# Generalized dilaton–axion models of inflation, de Sitter vacua and spontaneous SUSY breaking in supergravity

## Abstract

We propose the unified models of cosmological inflation, spontaneous SUSY breaking, and the dark energy (de Sitter vacuum) in \(N=1\) supergravity with the dilaton–axion chiral superfield *T* in the presence of an \(N=1\) vector multiplet with the alternative Fayet–Iliopoulos term. By using the Kähler potential as \(K=-\alpha \log (T+\overline{T})\) and the superpotential as a sum of a constant and a linear term, we find that viable inflation is possible for \(3\le \alpha \le \alpha _\mathrm{max}\) where \(\alpha _\mathrm{max}\approx 7.235\). Observations of the amplitude of primordial scalar perturbations fix the SUSY breaking scale in our models as high as \(10^{13}\ \hbox {GeV}\). In the case of \(\alpha >3\) the axion gets the tree-level (non-tachyonic) mass comparable to the inflaton mass.

## 1 Introduction

Supergravity is well motivated as the possible theoretical interface between (a) high-energy physics (well) beyond the Standard Model (SM) of elementary particles, (b) gravity beyond the Concordance (\(\Lambda \)CDM) Cosmological Model, and (c) string theory as the theory of quantum gravity whose low-energy effective action is described by supergravity. A phenomenological description of high energy particle physics and cosmology in supersymmetry (SUSY) and supergravity is known to be non-trivial, though many viable models exist, see e.g., the reviews [1, 2, 3, 4] and the references therein. No signs of SUSY at the Large Hadron Collider (LHC) may hint towards a high scale of SUSY phenomena. At such scales (indirect) cosmological probes of SUSY prevail over (direct) experimental probes at particle colliders. The early Universe is, therefore, the natural place for physical applications of supergravity.

A simultaneous description of cosmological inflation and dark energy (as the positive cosmological constant) in supergravity is another challenge due to the huge difference in the relevant scales and the need of (spontaneous) SUSY breaking. The standard approach in supergravity is based on the use of *chiral* \(N=1\) superfields in four spacetime dimensions with the input given by a Kähler potential *K* and a superpotential *W*. Then the scalar potential and the kinetic terms of the scalar field components are uniquely defined, and the phenomenological model building amounts to choosing both *K* and *W* in order to achieve a viable single-field inflation consistent with the Cosmic Microwave Background (CMB) observations and a de Sitter (dS) vacuum after inflation. There are several problems with that approach. First, the input given by *K* and *W* allows infinitely many choices. Second, it always leads to the multi-scalar framework so that one has to choose the inflaton direction in the field space and suppress the non-inflaton scalars during inflation in order to prevent spoiling of the inflaton slow roll and get enough number of e-foldings. Third, after inflation one has to get the hierarchy between the (high) SUSY breaking scale allowing large masses for the superpartners of the SM particles and the (low) dark energy scale given by the cosmological constant. Getting that hierarchy may require *two* different mechanisms of spontaneous SUSY breaking.

It is possible to reduce (and minimize) the number of scalars in the inflationary models by employing a massive (irreducible) \(N=1\) *vector* multiplet as the inflaton supermultiplet, instead of a chiral one [5, 6]. The massive vector multiplet has only *one* (real) physical scalar that can be identified with inflaton, while its fermionic superpartner can be identified with goldstino in the minimalistic setup for inflation in supergraviity (cf. Refs. [7, 8]). To avoid SUSY restoration after inflation in a Minkowski vacuum (it was the drawback of the first supergravity models with inflaton in a vector multiplet), one may either add the hidden sector described by a chiral (Polonyi) superfield as in Refs. [9, 10, 11] or introduce the alternative (new) Fayet–Iliopoulos (FI) terms as in Refs. [12, 13].^{1} Moreover, one can also combine both approaches and derive the supergravity-based inflationary models with inflaton in a massive vector multiplet in the presence of the FI term, with both F-type and D-type SUSY breaking needed for the hierarchy of scales [16, 17]. In all those cases, the canonical Kähler potential and a linear superpotential for Polonyi superfield were chosen, like the original Polonyi model [18].

