Challenges in Particle Physics and Cosmology

Abstract

We present the proposal that the superpartner of the goldstino (sgoldstino),which is the source of the supersymmetry breaking, acts as the inflaton. The so-called eta problem can be avoided by imposing a linear superpotential. The inflaton is charged under a gauged $U(1)_R$—symmetry. This creates an interesting class of small field inflation models with an inflationary plateau around the maximum of the scalar potential near the origin, where R-symmetry is restored as the inflaton rolls down to a minimum describing the current phase of the universe. The minimum has a positive tuneable vacuum energy, whereas the inflation can be caused by either an F- or a D-term. The models are consistent with cosmological evidence and anticipate a relatively low tensor-to-scalar ratio of primordial perturbations in the simplest scenario. We explored the inflaton’s decay modes after coupling it to the (supersymmetric) Standard Model, with the resulting reheating temperature being roughly $10^8$ GeV.

1 Introduction

Almost all research in theoretical high energy physics of the last thirty years has been concentrated on the quest for the theory that will replace the Standard Model (SM) as the proper description of nature at the energies beyond the TeV scale, despite its success on providing some of the most striking agreements between experimental data and theoretical predictions. Actually there is a variety of such arguments, both theoretical and experimental, that lead to the undoubtable conclusion that the Standard Model should be only an effective theory of a more fundamental one. We briefly mention some of the most important ones. Firstly, the Standard Model does not include gravity. It says absolutely nothing about one of the four fundamental forces of nature. Another is the popular “mass hierarchy” problem. Within the SM, the mass of the Higgs particle is extremely sensitive to any new physics at higher energies and its natural value is of order of the Planck mass, if the SM is valid up to that scale. This is several orders of magnitude higher than the electroweak scale implied by experiment. The way to cure this discrepancy within the SM requires an incredible fine tuning of parameters. Also, the SM does not explain why the charges of elementary particles are quantised. Other equally important reasons include that the SM does not describe the dark matter or the dark energy of the universe. It does not explain the observed neutrino masses and oscillations and does not predict any gauge coupling unification, suggested by experiments.

A possible direction beyond the SM of particle physics is based on postulating a new symmetry that exchanges particles with different spin and in particular bosons with fermions. Such a symmetry, called supersymmetry, offers a solution to the hierarchy problem predicting a light Higgs boson of a mass less than about 130 GeV, compatible with the experimental value of 125 GeV measured at the Large Hadron Collider (LHC) of CERN. Moreover, supersymmetry offers an excellent candidate for dark matter, the lightest supersymmetric particle. Another attractive feature is the gauge coupling unification a property that indicates that the theory might stay perturbative up to the Grand Unification scale. On the conceptual level, it provides a framework where elementary scalars can be naturally included and treated on the same footing as fermions. It is also an important ingredient of string theory which is a candidate for describing all forces of Nature, including gravity, in a consistent quantum framework. If supersymmetry is true, there will be a rich spectrum of new particles to search, although their masses related to its breaking scale is an unknown parameter.

On the other hand, observations far in the sky of the early universe probe high energy physics, providing complementary information on elementary particles and their fundamental interactions. In the recent years, several observational data have raised inflation to the level of a cornerstone of modern Cosmology, based on its own standard model, the so-called $\Lambda$CDM describing very accurately astrophysical and cosmological data, but requiring a composition of the Universe in terms of dark energy and dark matter, ingredients that are absent in the SM of particle physics. Moreover, many important questions, including the origin of inflation, the nature of the inflaton field and its associated symmetries protecting its mass, as well as the issue of initial conditions remain in need of an answer. Usually, inflation is described by a phenomenological model of a scalar field that drives the universe evolution at early times, due to an approximate flat region of the scalar potential corresponding to a constant positive energy and leading to an exponential expansion of at least 50–60 e-foldings. The value of the inflation scale is an unknown parameter related to the ratio of the tensor-to-scalar primordial fluctuations, while it is not clear whether the inflaton field corresponds to a fundamental (like the Higgs boson), composite or effective degree of freedom.

Theoretically, a fundamental theory of Nature like string theory should be able to explain both particle physics and cosmology at once. These phenomena include scales that range from the tiniest microscopic four-dimensional quantum gravity length $\sim 10^{-33}$ cm to the largest macroscopic lengths of the size of the visible Universe $\sim 10^{28}$ cm, which span a region of nearly 60 orders of magnitude. In addition to the known 4d Planck mass, the electroweak scale of the SM, and the dark energy scale, there are two unknown hypothetical scales related to the sypersymmetry breaking and the inflation energy described above. A first step towards the understanding of the physics origin of the above scales is to study possible connections between them. In particular, one can try to connect the two unknown scales, while imposing an additional constraint on the present (electroweak) vacuum corresponding to the presence of a tiny tuneable cosmological constant in order to accommodate the observed dark energy, without necessarily trying to explain it.

Although there hasn’t been any proof of low energy supersymmetry at the LHC at CERN, a sizable portion of the theoretical community believes supersymmetry should be involved at some very fundamental level. However, inflationary models in supergravity suffer in general from several problems, such as fine-tuning to satisfy the slow-roll conditions, large field initial conditions that break the validity of the effective field theory and stabilisation of the (pseudo) scalar companion of the inflaton arising from the fact that the number of bosonic components of superfields are always even. A solution to all three problems was recently proposed in [1, 2] by identifying the inflaton with the superpartner of the Goldstone fermion of spontaneous supersymmetry breaking (called sgoldstino), providing a direct connect between supersymmetry and inflation.

