV}})\) | \(m_0\) \(({\mathrm{TeV}})\) | \(|\mu | \) \(({\mathrm{TeV}})\) | \(\mathrm{a}_{3/2}\) | \(\lambda _\mu ~(10^{-6})\) | ||||

\(K=K_1\) | \(K=K_2\) | \(K=K_3\) | \(K=K_4,~K_5\) | |||||

| 9.9244 | 9.136 | 1.409 | 1.086 | 2.946 | 2.867 | 3.28 | 3.195 |

\({\tilde{\tau }}_1-\chi \) coannihilation | 1.2271 | 1.476 | 2.62 | 0.831 | | | 49.35 | 48.06 |

\({\tilde{t}}_1-\chi \) coannihilation | 9.965 | 4.269 | 4.073 | 2.33 | 8.58 | 8.27 | 9.461 | 9.214 |

\({\tilde{\chi }}^\pm _1-\chi \) coannihilation | 9.2061 | 9.000 | 0.983 | 1.023 | 2.215 | 2.156 | 2.46 | 2.4 |

#### 5.2.2 Generation of the \(\mu \) term of MSSM

*S*has been rotated in the real axis by an appropriate

*R*-transformation. The extremum condition for \(\left\langle {V_\mathrm{tot}(S)} \right\rangle \) in Eq. (71a) w.r.t

*S*leads to a non-vanishing \(\left\langle {S} \right\rangle \) as follows

*n*and \(\lambda _\mu \) since \(\lambda /c_{{\mathcal {R}}}^2\) is fixed for frozen

*n*by virtue of Eqs. (48a)–(48c). As a consequence, we may verify that any \(|\mu |\) value is accessible for the \(\lambda _\mu \)’s allowed by Eq. (62) without any ugly hierarchy between \(m_{3/2}\) and \(\mu \).

#### 5.2.3 Link to the MSSM phenomenology

The subgroup, \({\mathbb {Z}}_2^{R}\) of \(U(1)_R\) – which remains unbroken after the consideration of the SUSY breaking effects in Eq. (69) – combined with the \({\mathbb {Z}}_2^{\mathrm{f}}\) fermion parity yields the well-known *R*-parity. This symmetry guarantees the stability of the *lightest SUSY particle* (LSP), providing thereby a well-motivated *cold dark matter* (CDM) candidate.

*Constrained MSSM*(CMSSM), which employs the following free parameters

*N*and \(c_{{\mathcal {R}}}\) required by each

*K*and, besides the ones written in italics, are comfortably compatible with Eq. (62). Therefore, the whole inflationary scenario can be successfully combined with all the allowed regions CMSSM besides the \({\tilde{\tau }}_1-\chi \) coannihilation region for \(K=K_1\) and \(K_2\). On the other hand, the \(m_{3/2}\)’s used in all the CMSSM regions can be consistent with the gravitino limit on reheat temperature \(T_{\mathrm{rh}}\), under the assumption of the unstable \(\widetilde{G}\), for the \(T_{\mathrm{rh}}\) values necessitated for satisfactory leptogenesis – see Sect. 5.3.2.

### 5.3 Non-thermal leptogenesis and neutrino masses

Besides the generation of \(\mu \) term, our post-inflationary setting offers a graceful exit from the inflationary phase (see Sect. 5.3.1) and explains the observed BAU (see Sect. 5.3.2) consistently with the \(\widetilde{G}\) constraint and the low energy neutrino data. Our results are summarized in Sect. 5.3.3.

#### 5.3.1 Inflaton mass and decay

*M*which is bounded from above by the requirement \(\left\langle {f_{{\mathcal {R}}}} \right\rangle =1\) ensuring the establishment of the conventional Einstein gravity at the vacuum. This bound is translated to an upper bound on the mass \(\left\langle {M_{BL}} \right\rangle \) that the \(B-L\) gauge boson acquires for \(\phi =\left\langle {\phi } \right\rangle \) – see Table 2. Namely we obtain \(\left\langle {M_{BL}} \right\rangle \le 10^{14}~{\mathrm{GeV}}\), which is lower than the value \(M_{\mathrm{GUT}}\simeq 2\cdot 10^{16}~{\mathrm{GeV}}\) dictated by the unification of the MSSM gauge coupling constants – cf. Refs. [27, 33, 34]. However, since \(U(1)_{B-L}\) gauge symmetry does not disturb this unification, we can treat \(\left\langle {M_{BL}} \right\rangle =gM\) as a free parameter with \(g\simeq 0.5-0.7\) being the value of the GUT gauge coupling at the scale \(\left\langle {M_{BL}} \right\rangle \).