*T*by replacing the canonical (free) Kähler potential by the generalized “no-scale” one as follows [19]:

*T*can be identified with the volume modulus of the compactified manifold in heterotic string theory. It is remarkable that the same Kähler potential with \(\alpha =3\) also arises in the modified

*F*(

*R*) supergravity after its dualization [23, 24, 25].

It is, however, also known that the case of \(\alpha =3\) in Eq. (1) with just a *single* chiral superfield is not viable for cosmological applications because it does not allow stable dS vacua and cannot be used for realizing Starobinsky inflation [26] with *any* choice of the superpotential *W* [3, 27, 28, 29, 30], although there are single field models with generalized \(\alpha \) (\(\alpha \)-attractors) leading to a supersymmetric Minkowski vacuum [31, 32, 33].

*two chiral*superfields and the Kähler potential

*T*is the volume modulus, and \(\Phi ^i\) are the matter chiral superfields parametrizing the non-linear sigma-model tangent space \(SU(p,1;\mathbb {C})/SU(p)\times U(1)\), with the suitable superpotential.

In this paper we use a single “dilaton–axion” chiral superfield *T* with the Kähler potential (1) but introduce a single *vector* multiplet in addition. We demonstrate that it leads to the viable set of cosmological models describing inflation, dS vacua and spontaneous SUSY breaking.

*dual*description in terms of the standard supergravity coupled to a massive vector multiplet or, equivalently, a massless vector multiplet and a Stückelberg chiral multiplet with the Kähler potential [6]

In this paper we find that a non-vanishing D-term allows us to introduce the new inflationary models based on the Kähler potentials having the form (1) with a single chiral superfield and a single vector superfield. The other examples of the D-term based on the alternative FI terms [14, 15] can be found in Refs. [13, 38, 39, 40, 41]. Those FI terms provide a tunable positive cosmological constant or dS uplifting of the vacuum after inflation [12, 16, 17, 42]. Our inflationary models in this paper have the Kähler potential (1) and the Polonyi-type linear superpotential (without gauging the shift symmetry of *K*) leading to the spontaneous F-type SUSY breaking. In addition, the simplest alternative FI term leads to another D-type SUSY breaking and uplifts an Anti-dS (AdS) minimum of the F-term scalar potential to a dS minimum.

The paper is organized as follows. Our setup is given in Sect. 2. In Sect. 3 we study vacua and SUSY breaking. In Sect. 4 we study inflation in our framework and analyze in detail the models with integer \(\alpha \). In particular, we derive the explicit values of the dilaton and axion masses and the SUSY breaking parameters by fixing the inflationary observables with the CMB observational data. We conclude in Sect. 5. The basic formulae about the standard \(N=1\) supergravity and the alternative FI term are given in Appendix. We set the reduced Planck mass as \(M_\mathrm{Pl}=\kappa ^{-1}=1\) unless otherwise stated.

## 2 The setup

*T*is parametrized as

*t*. The F-term scalar potential reads

*R*,

*I*stand for the real and imaginary parts. It is convenient to trade the complex parameter \(\lambda \) for the two real ones, \(\omega _1\) and \(\omega _2\) defined above.

*D*) of the vector multiplet, one gets a positive contribution

*g*is the gauge coupling, and \(\xi \) is the real FI constant.

^{2}More details about the alternative FI term and the bosonic action of the standard \(N=1\) supergravity can be found in Appendix.

## 3 Vacua and SUSY breaking

*A*and \(V_t\), thus making the potential

*t*-independent. In the next Section we consider separately the case of \(\alpha =3\) and then turn to \(\alpha \ne 3\). When \(\alpha \ne 3\), we can use the solution to Eq. (14) as \(t_0=-\omega _2/(2|\mu |^2)\) and rewrite

*A*as

*A*becomes negative when \(\alpha <3\). Given negative

*A*, the potential (12) is unbounded from below because

*A*multiplies the highest power of

*x*(for \(\alpha <3\) the potential also becomes unstable in the

*t*-direction). Therefore, we restrict ourselves to \(\alpha \ge 3\) in what follows.

### 3.1 The case \(\alpha =3\)

*t*-flat.