The model has a gauged R-symmetry and generalises models of the so-called “minimal inflation”. The inflaton appears as a single scalar field in the low-energy spectrum after its pseudo-scalar partner is absorbed by the R-gauge field, which becomes massive. The superpotential is linear and the slow-roll conditions are automatically satisfied. Also, since inflation arises at a plateau around the maximum of the scalar potential (hill-top) no large field excursion is required, while initial conditions are imposed by symmetry around the point where R-symmetry is restored. Moreover, this model allows the presence of a realistic minimum describing our present Universe with an infinitesimal positive vacuum energy arising due to a cancellation between F- and D-term contributions to the scalar potential. This proposal was studied using an effective field theory approach, in perturbation around the origin of the inflaton field potential where R-symmetry is restored. Both cases have been analysed in detail, corresponding to inflation dominated by F-term or D-term supersymmetry breaking. The second case is possible only in the presence of a new Fayet-Iliopoulos (FI) term constructed recently [3, 4].

The Outline of this chapter is the following. In Sect. 2, we briefly review the framework of inflation by supersymmetry breaking and explain the coupling of the supersymmetry breaking sector to the supersymmetric extension of the Standard Model (MSSM). We then estimate the reheating temperature in Sect. 3. In Sect. 4, we review the new FI term and analyse its consequences in models of inflation driven by supersymmetry breaking.

2 Inflation by Supersymmetry Breaking

Starting with a class of models with gauged $U(1)_R$ phase symmetry given by the Kähler potential and superpotential,

$$\begin{aligned} \mathcal {K}(X,\overline{X},\phi ,\overline{\phi })=\sum \phi \overline{\phi }+J(X\overline{X})~,\end{aligned}$$

(1)

$$\begin{aligned} \mathcal {W}(X,\phi ) = \kappa [f\kappa ^{-3}+\Omega (\phi )] X~, \end{aligned}$$

(2)

where J is the Kähler potential for the inflaton/sgoldstino superfield X. $\phi$ collectively refer to matter superfields. f is a dimensionless real constant in the superpotential, and $\Omega$ describes to the MSSM part

$$\begin{aligned} \Omega =\hat{y}_u\bar{u}Q H_u-\hat{y}_d\bar{d}Q H_d-\hat{y}_e\bar{e}L H_d+\hat{\mu }H_uH_d~. \end{aligned}$$

(3)

Note that $\bar{u},\bar{d},\bar{e},Q,L,H_u,H_d$ are chiral superfields. The corresponding Standard Model matter fields (quarks, leptons and Higgs) are denoted with the same character, while tildes referred to their superpartners (squarks, sleptons and Higgsinos). The un-normalized Yukawa couplings y and the $\mu$-parameter are denoted by hats which will be removed after proper rescaling, once X settles at the minimum.

The total gauge group of the model is,

$$\begin{aligned} SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_R~. \end{aligned}$$

(4)

Squarks, sleptons, and Higgs scalars are neutral under $U(1)_R$, while X carries the same R-charge as the superpotential. The R-charges of the MSSM fermions are fixed as shown in Table 1.

Table 1 MSSM and $U(1)_R$ charges of the fermions

In this class of models, the X-dependent part of the potential drives inflation, after which X and its auxiliary field $F^X$ settle at non-zero vacuum expectation values (VEVs), spontaneously breaking both supersymmetry (SUSY) and $U(1)_R$. At the minimum of the potential, the gravitino mass and the the auxiliary fields of X and $U(1)_R$ are given by,

$$\begin{aligned} {\begin{matrix} m_{3/2} &{}=f\langle e^{\kappa ^2J/2}|X|\rangle ~,\\ \langle F^X\rangle &{}=-f\langle e^{\kappa ^2J/2}J^{X\bar{X}}(\kappa ^{-2}+J_XX)\rangle ~,\\ \langle \mathcal{D}_R\rangle &{}=g\langle \kappa ^{-2}+J_XX\rangle ~, \end{matrix}} \end{aligned}$$

(5)

where we assume that matter fields $\phi$ vanish at the minimum.

Due to the overall factor of $e^\mathcal {K}$ in the F-term potential, as well as the coupling of $\Omega$ to X as described in Eq. (2), the Yukawa couplings $\hat{y}$ and the parameter $\hat{\mu }$ in (3) are related to their properly normalized versions y and $\mu$ by

$$\begin{aligned} \{\hat{y},\hat{\mu }\}=\left\langle \frac{e^{-\kappa ^2J/2}}{\kappa |X|}\right\rangle \times \{y,\mu \}~. \end{aligned}$$

(6)

It should be noticed that X and J(X, X) take non-vanishing VEVs at the minimum, which makes this rescaling hold.

The scalar potential can be written as $\mathcal V= \mathcal V_F+\mathcal V_D$ where¹

$$\begin{aligned} \mathcal {V}_F=e^{\kappa ^2 \mathcal {K}}\left\{ \mathcal {K}^{I\bar{J}}D_I \mathcal {W} D_{\bar{J}}\overline{\mathcal {W}}-3\kappa ^2|\mathcal {W}|^2\right\} ~,\end{aligned}$$

(7)

$$\begin{aligned} \mathcal {V}_D=\frac{1}{2}\textrm{Re}(\mathcal {F}^{AB})\mathcal{{D}}_{A}\mathcal{{D}}_{B}~. \end{aligned}$$

(8)

In our notation, the indices I, J run through all the chiral (super)fields, while A, B are the gauge group indices. The relevant part of supergravity Lagrangian that we use here can be found in the Appendix of Ref. [5] and its derivation in Ref. [6].