*X*,

*Y*,

*Z*involved in a typical trilinear superpotential term \(W_y=yXYZ\). Here, \(\psi _X, \psi _{Y}\) and \(\psi _{Z}\) are the chiral fermions associated with the superfields

*X*,

*Y*and

*Z*whose scalar components are denoted with the superfield symbols and \(y=y_3\simeq (0.4-0.6)\) is a Yukawa coupling constant of the third generation.

#### 5.3.2 Lepton-number and gravitino abundances

*normal [inverted] ordered*(NO [IO])

*neutrino masses*, \(m_{i\mathrm \nu }\)’s. Furthermore, the sum of \(m_{i\mathrm \nu }\)’s is bounded from above at 95% c.l. by the data [39]

Parameters yielding the correct BAU for \(K=K_1\) or \(K_2\), \(\left\langle {M_{BL}} \right\rangle =10^{12}~{\mathrm{GeV}}\), \(n=0\), \(\lambda _\mu =10^{-6}\), \(y_3=0.5\) and various neutrino mass schemes

Cases | A | B | C | D | E | F | G |
---|---|---|---|---|---|---|---|

Normal | Almost | Inverted | |||||

Parameters | Hierarchy | Degeneracy | Hierarchy | ||||

Low scale parameters (masses in \({\mathrm{eV}}\)) | |||||||

\(m_{1\mathrm \nu }/0.1\) | 0.01 | 0.1 | 0.5 | 0.6 | 0.7 | 0.51 | 0.5 |

\(m_{2\mathrm \nu }/0.1\) | 0.09 | 0.13 | 0.51 | 0.61 | 0.71 | 0.52 | 0.51 |

\(m_{3\mathrm \nu }/0.1\) | 0.5 | 0.51 | 0.71 | 0.78 | 0.5 | 0.1 | 0.05 |

\(\sum _im_{i\mathrm \nu }/0.1\) | 0.6 | 0.74 | 1.7 | 2 | 1.9 | 1.1 | 1 |

\(\varphi _1\) | \(-\pi /9\) | \(-2\pi /3\) | \(\pi \) | \(\pi /2\) | \(\pi /2\) | \(3\pi /5\) | \(2\pi /3\) |

\(-\varphi _2\) | \(\pi /2\) | 0 | \(-\pi /2\) | \(2\pi /3\) | \(2\pi /3\) | \(\pi /2\) | \(\pi /2\) |

Leptogenesis-scale mass parameters in \({\mathrm{GeV}}\) | |||||||

\(m_{1\mathrm D}/0.1\) | 0.38 | 0.69 | 3.23 | 2.25 | 0.94 | 50 | 1.9 |

\(m_{2\mathrm D}\) | 1 | 5 | 0.2 | 0.15 | 0.3 | 0.097 | 0.2 |

\(m_{3\mathrm D}\) | 1 | 10 | 5 | 10 | 9 | 6 | 5 |

\(M_{1N^c}/10^{8}\) | 5.4 | 8.4 | 9.9 | 4.7 | 2.2 | 7.9 | 12.6 |

\(M_{2N^c}/10^{9}\) | 13 | 755 | 2.5 | 0.96 | 1.57 | 219 | 3.5 |

\(M_{3N^c}/10^{11}\) | 3.2 | 18.8 | 1.6 | 6.1 | 5 | 12.7 | 12.1 |

Decay channels of \(\widehat{\delta \phi }\) | |||||||

\(\widehat{\delta \phi }\rightarrow \) | \(N^c_{1}\) | \(N^c_{1}\) | \(N^c_{1}\) | \(N^c_{1,2}\) | \(N^c_{1,2}\) | \(N^c_{1}\) | \(N^c_{1}\) |