The bosonic sector also includes a massless axion *t* and a massless vector. The vector can be made massive via (additional) super-Higgs effect. A massless scalar is phenomenologically problematic, but the mass of *t* may be generated either by quantum corrections when \(\alpha =3\) (as is usually assumed in the “no-scale” supergravity models), or already at the tree level when \(\alpha >3\), as we are going to show in the next Subsection.

### 3.2 The case \(\alpha >3\): vacuum solutions

*x*requires \(\gamma /\omega _1\) to be positive. Since we have \(\alpha >3\), the \(\gamma _-\) is always negative, while the sign of \(\gamma _+\) depends on the choice of \(\alpha \). More specifically, we find

^{3}

### 3.3 The case \(\alpha >3\): SUSY breaking and scalar masses

*t*, we first get

*t*is not canonical at the \(\phi \)-minimum because the \(\phi _0\) is non-vanishing and

*t*for all values of \(\phi \), it is certainly possible at the reference point \(\phi _0\) by the rescaling

*x*solution (32). We find

In Fig. 2 we plot the mass ratios for \(3<\alpha <\alpha _*\). Figure 2a shows that with \(\gamma _+\) corresponding to a positive \(\omega _1\) axion is lighter than inflaton if \(\alpha <(5+\sqrt{33})/2\approx 5.37\), whereas beyond this point axion becomes heavier. Gravitino (with \(\gamma _+\)) is slightly lighter than inflaton in the range \(3.8\lessapprox \alpha \lessapprox 5.27\), whereas outside this range gravitino becomes heavier.^{4}

## 4 Inflation

In order to study inflation, let us restore the gravitational constant \(\kappa \equiv \sqrt{8\pi G}=M_P^{-1}\). We choose the Kähler potential and the chiral field *T* to be dimensionless, whereas the superpotential has the mass dimension three, \([W]=M^3\). It follows that \([\lambda ]=[\mu ]=M^3\) and \([\omega _1]=M^6\), where \([\ldots ]\) stands for the mass dimension of the corresponding quantity. We also set \([g\xi ]=M^0\) and \([\phi ]=[\varphi ]=M\).

*x*-solution (32). Restoring \(\kappa \) results in the potential

*i*means evaluation at the initial value of the inflaton, \(\varphi _i\) i.e., at the horizon crossing. The number of e-foldings between \(\varphi _i\) and \(\varphi _f\) is given by

*r*, and \(A_s\) are [44]

*r*depend only on \(\alpha \) and \(\mathrm{sgn}(\omega _1)\) (and not on the value of \(\omega _1\)) which determine the shape of the scalar potential. The observed value of \(A_s\) (\(\sim 10^{-9}\)) can be used to fix the composite parameter \(|\mu |^{2(\alpha -1)}/\omega _1^{\alpha -2}\) that is related to the inflaton mass via Eq. (48).

^{5}A more precise value of \(\alpha _\mathrm{max}\) can be derived by finding \(\varphi _i\) that solves the condition \(n_s(\varphi _i)=0.9607\) and substituting this value in Eq. (54) to solve \(N_e(\alpha )=60\). This results in

As we show below, the tensor-to-scalar ratio *r* decreases with increasing \(\alpha \) and is always compatible with the limit \(r<0.064\).

### 4.1 The case \(3\le \alpha \le \alpha _*\): starobinsky-like inflation

Let us divide our models into two classes for \(3\le \alpha \le \alpha _*\) and \(\alpha _*<\alpha \le \alpha _\mathrm{max}\), respectively. The reason is that in the range \(3\le \alpha \le \alpha _*\) the inflationary potential is truly Starobinsky-like and has a single extremum, namely, the global minimum and the infinite plateau asymptotically approaching a constant positive height. In contrast, if \(\alpha >\alpha _*\) the potential has a local maximum, which means that we get the hilltop inflationary models.

*r*for \(3\le \alpha \le \alpha _*\) by setting \(N_e=55\). In this subsection, we take \(\alpha =3,4,5,6\) (\(\alpha =3\) is the Starobinsky case) and, in addition, we include the upper limit \(\alpha =\alpha _* \equiv (7+\sqrt{33})/2\). The results of our numerical calculations of \(n_s\) and

*r*are in Table 1, and the corresponding scalar potentials for the chosen values of \(\alpha \) are in Fig. 4.