For the gauge kinetic matrix, we use $\mathcal {F}^{AB}\equiv \mathcal {F}_{AB}^{-1}$. Kähler covariant derivatives are defined as $D_I \mathcal {W}\equiv \mathcal {W}_I+\kappa ^2 \mathcal {K}_I \mathcal {W}$, where the indices denote the respective partial derivatives. The Killing potential and Killing vector are related by

$$\begin{aligned} \mathcal{D}_{A}=ik_{A}^I\left( \mathcal {K}_I+\kappa ^{-2}\frac{\mathcal {W}_I}{\mathcal {W}}\right) ~, \end{aligned}$$

(9)

where the gauge couplings and charges are included in the Killing vectors $k^I_{A}$. For example, if X transforms under $U(1)_R$ as $X\rightarrow Xe^{-igq\vartheta }$ (where $\vartheta$ is a transformation parameter and q is its R-charge), its Killing vector is $k^X_R=-igq X$. The gauge couplings of $U(1)_R$, $U(1)_Y$, $SU(2)_L$, and $SU(3)_c$ are g, $g_1$, $g_2$, and $g_3$, respectively.

We use a convention where the superpotential transforms under $U(1)_R$ with unit R-charge, $\mathcal {W}\rightarrow \mathcal {W}e^{-ig\vartheta }$, and the fermionic superspace coordinate transforms with half-unit R-charge, $\theta \rightarrow \theta e^{-ig\vartheta /2}$. Then X has unit R-charge, while its fermionic partner has half-unit R-charge. For a scalar field with R-charge q, its fermionic partner has R-charge $q-1/2$. With this convention, the fermion charges under the total gauge group of our model are summaised in Table 1.

Let us focus on the possibility of inflation in this model by ignoring matter fields so that $\mathcal {K} = J$ and $\mathcal {W} = \kappa ^{-2} f X$. Here we would like to introduce a simpler choice of J with finite number of perturbative corrections, namely,

$$\begin{aligned} J=X\overline{X}+A \kappa ^2(X\overline{X})^2+ B \kappa ^4(X\overline{X})^3~, \end{aligned}$$

(10)

where A and B are dimensionless parameters. The above form can be viewed as a perturbative expansion around the canonical kinetic terms with coefficients less than one. The scalar potential then reads,

$$\begin{aligned} \mathcal {V}&=\frac{f^2}{\kappa ^4} e^{\left( |\kappa X|^2+A|\kappa X|^4+B|\kappa X|^6\right) }\left\{ \frac{(1+|\kappa X|^2+2A|\kappa X|^4+3B|\kappa X|^6)^2}{1+4A|\kappa X|^2+9B|\kappa X|^4}-3|\kappa X|^2\right\} \nonumber \\ &\quad +\frac{g^2}{2\kappa ^4}\left( 1+|\kappa X|^2+2\,A |\kappa X|^4+3 B |\kappa X|^6\right) ^2~, \end{aligned}$$

(11)

where we set gauge kinetic function $\mathcal {F} =1$ for now. Note that we can write

$$\begin{aligned} X = \rho e^{i \theta }. \end{aligned}$$

(12)

As a result, we may identify the field $\rho$ as the inflaton and conclude that the scalar potential is just a function of the modulus $\rho$:

$$\begin{aligned} \mathcal {V}&=\frac{f^2}{\kappa ^4} \exp \left( \kappa ^2 \rho ^2+A \kappa ^4\rho ^4+B\kappa ^6 \rho ^6\right) \nonumber \\ &\quad \left\{ \frac{(1+\kappa ^2\rho ^2+2A\kappa ^4\rho ^4+3B\kappa ^6\rho ^6)^2}{1+4A \kappa ^2\rho ^2+9B\kappa ^4\rho ^4}-3\kappa ^2\rho ^2\right\} \nonumber \\ &\quad +\frac{g^2}{2\kappa ^4}\left( 1+\kappa ^2\rho ^2+2\,A \kappa ^4\rho ^4+3 B \kappa ^6\rho ^6\right) ^2~. \end{aligned}$$

(13)

The $U(1)_R$ gauge field absorbs the phase $\theta$ by the standard Brout-Englert-Higgs mechanism. Nevertheless, we introduce the canonically normalized field $\chi$, which satisfies the following relation, in order to determine the slow-roll parameters:

$$\begin{aligned} \frac{d\chi }{d\rho } = \sqrt{2 \mathcal{K}_{X \bar{X}} }. \end{aligned}$$

(14)

Fig. 1

Scalar potential (11) for the parameter set (23). Both non-canonical ($\rho$) and canonical ($\chi$) parametrizations are shown, where the latter is found numerically. The markers represent the start and end of 60 e-folds of inflation (the starting point of inflation almost coincides for the two curves)

The slow-roll parameters can be defined in terms of the canonical field $\chi$ as:

$$\begin{aligned} \epsilon = \frac{1}{2\kappa ^2} \left( \frac{d\mathcal {V}/d\chi }{\mathcal {V}}\right) ^2, \quad \eta = \frac{1}{\kappa ^2} \frac{d^2\mathcal {V}/d\chi ^2}{\mathcal {V}}. \end{aligned}$$

(15)