Resulting | |||||||

\(Y_B/10^{-11}\) | 8.7 | 8.6 | 8.6 | 8.6 | 8.5 | 8.7 | 8.6 |

Resulting \(T_{\mathrm{rh}}\) (in \({\mathrm{GeV}}\)) | |||||||

\(T_{\mathrm{rh}}/10^{7}\) | 2.8 | 3 | 3 | 3.1 | 2.7 | 2.9 | 3.3 |

*nucleosynthesis*(BBN), which is estimated to be [62, 63, 64, 65, 66, 67]

#### 5.3.3 Results

*K*. Although this amount of uncertainty does not cause any essential alteration of the final outputs, we mention just for definiteness that we take throughout \(K=K_1\) corresponding to \(\widehat{m}_{\mathrm{\delta \phi }}=3.3\cdot 10^{10}~{\mathrm{GeV}}\). We consider NO (cases A and B), almost degenerate (cases C, D and E) and IO (cases F and G) \(m_{i\mathrm \nu }\)’s. In all cases, the current limit in Eq. (84) is safely met. This is more restrictive than the \(90\%\) c.l. upper bound arising from the effective electron neutrino mass \(m_{\beta }\) in \(\beta \)-decay [71, 72, 73] by various experiments. Indeed, the current upper bounds on \(m_{\beta }\) are comfortably satisfied by the values found in our set-up

As a bottom line, nTL is a realistic possibility within our setting. It can be comfortably reconciled with the \(\widetilde{G}\) constraint even for \(m_{3/2}\sim 1~{\mathrm{TeV}}\) as deduced from Eqs. (89b) and (86) adopting a sufficiently low \(\left\langle {M_{BL}} \right\rangle \).

## 6 Conclusions

Motivated by the fact that a strong linear non-minimal coupling of the inflaton to gravity does not cause any problem with the validity of the effective theory up to the Planck scale, we explored the possibility to attain observationally viable nMI (i.e. non-minimal inflation) in the context of standard SUGRA by strictly employing this coupling. We showed that nMI is easily achieved, for \(p\le 4\) in the superpotential of Eq. (8), by conveniently adjusting the prefactor \((-N)\) of the logarithmic part of the relevant Kähler potentials *K* given in Eqs. (9a)–(9e), where the relevant functions \(F_{{\mathcal {R}}}\) and \(F_{-}\) are shown in Eq. (10) for a gauge-singlet inflaton. For appropriately selected integer *N*’s – i.e., setting \(n=0\) in Eq. (20) –, our models retain the predictive power of well-known universal attractor models – which employ a non-minimal coupling functionally related to the potential – and yield similar results. Allowing for non-integer *N* values, this predictability is lost since the observables depend on the adopted *K* and *n* in Eq. (20) and may yield any \(n_{\mathrm{s}}\) in its allowed region and \(0.0013\le r\le 0.02\).

This scheme works also for a gauge non-singlet inflaton employing the superpotential shown in Eq. (24) and the functions \(F_{{\mathcal {R}}}\) and \(F_{-}\) in Eq. (25). Embedding these models within a \(B-L\) extension of MSSM, we showed that a \(\mu \) term is easily generated and the baryon asymmetry in the Universe is naturally explained via non-thermal leptogenesis. The \(B-L\) breaking scale \(\left\langle {M_{BL}} \right\rangle \), though, has to take values lower than the MSSM unification scale and so, the present scheme is similarly predictive with that of Refs. [33, 34] which employs one more parameter in the Kähler potentials but allows for \(\left\langle {M_{BL}} \right\rangle \)’s fixed by the gauge coupling unification within MSSM. Our scenario can be comfortably tolerated with almost all the allowed regions of the CMSSM with gravitino as low as \(1~{\mathrm{TeV}}\). Moreover, leptogenesis is realized through the out-of equilibrium decay of the inflaton to the right-handed neutrinos \(N_1^c\) and/or \(N_2^c\) with masses lower than \(2.32\cdot 10^{10}~{\mathrm{GeV}}\), and a reheat temperature \(T_{\mathrm{rh}}\le 10^9~{\mathrm{GeV}}\) taking \(\left\langle {M_{BL}} \right\rangle \le 10^{13}~{\mathrm{GeV}}\).

## Notes

### Acknowledgements

I would like to acknowledge A. Riotto for a useful discussion and encouragement.

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