The predictions for the inflationary parameters (\(n_s\), *r*), and the values of \(\varphi \) at the horizon crossing (\(\varphi _i\)) and at the end of inflation (\(\varphi _f\)), in the case \(3\le \alpha \le \alpha _*\) with both signs of \(\omega _1\). The \(\alpha \) parameter is taken to be integer, except of the upper limit \(\alpha _*\equiv (7+\sqrt{33})/2\)

\(\alpha \) | 3 | 4 | 5 | 6 | \(\alpha _*\) | ||
---|---|---|---|---|---|---|---|

\(\mathrm{sgn}(\omega _1)\) | − | \(+\) | − | \(+\)/− | \(+\) | − | − |

\(n_s\) | 0.9650 | 0.9649 | 0.9640 | 0.9639 | 0.9634 | 0.9637 | 0.9632 |

| 0.0035 | 0.0010 | 0.0013 | 0.0007 | 0.0005 | 0.0004 | 0.0003 |

\(-\kappa \varphi _i\) | 5.3529 | 3.5542 | 3.9899 | 3.2657 | 3.0215 | 2.7427 | 2.5674 |

\(-\kappa \varphi _f\) | 0.9402 | 0.7426 | 0.8067 | 0.7163 | 0.6935 | 0.6488 | 0.6276 |

The inflaton mass is \(m_\varphi \sim 10^{13}~\mathrm{GeV}\) irrespectively of the choice of \(\alpha \) and \(\mathrm{sgn}(\omega _1)\).

### 4.2 The case \(\alpha >\alpha _*\): hilltop inflation

The viable hilltop inflationary models are limited to \(\alpha _*< \alpha \le \alpha _\mathrm{max}\) with \(\alpha _*=(7+\sqrt{33})/2\approx 6.372\) and \(\alpha _\mathrm{max}\approx 7.235\). Let us consider \(\alpha =7\), because it is the only integer between \(\alpha _*\) and \(\alpha _\mathrm{max}\).

### 4.3 SUSY breaking scale

The masses of inflaton, axion and gravitino, and the VEVs of *F*- and *D*-fields derived from our models by fixing the amplitude \(A_s\) according to PLANCK data – see Eq. (57). The value of \(\langle F_T\rangle \) for a positive \(\omega _1\) is not fixed by \(A_s\)

The most important prediction of our models (apart from the existence of the upper limit \(\alpha _\mathrm{max}\)) for integer \(\alpha \) is the very high SUSY breaking scale parametrized by the superheavy gravitino mass \(m_{3/2}\) of the order of \(10^{12}\) to \(10^{13}\ \hbox {GeV}\). For fractional \(\alpha \), if \(\omega _1>0\), the SUSY breaking scale can be arbitrarily high as \(\alpha \) approaches 3 or \(\alpha _*\).

## 5 Conclusion

We showed that, unless \(\alpha \ge 3\), the scalar potential is unstable. The choice \(\alpha =3\) leads to the Starobinsky potential for the dilaton \(\varphi \), while the axion direction is flat, i.e. the axion mass has to be generated by quantum corrections. On the other hand, for \(\alpha >3\) the axion has a positive non-vanishing mass squared and is automatically stabilized. Once the axion acquires a VEV, those models lead to the effective single-field inflation where inflaton is identified with dilaton. We found that the shape of the potential, and thus the inflationary observables \(n_s\) and *r*, are controlled by \(\alpha \) and the sign of the real parameter \(\omega _1\equiv \overline{\lambda }\mu +\lambda \overline{\mu }\), whereas the amplitude of scalar perturbations is related to the value of the composite parameter \(\Lambda ^6=|\mu |^{2(\alpha -1)}/ |\omega _1|^{\alpha -2}\). In particular, when \(3\le \alpha \le \alpha _*\) (\(\alpha _*\approx 6.372\)), the derived inflation is of the Starobinsky type where the inflaton rolls down an infinite plateau, while for \(\alpha >\alpha _*\) the potential has a local maximum (hilltop).