The number of e-folds N during inflation is determined by

$$\begin{aligned} N = \kappa ^2\int \limits ^{\chi _\textrm{end}}_{\chi _{*}}\frac{\mathcal {V}}{\partial _{\chi }\mathcal {V}} d \chi = \kappa ^2\int \limits ^{\rho _\textrm{end}}_{\rho _{*}}\frac{\mathcal {V}}{\partial _{\rho }\mathcal {V}}\left( \frac{d\chi }{d\rho } \right) ^2\,d \rho , \end{aligned}$$

(16)

where we choose $|\eta (\chi _\textrm{end})| =1$ for this section. The scalar potential as a function of non-canonical ($\rho$) and canonical ($\chi$) parametrizations are shown in Fig. 1. Notice that our model is classified as hilltop inflation which may encounter the so-called overshoot problems. It was shown in [7] that initial condition for models of this type can be determined quantum mechanically such that the inflaton is driven toward the slow-roll attractor solution exponentially fast and the overshoot problem is partially evade.

Since inflation arises near the maximum $\kappa \rho =0$, we expand

$$\begin{aligned} \epsilon &= 4 \left( \frac{-4\,A + y^2}{2 + y^2} \right) ^2(\kappa \rho )^2 + \mathcal O(\rho ^4), \end{aligned}$$

(17)

$$\begin{aligned} \eta &= 2 \left( \frac{-4\,A + y^2}{2 + y^2} \right) + \mathcal O(\rho ^2) , \end{aligned}$$

(18)

where we defined $y = g/f$. The above equation implies $\epsilon \simeq \eta ^2(\kappa \rho )^2 \ll \eta$.

We concentrate on the special case $y \rightarrow 0$, when the F-term contribution to the scalar potential is dominating, in order to keep things simple. We may impose certain restrictions on the coefficient A of the quadratic term of the Kähler potential defined in (10) by taking into account the behavior near the origin. We can readily demonstrate that $A > 0$ is necessary for the scalar potential to have a local maximum at $\rho = 0$. Additionally, an upper bound $A \ll 0.25$ is established by the slow-roll condition $|\eta | \ll 1$. With these specifications, the restriction on A is

$$\begin{aligned} 0<A \ll 0.25. \end{aligned}$$

(19)

By choosing $A \sim 0.005$, we can obtain a theoretical prediction $\eta \sim -0.02$ that is consistent with the observational data from the CMB.² Note also that the amplitude of density fluctuations $A_s$, the spectral index $n_s$ and the tensor-to-scalar ratio r can all be expressed in terms of the slow-roll parameters:

$$\begin{aligned} \mathcal A_s =&\quad \frac{\kappa ^4 \mathcal {V}_*}{24 \pi ^2 \epsilon _*},\end{aligned}$$

(20)

$$\begin{aligned} n_s =&\quad 1+2\eta _* - 6\epsilon _* \simeq 1+ 2\eta _*,\end{aligned}$$

(21)

$$\begin{aligned} r =&\quad 16 \epsilon _*, \end{aligned}$$

(22)

all evaluated at the horizon exit $\rho _*$.

The following parameter settings are selected as a specific example

$$\begin{aligned} A=0.139, \quad B=0.6, \quad y=0.7371, \quad f=2.05\times {10^{-7}}, \end{aligned}$$

(23)

which leads to the following predictions for the CMB observational data

$$\begin{aligned} {\mathcal {A}}_s = 2.1\times {10^{-9}}, \quad n_s=0.9543, \quad r=1.72\times {10^{-6}}, \end{aligned}$$

(24)

for 60 e-fold. The Hubble parameter during inflation is

$$\begin{aligned} {H_\textrm{inf}= \kappa \sqrt{\mathcal {V}_*/3} = 3.25\times 10^{11}~\textrm{GeV}.} \end{aligned}$$

(25)

Figure 1 shows the scalar potential, with the canonically normalized scalar $\chi$, which was obtained numerically, represented in orange and the non-canonical scalar $\rho$ in blue. The corresponding inflaton VEV is $\langle \kappa \rho \rangle =0.89$.

3 Reheating After Inflation by Supersymmetry Breaking

In our model defined by (1) and (2) (with general J) soft scalar masses are universal,

$$\begin{aligned} m_Q^2=m_u^2=m_d^2=m_L^2=m_e^2=m_{H_u}^2=m_{H_d}^2=m^2_0~, \end{aligned}$$

(26)

where $m_0^2$ is given by

$$\begin{aligned} m_0^2=\kappa ^2\langle J_{X\bar{X}}F^X\overline{F}^X\rangle -2m_{3/2}^2~. \end{aligned}$$

(27)

Additionally, for the MSSM $\mu$-parameter, in order to avoid extremely fine-tuning the Higgs boson mass, we assume $|\mu | \ll |m_0|$ (since $m_0$ is close to the inflationary scale). From the Minkowski minimum’s condition, we derive the relation,

$$\begin{aligned} \langle V\rangle =\langle J_{X\bar{X}}F^X\overline{F}^X\rangle -3\kappa ^{-2}m_{3/2}^2+\tfrac{1}{2}\langle \mathcal{D}_R\rangle ^2=0~. \end{aligned}$$

(28)

The relation (28) allows us to rewrite $m_0^2$ in terms of the D-term contribution,

$$\begin{aligned} m_0^2=m_{3/2}^2-\tfrac{\kappa ^2}{2}\langle \mathcal{D}_R\rangle ^2~, \end{aligned}$$

(29)

and this leads to the requirement $m_{3/2}>\kappa \langle \mathcal{D}_R\rangle /\sqrt{2}$, in order to avoid tachyonic instabilities in the MSSM sector.