One of our main results is the upper limit on \(\alpha \): by analyzing the dependence of \(n_s\) on \(\alpha \) (Fig. 3), we found that \(\alpha _\mathrm{max}\approx 7.235\) is the maximum value that can reproduce the observed spectral tilt \(n_s=0.9649\pm 0.0042\). More precise observations of \(n_s\) may further reduce the value of \(\alpha _\mathrm{max}\).

Another important prediction of our models is the (very) high-scale SUSY breaking, so that for integer \(\alpha \) the gravitino mass is roughly of the order of the inflaton mass, \(m_{3/2}\sim m_\varphi \sim 10^{13}\ \hbox {GeV}\) (for fractional \(\alpha \), \(m_{3/2}\) can be arbitrarily high). In comparison, the scale of the D-term is \(\sqrt{|\langle D\rangle |}=\kappa ^{-1}\sqrt{g|\xi |}\sim 10^{15.5}\ \hbox {GeV}\). We explicitly derived the masses of dilaton, axion and gravitino, together with the SUSY breaking parameters \(\langle F_T\rangle \) and \(\langle D\rangle \) for \(\alpha =3,4,5,6,7\) (see Table 2). It is interesting that the models with a negative \(\omega _1\) have the vanishing F-terms \(\langle F_T\rangle =0\) (except for \(\alpha =3\)), so that SUSY is broken purely by the D-term. Those models may be interesting in connection to the universality of scalar masses in the Supersymmetric Standard Model due to the vanishing F-terms, see e.g., Refs. [45, 46], though more research is needed in this direction. The axions and gravitinos in our models can be used as the superheavy dark matter along the lines of Refs. [11, 47, 48, 49, 50, 51].

## Footnotes

- 1.
- 2.
The model defined by Eqs. (5) and (6) in the presence of the alternative FI term and the linear gauge kinetic function \(f(g)=T\) leads to the

*vanishing*scalar potential [see Eqs. (2) and (8)] when \(\alpha =3\) and \(\omega _1=0\) [43], which is the defining property of the “no-scale” supergravity. - 3.
\(\alpha =\frac{1}{2}(7+\sqrt{33})\) is one of the two roots of the polynomial \(\alpha ^2-7\alpha +4\) that yields \(\gamma _+=x_+=0\). Another root is \(\alpha =\frac{1}{2} (7-\sqrt{33})<3\) so that it is excluded from the analysis.

- 4.
The point \(\alpha =(5+\sqrt{33})/2\approx 5.37\) can be found by solving \(\Delta _+=1\) that yields a quadratic equation for \(\alpha \), while the points \(\alpha \approx 3.8\) and \(\alpha \approx 5.27\) are found numerically by solving a quartic equation coming from \(\Gamma _+=1\).

- 5.
In fact, the \(n_s\) decreases quite rapidly after \(\alpha _\mathrm{max}\). For example, already for \(\alpha =10\) we have \(n_s\approx 0.3614\).

- 6.
A derivation of this action from curved superspace can be found e.g., in Ref. [55].

## Notes

### Acknowledgements

Y.A. and A.C. were supported by the CUniverse research promotion project of Chulalongkorn University under the grant reference CUAASC. Y.A. was also supported by the Ministry of Education and Science of the Republic of Kazakhstan under the grant reference BR05236322. S.V.K. was supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan, and the Competitiveness Enhancement Program of Tomsk Polytechnic University in Russia.