For the bilinear $H_u H_d$ coupling we have

$$\begin{aligned} e^{-1}\mathcal{L}\supset -B_0\mu H_uH_d+\mathrm{h.c.}~, \end{aligned}$$

(30)

where

$$\begin{aligned} B_0=\frac{\kappa ^2\langle J_{X\bar{X}}F^X\overline{F}^X\rangle -m^2_{3/2}}{m_{3/2}}~. \end{aligned}$$

(31)

The Green-Schwarz mechanism of anomaly cancellation, which generates the MSSM gaugino masses at one loop, involves the appropriate $U(1)_R$ transformations of the following terms depending on the imaginary part of the gauge kinetic matrix to cancel the gauge anomalies resulting from triangle diagrams involving the fermions (all fermions in the model carry non-zero R-charges):

$$\begin{aligned} e^{-1}\mathcal{L}\supset \tfrac{1}{8}\textrm{Im}(\mathcal {F}_{AB})\epsilon ^{mnkl}F^{A}_{mn}F^{B}_{kl}~. \end{aligned}$$

(32)

The gauge kinetic matrix takes the form,

$$\begin{aligned} \mathcal {F}_{AB}= \begin{pmatrix} \mathcal {F}_R &{} &{} &{} \\ &{} \mathcal {F}_1 &{} &{} \\ &{} &{} \mathcal {F}_2 &{} \\ &{} &{} &{} \mathcal {F}_3 \end{pmatrix}~, \end{aligned}$$

(33)

where $\mathcal {F}_{R,1,2,3}$ are gauge kinetic functions for $U(1)_R$, $U(1)_Y$, $SU(2)_L$, and $SU(3)_c$, respectively. To cancel the anomalies we fix these kinetic functions as,

$$\begin{aligned} \mathcal {F}_R &=1+\beta _R\log ({\kappa \rho })~,\end{aligned}$$

(34)

$$\begin{aligned} \mathcal {F}_{a} &=1+\beta _{a}\log ({\kappa \rho })~, \end{aligned}$$

(35)

where $a = 1, 2, 3$ represents the gauge groups from the Standard Model. Here, $\beta$ are constants that we calculate using the techniques outlined in Refs. [10,11,12]. As a result,

$$\begin{aligned} \beta _R=-\frac{g^2}{3\pi ^2}~,~~~\beta _1=-\frac{11g_1^2}{8\pi ^2}~,~~~\beta _2=-\frac{5g_2^2}{8\pi ^2}~,~~~\beta _3=-\frac{3g_3^2}{8\pi ^2}~, \end{aligned}$$

(36)

where $\beta _R$ is found from the cancellation of $U(1)_R^3$ anomaly, $\beta _1$ from $U(1)_R\times U(1)_Y^2$ anomaly, $\beta _2$ from $U(1)_R\times [SU(2)_L]^2$ anomaly, and $\beta _3$ from $U(1)_R\times [SU(3)_c]^2$ anomaly.

The values of $\beta _{a}$ are the same as in the model of Ref. [13], because the MSSM fermions in the two models have the same R-charges, while $\beta _R$ is different due to the difference in the hidden sector fermion (inflatino) R-charges. Since $g/\kappa$ in our models is not far from the Hubble scale (e.g. the parameter choice (23) leads to $g\sim 10^{-7}$), we have

$$\begin{aligned} \mathcal {F}_R=1+\beta _R\log (\kappa \rho )\approx 1~, \end{aligned}$$

(37)

if $\kappa \rho$ is around unity. Gauged $U(1)_R$ also leads to a gravitational anomaly which can be cancelled in a similar fashion (see for example Refs. [10,11,12]).

We now come to the MSSM gaugino masses,

$$\begin{aligned} m_{ab}=\frac{1}{2}\left| \langle F^X\partial _X \mathcal {F}_{ab}\rangle \right| =\frac{f}{2}\left| \left\langle e^{\kappa ^2J/2}J^{X\bar{X}}(\kappa ^{-2}+J_{\bar{X}}\overline{X})\partial _X \mathcal {F}_{ab} \right\rangle \right| ~. \end{aligned}$$

(38)

Using Eq. (35) we get,

$$\begin{aligned} m_{a}=\left| \frac{\langle \kappa F^X\rangle \beta _{a}}{2\langle \kappa X\rangle }\right| ~, \end{aligned}$$

(39)

where we denote $m_{a}\equiv m_{aa}$. Finally, $m_{a}$ should be rescaled after taking into account non-canonical kinetic terms of the gaugini,

$$\begin{aligned} &e^{-1}\mathcal{L}\supset -\tfrac{i}{2}\langle \textrm{Re} \mathcal {F}_{a}\rangle \lambda ^{a}\sigma ^mD_m\bar{\lambda }^{a}+\mathrm{h.c.} \nonumber \\ &\quad =-\tfrac{i}{2}(1+\beta _{a}\log \langle \kappa X\rangle )\lambda ^{a}\sigma ^mD_m\bar{\lambda }^{a}+\mathrm{h.c.} \end{aligned}$$

(40)

However, if $|\beta _a|\log \langle \kappa X\rangle \ll 1$, as in the models that we consider here, the rescaling of the gaugini can be neglected. The trilinear couplings between the MSSM scalars are

$$\begin{aligned} &e^{-1}\mathcal{L}\supset -A_0(y_u\bar{\tilde{u}}\tilde{Q}H_u-y_d\bar{\tilde{d}}\tilde{Q}H_d-y_e\bar{\tilde{e}}\tilde{L}H_d) \nonumber \\ &\quad -\mu (y_u\bar{\tilde{u}}\tilde{Q}\overline{H}_d-y_d\bar{\tilde{d}}\tilde{Q}\overline{H}_u-y_e\bar{\tilde{e}}\tilde{L}\overline{H}_u)+\mathrm{h.c.}~, \end{aligned}$$

(41)

where for $A_0$ we have

$$\begin{aligned} A_0=\frac{\kappa ^2\langle J_{X\bar{X}}F^X\overline{F}^X\rangle }{m_{3/2}}~, \end{aligned}$$

(42)

which is related to $B_0$ from Eq. (31) as $A_0=B_0+m_{3/2}$. Here we provide explicit values of the MSSM soft parameters for the parameter set (23), as well as the mass spectrum of the model. The results are summarized in Table 2, where we take one-loop values of the Standard Model gauge couplings³ at the reheating temperature $10^{8}$ GeV (estimated below),

$$\begin{aligned} g_1=0.5~,~~~g_2=0.59~,~~~g_3=0.72~. \end{aligned}$$

(43)

As for the $U(1)_R$ gauge boson, its mass generated by the Higgs mechanism is $9.61\times 10^{11}$ GeV, close to the inflaton mass.

The parameters $A_0$ and $B_0$ are estimated as

$$\begin{aligned} A_0=1.6\times 10^{12}~\textrm{GeV}~,~~~B_0=8.46\times 10^{11}~\textrm{GeV}~. \end{aligned}$$

(44)

Table 2 Masses (in GeV) of inflaton, inflatino, gravitino, and MSSM sparticles derived from our model with parameter set (23)

For the parameters set in (23), the masses of MSSM scalars, gaugini, and inflatino are less than half the masses of inflaton ($< m_{\rho }/2$), therefore inflaton $\rho$ can decay into one of them. However, the inflaton mass is less than double that of the gravitino mass, $m_\rho < 2m_{3/2}$, the former cannot perturbatively decay into two gravitini. As shown in [5], the total decay rate is

$$\begin{aligned} \Gamma _\textrm{tot} =6.54\times 10^{-3}~\textrm{GeV}~, \end{aligned}$$

(45)

and the estimated reheating temperature is

$$\begin{aligned} T_\textrm{reh}\simeq \sqrt{M_P \Gamma _\textrm{tot}}=1.26\times 10^{8}~\textrm{GeV}~. \end{aligned}$$

(46)

4 Inflation by Supersymetry Breaking with the New Fayet-Iliopoulos Term

In the previous sections, we discuss a class of minimal inflation models in supergravity that identify the inflaton with the goldstino superpartner in the presence of a gauged R-symmetry. We notice that the D-term has a constant Fayet-Iliopoulos (FI) contribution but plays no role during the inflation and can be neglected, while the pseudoscalar partner of the inflaton is absorbed by the $U(1)_R$ gauge field that becomes massive away from the origin.

In this section, we present the formalism that allows to generalize D-terms in supergravity with the same bosonic component given by a Fayet-Iliopoulos (FI) term, proportional to a D-auxiliary field of an abelian vector multiplet with a general scalar field dependent coefficient. We then discuss their physical applications in constructing models of D-term inflation. In the following, we use the notation in [14] and $\kappa$ is set to 1 for simplicity.

4.1 Component Action of New FI Term in Superconformal Tensor Calculus

In 2017, a new type of the FI term in supergravity was proposed in [3] (see also in [4]) of the form $\mathcal {L}_{\text {FI}}=\xi _2\, D+$ fermions, that can be coupled to supergravity without gauging the R-symmetry. This new term contains the inverse powers of some auxiliary field. It is non-singular when the D-auxiliary field has a non-vanishing vacuum expectation value (VEV). The supergravity Lagrangian that gives the new FI term is given by

$$\begin{aligned} \mathcal {L}_{\text {FI}} &= \xi _2 \left[ S_0 \bar{S}_0 \frac{w^2 \bar{w}^2}{ \bar{T}(w^2) T ( \bar{w}^2) } (V)_D \right] _D , \end{aligned}$$

(47)

where $\xi _2$ is a constant parameter. In the superconformal formalism, the chiral compensator field $S_0$, with Weyl and chiral weights $(\delta ,w^\prime ) = (1,1)$, has components $S_0 = \left( s_0, P_L \Omega _0, F_0 \right)$. The vector multiplet $V = \left( v , \zeta , \mathcal {H}, v_\mu , \lambda , D \right)$ has weights (0, 0). We will use the Wess-Zumino gauge in which the first components $v = \zeta = \mathcal {H} = 0$. The multiplet $w^2$ with weights (1, 1) is given by

$$\begin{aligned} w^2 = \frac{ \bar{\lambda }P_L \lambda }{S_0^2}, \ \ \ \ \ \ \bar{w}^2 = \frac{ \lambda P_R \bar{\lambda }}{\bar{S}_0^2}, \end{aligned}$$

(48)

where we have (in the components form)

$$\begin{aligned} \bar{\lambda }P_L \lambda = \Big ( \bar{\lambda }P_L \lambda \ \ ; \ \ \sqrt{2} P_L \big ( - \frac{1}{2} \gamma \cdot \hat{F} + i D \big ) \lambda \ \ ; \ \ 2 \bar{\lambda } P_{L} {\not \!\!\mathcal{D}} \lambda + \hat{F}^- \cdot \hat{F}^- - D^2 \Big ) . \end{aligned}$$