## References

- 1.H.P. Nilles, Supersymmetry, supergravity and particle physics. Phys. Rep.
**110**, 1–162 (1984)ADSCrossRefGoogle Scholar - 2.I.J.R. Aitchison,
*Supersymmetry in Particle Physics. An Elementary Introduction*(Cambridge University Press, Cambridge, 2007)CrossRefGoogle Scholar - 3.S.V. Ketov, Supergravity and early universe: the meeting point of cosmology and high-energy physics. Int. J. Mod. Phys. A
**28**, 1330021 (2013). arXiv:1201.2239 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 4.S.V. Ketov, MYu. Khlopov, Cosmological probes of supersymmetric field theory models at superhigh energy scales. Symmetry
**11**(4), 511 (2019)CrossRefGoogle Scholar - 5.F. Farakos, A. Kehagias, A. Riotto, On the Starobinsky model of inflation from supergravity. Nucl. Phys. B
**876**, 187–200 (2013). arXiv:1307.1137 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 6.S. Ferrara, R. Kallosh, A. Linde, M. Porrati, Higher order corrections in minimal supergravity models of inflation. JCAP
**1311**, 046 (2013). arXiv:1309.1085 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 7.S.V. Ketov, T. Terada, Generic scalar potentials for inflation in supergravity with a single chiral superfield. JHEP
**12**, 062 (2014). arXiv:1408.6524 [hep-th]ADSCrossRefGoogle Scholar - 8.S.V. Ketov, T. Terada, Inflation in supergravity with a single chiral superfield. Phys. Lett. B
**736**, 272–277 (2014). arXiv:1406.0252 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 9.Y. Aldabergenov, S.V. Ketov, SUSY breaking after inflation in supergravity with inflaton in a massive vector supermultiplet. Phys. Lett. B
**761**, 115–118 (2016). arXiv:1607.05366 [hep-th]ADSCrossRefGoogle Scholar - 10.Y. Aldabergenov, S.V. Ketov, Higgs mechanism and cosmological constant in \(N=1\) supergravity with inflaton in a vector multiplet. Eur. Phys. J. C
**77**(4), 233 (2017). arXiv:1701.08240 [hep-th]ADSCrossRefGoogle Scholar - 11.A. Addazi, S.V. Ketov, MYu. Khlopov, Gravitino and Polonyi production in supergravity. Eur. Phys. J. C
**78**(8), 642 (2018). arXiv:1708.05393 [hep-ph]ADSCrossRefGoogle Scholar - 12.Y. Aldabergenov, S.V. Ketov, Removing instability of inflation in Polonyi–Starobinsky supergravity by adding FI term. Mod. Phys. Lett. A
**91**(05), 1850032 (2018). arXiv:1711.06789 [hep-th]MathSciNetCrossRefGoogle Scholar - 13.Y. Aldabergenov, S.V. Ketov, R. Knoops, General couplings of a vector multiplet in \(N=1\) supergravity with new FI terms. Phys. Lett. B
**785**, 284–287 (2018). arXiv:1806.04290 [hep-th]ADSCrossRefGoogle Scholar - 14.N. Cribiori, F. Farakos, M. Tournoy, A. van Proeyen, Fayet–Iliopoulos terms in supergravity without gauged R-symmetry. JHEP
**04**, 032 (2018). arXiv:1712.08601 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 15.S.M. Kuzenko, Taking a vector supermultiplet apart: alternative Fayet–Iliopoulos-type terms. Phys. Lett. B
**781**, 723–727 (2018). arXiv:1801.04794 [hep-th]ADSCrossRefGoogle Scholar - 16.H. Abe, Y. Aldabergenov, S. Aoki, S.V. Ketov, Massive vector multiplet with Dirac–Born–Infeld and new Fayet–Iliopoulos terms in supergravity. JHEP
**09**, 094 (2018). arXiv:1808.00669 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 17.H. Abe, Y. Aldabergenov, S. Aoki, S.V. Ketov, Polonyi–Starobinsky supergravity with inflaton in a massive vector multiplet with DBI and FI terms. Class. Quant. Gravit.
**36**(7), 075012 (2019). arXiv:1812.01297 [hep-th]ADSCrossRefGoogle Scholar - 18.J. Polonyi, Generalization of the massive scalar multiplet coupling to the supergravity. Hungary Central Inst. Res. KFKI-77-93 (1977, REC. JUL 1978), p. 5 (unpublished)Google Scholar
- 19.J.R. Ellis, C. Kounnas, D.V. Nanopoulos, Phenomenological SU(1,1) supergravity. Nucl. Phys. B