(49)

The kinetic terms for the gauge multiplet in the supergravity Lagrangian are given by

$$\begin{aligned} \mathcal {L}_{\text {kin}} = - \frac{1}{4} \left[ \bar{\lambda }P_L \lambda \right] _F + \text {h.c. }. \end{aligned}$$

(50)

The operator T ($\bar{T}$) in (47) is defined in [15, 16], and can be used to define a chiral (antichiral) multiplet. For example, the chiral multiplet $T(\bar{w}^2)$ has weights (2, 2). This corresponds to the usual chiral projection operator $\bar{D}^2$ in the case of global supersymmetry. Note that we will drop the notation of $\text {h.c.}$ and implicitly assume its presence for every $[ \ \ ]_F$ term in the Lagrangian. Finally, the multiplet $(V)_D$ is a (2, 0) linear multiplet. Its components are given by

$$\begin{aligned} (V)_D = \left( D, {\not \!\!\mathcal{D}} \lambda , 0 , \mathcal{D}^b \hat{F}_{ab}, - {\not \!\!\mathcal{D}} {\not \!\!\mathcal{D}} \lambda , - \Box ^C {D} \right) . \end{aligned}$$

(51)

The component ${\not \!\!\mathcal{D}} \lambda$ and the covariant field strength $\hat{F}_{ab}$ are defined in Eq. (17.1) of [14]. In our case, we have

$$\begin{aligned} \hat{F}_{ab} &= e_a^{\ \mu } e_b^{\ \nu } \left( 2 \partial _{[\mu } A_{\nu ]} + \bar{\psi }_{[\mu } \gamma _{\nu ]} \lambda \right) ~, \nonumber \\ \mathcal{D}_\mu \lambda &= \left( \partial _\mu - \frac{3}{2} b_\mu + \frac{1}{4} w_\mu ^{ab} \gamma _{ab} - \frac{3}{2} i \gamma _* \mathcal {A}_\mu \right) \lambda -\left( \frac{1}{4} \gamma ^{ab} \hat{F}_{ab} + \frac{1}{2} i \gamma _* D \right) \psi _\mu , \end{aligned}$$

(52)

where $e_a^{\ \mu }$ is the vierbein, with frame indices a, b and coordinate indices $\mu , \nu$. The gauge fields $w_\mu ^{ab}$, $b_\mu$, and $\mathcal {A}_\mu$ correspond to Lorentz transformations, dilatations, and $T_R$ symmetry of the conformal algebra respectively, while $\psi _\mu$ denotes the gravitino. The conformal d’Alembertian operator is defined by $\Box ^C \equiv \eta ^{ab} \mathcal{D}_a \mathcal{D}_b$.

Let us consider first the case of pure supergravity coupled to a U(1) gauge multiplet with the FI term in (47). The supergravity Lagrangian can be written as

$$\begin{aligned} \mathcal {L} = -3 \left[ S_0 \bar{S}_0 \right] _D + \left[ S_0^3 W_0 \right] _F -\frac{1}{4} \left[ \bar{\lambda }P_L \lambda \right] _F + \mathcal {L}_{\text {FI}} . \end{aligned}$$

(53)

While D appears just linearly in the bosonic part, one can show that the fermionic part becomes singular if D vanishes. This implies that the supersymmetry must be broken (D-term supersymmetry breaking). The Goldstino is then the U(1) Gaugino. By setting $S_0=\bar{S}_0 = 1$, integrating out the auxiliary fields and choosing the unitary gauge in which the Goldstino is set to zero, the Lagarangian in component form is

$$\begin{aligned} e^{-1}\mathcal {L} &= \frac{1}{2}\left( R - \bar{\psi }_\mu \gamma ^{\mu \nu \rho }D_\nu \psi _\rho + m_{3/2}\bar{\psi }_\mu \gamma ^{\mu \nu }\psi _\nu \right) \nonumber \\ &\quad -\frac{1}{4}F^{\mu \nu }F_{\mu \nu } - \left( -3\,m^2_{3/2}+\frac{1}{2}\xi ^2_2\right) , \end{aligned}$$

(54)

with a constant superpotential $W_0 = m_{3/2}$. Any non-vanishing value of $\xi _2$ breaks supersymmetry and increases the vacuum energy by a constant term $\mathcal {V}_{FI} = \xi ^2_2/2$. Noting that the FI term in Eq. (47) does not necessitate the gauging of an R-symmetry, it is also important to remark that it breaks the Kähler invariance.