**241**, 406–428 (1984)ADSCrossRefGoogle Scholar - 20.E. Cremmer, S. Ferrara, C. Kounnas, D.V. Nanopoulos, Naturally vanishing cosmological constant in \(\text{ N }=1\) supergravity. Phys. Lett.
**133B**, 61 (1983)ADSMathSciNetCrossRefGoogle Scholar - 21.J.R. Ellis, A.B. Lahanas, D.V. Nanopoulos, K. Tamvakis, No-scale supersymmetric standard model. Phys. Lett.
**134B**, 429 (1984)ADSCrossRefGoogle Scholar - 22.J.R. Ellis, C. Kounnas, D.V. Nanopoulos, No scale supersymmetric guts. Nucl. Phys. B
**247**, 373–395 (1984)ADSCrossRefGoogle Scholar - 23.S. Cecotti, Higher derivative supergravity is equivalent to standard supergravity coupled to matter. 1. Phys. Lett. B
**190**, 86–92 (1987)ADSMathSciNetCrossRefGoogle Scholar - 24.S.J. Gates Jr., S.V. Ketov, Superstring-inspired supergravity as the universal source of inflation and quintessence. Phys. Lett. B
**674**, 59–63 (2009). arXiv:0901.2467 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 25.S. Ketov, \(F(R)\) supergravity. AIP Conf. Proc.
**1241**(1), 613–619 (2010). arXiv:0910.1165 [hep-th]ADSCrossRefGoogle Scholar - 26.A.A. Starobinsky, A new type of isotropic cosmological models without singularity. Phys. Lett. B
**91**, 99–102 (1980). [771(1980)]ADSCrossRefGoogle Scholar - 27.J. Ellis, M.A.G. Garcia, D.V. Nanopoulos, K.A. Olive, Phenomenological aspects of no-scale inflation models. JCAP
**1510**(10), 003 (2015). arXiv:1503.08867 [hep-ph]ADSMathSciNetCrossRefGoogle Scholar - 28.J. Ellis, M.A.G. Garcia, D.V. Nanopoulos, K.A. Olive, No-scale inflation. Class. Quant. Gravit.
**33**(9), 094001 (2016). arXiv:1507.02308 [hep-ph]ADSMathSciNetCrossRefGoogle Scholar - 29.J. Ellis, D.V. Nanopoulos, K.A. Olive, From \(R^2\) gravity to no-scale supergravity. Phys. Rev. D
**97**(4), 043530 (2018). arXiv:1711.11051 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 30.J. Ellis, B. Nagaraj, D.V. Nanopoulos, K.A. Olive, De Sitter vacua in no-scale supergravity. JHEP
**11**, 110 (2018). arXiv:1809.10114 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 31.R. Kallosh, A. Linde, D. Roest, Superconformal inflationary \(\alpha \)-attractors. JHEP
**11**, 198 (2013). arXiv:1311.0472 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 32.D. Roest, M. Scalisi, Cosmological attractors from \(\alpha \)-scale supergravity. Phys. Rev. D
**92**, 043525 (2015). arXiv:1503.07909 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 33.A. Linde, Single-field \(\alpha \)-attractors. JCAP
**1505**, 003 (2015). arXiv:1504.00663 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 34.J. Ellis, D.V. Nanopoulos, K.A. Olive, S. Verner, A general classification of Starobinsky-like inflationary avatars of SU(2,1)/SU(2) \(\times \) U(1) no-scale supergravity. JHEP
**03**, 099 (2019). arXiv:1812.02192 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 35.J. Ellis, D.V. Nanopoulos, K.A. Olive, S. Verner, Unified No-Scale Attractors. arXiv:1906.10176 [hep-th]
- 36.S. Ferrara, R. Kallosh, A. Linde, M. Porrati, Minimal supergravity models of inflation. Phys. Rev. D
**88**(8), 085038 (2013). arXiv:1307.7696 [hep-th]ADSCrossRefGoogle Scholar - 37.I. Antoniadis, A. Chatrabhuti, H. Isono, R. Knoops, Inflation from supergravity with gauged R-symmetry in de Sitter vacuum. Eur. Phys. J. C
**76**(12), 680 (2016). arXiv:1608.02121 [hep-ph]ADSCrossRefGoogle Scholar - 38.F. Farakos, A. Kehagias, A. Riotto, Liberated \( \cal{N} = 1\) supergravity. JHEP
**06**, 011 (2018). arXiv:1805.01877 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 39.I. Antoniadis, A. Chatrabhuti, H. Isono, R. Knoops, The cosmological constant in supergravity. Eur. Phys. J. C
**78**(9), 718 (2018). arXiv:1805.00852 [hep-th]ADSCrossRefGoogle Scholar - 40.N. Cribiori, F. Farakos, M. Tournoy, Supersymmetric Born–Infeld actions and new Fayet–Iliopoulos terms. JHEP
**03**, 050 (2019). arXiv:1811.08424 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 41.I. Antoniadis, J.-P. Derendinger, F. Farakos, G. Tartaglino-Mazzucchelli, New Fayet–Iliopoulos terms in \(\cal{N}=2\) supergravity. arXiv:1905.09125 [hep-th]
- 42.I. Antoniadis, A. Chatrabhuti, H. Isono, R. Knoops, Fayet–Iliopoulos terms in supergravity and D-term inflation. Eur. Phys. J. C
**78**(5), 366 (2018). arXiv:1803.03817 [hep-th]ADSCrossRefGoogle Scholar - 43.Y. Aldabergenov, No-scale supergravity with new Fayet–Iliopoulos term. Phys. Lett. B
**795**, 366–370 (2019). arXiv:1903.11829 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 44.Planck Collaboration, Y. Akrami et al., Planck 2018 Results. X. Constraints on Inflation. arXiv:1807.06211 [astro-ph.CO]
- 45.G.R. Dvali, A. Pomarol, Supersymmetry breaking with vanishing F terms in supergravity theories. Phys. Lett. B
**410**, 160–166 (1997). arXiv:hep-ph/9706429 [hep-ph]ADSMathSciNetCrossRefGoogle Scholar - 46.E. Dudas, S.K. Vempati, Large D-terms, hierarchical soft spectra and moduli stabilisation. Nucl. Phys. B
**727**, 139–162 (2005). arXiv:hep-th/0506172 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 47.A. Addazi, A. Marciano, S.V. Ketov, MYu. Khlopov, Physics of superheavy dark matter in supergravity. Int. J. Mod. Phys. D
**27**(06), 1841011 (2018)ADSMathSciNetCrossRefGoogle Scholar - 48.V. Kuzmin, I. Tkachev, Matter creation via vacuum fluctuations in the early universe and observed ultrahigh-energy cosmic ray events. Phys. Rev. D
**59**, 123006 (1999). arXiv:hep-ph/9809547 [hep-ph]ADSCrossRefGoogle Scholar - 49.D.J.H. Chung, E.W. Kolb, A. Riotto, Nonthermal supermassive dark matter. Phys. Rev. Lett.
**81**, 4048–4051 (1998). arXiv:hep-ph/9805473 [hep-ph]ADSCrossRefGoogle Scholar - 50.D.J.H. Chung, E.W. Kolb, A. Riotto, Superheavy dark matter. Phys. Rev. D
**59**, 023501 (1999). arXiv:hep-ph/9802238 [hep-ph]ADSCrossRefGoogle Scholar - 51.D.J.H. Chung, E.W. Kolb, A. Riotto, Production of massive particles during reheating. Phys. Rev. D
**60**, 063504 (1999). arXiv:hep-ph/9809453 [hep-ph]ADSCrossRefGoogle Scholar - 52.M.J. Duff, S. Ferrara, Generalized mirror symmetry and trace anomalies. Class. Quant. Gravit.
**28**, 065005 (2011). arXiv:1009.4439 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 53.M.J. Duff, S. Ferrara, Four curious supergravities. Phys. Rev. D
**83**, 046007 (2011). arXiv:1010.3173 [hep-th]ADSCrossRefGoogle Scholar - 54.S. Ferrara, R. Kallosh, Seven-disk manifold, \(\alpha \)-attractors, and \(B\) modes. Phys. Rev. D
**94**(12), 126015 (2016). arXiv:1610.04163 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 55.J. Wess, J. Bagger,
*Supersymmetry and supergravity*(Princeton University Press, Princeton, 1992)zbMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}