Let us now couple the FI-term given by Eq. (47) to additional matter fields charged under the U(1). For simplicity, we focus on a single chiral multiplet X. The Lagrangian is given by

$$\begin{aligned} \mathcal {L} = -3 \left[ S_0 \bar{S}_0 e^{- \frac{1}{3} \mathcal {K}(X ,\bar{X}) } \right] _D + \left[ S_0^3 \mathcal {W}(X) \right] _F -\frac{1}{4} \left[ \mathcal {F}(X) \bar{\lambda }P_L \lambda \right] _F + \mathcal {L}_{\text {FI}} . \end{aligned}$$

(55)

Here $\mathcal {K}(X, \bar{X})$, $\mathcal {W}(X)$ and $\mathcal {F}(X)$ are a Kähler potential, a superpotential and a gauge kinetic function respectively. In Eq. (55), the first three terms are the typical supergravity Lagrangian [14]. Let us assume that the multiplet X transforms via

$$\begin{aligned} V &\rightarrow V + i\Lambda -i \bar{\Lambda }, \nonumber \\ X &\rightarrow X e^{- iq \Lambda }, \end{aligned}$$

(56)

where $\Lambda$ stands for a gauge multiplet parameter. Since the superpotential does not transform in the case we take into consideration, the U(1) is not an R-symmetry. The superpotential for a model with a single chiral multiplet must be constant

$$\begin{aligned} \mathcal {W}(X) = F . \end{aligned}$$

(57)

The Kähler potential must depend on $X e^{qV} \bar{X}$ for the supergravity action to remain gauge invariant. However, we omit the $e^{qV}$ factors in the following for notational simplicity.

In fact, we can consistently include the FI-term $\mathcal {L}_{\text {FI}}$ into the theory in this scenario, similar to [3], and the resulting D-term potential gains an additional term proportional to $\xi _2$

$$\begin{aligned} \mathcal {V}_D &= \frac{1}{2} \text {Re}\left( \mathcal {F}(X) \right) ^{-1} \left( i k_X \partial _X \mathcal {K} + \xi _2 e^{\frac{1}{3} \mathcal {K}} \right) ^2 , \end{aligned}$$

(58)

where the Killing vector is $k_X = -iqX$. For a constant superpotential (57), the F-term potential reduces to

$$\begin{aligned} \mathcal {V}_F &= |F|^2 e^{\mathcal {K}(X, \bar{X})} \left( -3 + g^{X \bar{X}} \partial _X \mathcal {K} \partial _{\bar{X}} \mathcal {K} \right) . \end{aligned}$$

(59)

It is clear from Eq. (58) that the D-term contribution to the scalar potential acquires another constant contribution if the Kähler potential contains a term proportional to $\xi _1 \log (X \bar{X})$. For instance, the D-term potential becomes

$$\begin{aligned} \mathcal {V}_D &= \frac{1}{2} \text {Re}\left( \mathcal {F}(X) \right) ^{-1} \left( q X \bar{X} + q \xi _1 + \xi _2 e^{\frac{1}{3} \mathcal {K}} \right) ^2 , \end{aligned}$$

(60)

if we choose

$$\begin{aligned} \mathcal {K} (X , \bar{X}) = X \bar{X} + \xi _1 \ln (X \bar{X}). \end{aligned}$$

(61)

In a non-R-symmetric Kähler frame, the term proportional to $\xi _1$ is the normal FI term, and it can be consistently included in the model alongside the new FI term proportional to $\xi _2$. In the absence of the additional term, we may recast the model in the form

$$\begin{aligned} \mathcal {K}(X, \bar{X}) &= X \bar{X} , \nonumber \\ \mathcal {W}(X) &= m_{3/2} X , \end{aligned}$$

(62)

where $m_{3/2} = F$ by using the Kähler transformation

$$\begin{aligned} \mathcal {K}(X , \bar{X}) &\rightarrow \mathcal {K}(X ,\bar{X}) + \mathcal {J}(X) + \bar{\mathcal {J}}(\bar{X}), \nonumber \\ \mathcal {W}(X) &\rightarrow \mathcal {W}(X) e^{-\mathcal {J}(X)}, \end{aligned}$$

(63)

with $\mathcal {J}(X) = - \xi _1 \ln X$. At the classical level, the two models have the same Lagrangian.⁴ However, in the Kähler frame of Eqs. (62), the superpotential undergoes a nontrivial transformation due to the gauge symmetry. The gauge symmetry changes into an R-symmetry as a result.

Due to the additional term  (47) violating the theory’s Kähler invariance, the two models connected by a Kähler transformation are no longer comparable. The model written in the Kähler frame where the gauge symmetry becomes an R-symmetry in Eqs. (62) can not be consistently coupled to $\mathcal {L}_{\text {FI}}$. In  [17] and [19], a generalized Kähler invariant FI term has been developed.

4.2 The Scalar Potential in a Non R-Symmetry Frame

The two types of FI terms that were reviewed in the previous section are taken into consideration as we work in the Kähler frame in which the superpotential does not transform. We repeat the Kähler potential in Eq. (61) for convenience and reintroduce the inverse reduced Planck mass $\kappa$:

$$\begin{aligned} \mathcal {K} = X\bar{X} + \kappa ^{-2}\xi _1 \ln \kappa ^2 X\bar{X}. \end{aligned}$$

(64)

The superpotential and the gauge kinetic function are set to be constant⁵:

$$\begin{aligned} \mathcal {W} = \kappa ^{-3} F ,~~~~ \mathcal {F}(X) = 1. \end{aligned}$$

(65)

After performing a change of the field variable $X = \rho e^{i\theta }$, the scalar potential’s F-term contribution is provided by

$$\begin{aligned} \mathcal {V}_F = \frac{1}{\kappa ^4}F^2 e^{\kappa ^2\rho ^2}(\kappa \rho )^{2 b} \left[ \frac{\left( b+\kappa ^2\rho ^2\right) ^2}{(\kappa \rho )^2}-3\right] , \end{aligned}$$

(66)

and the D-term contribution is

$$\begin{aligned} \mathcal {V}_D = \frac{q^2}{2 \kappa ^4} \left( b+(\kappa \rho )^2+\xi (\kappa \rho )^{\frac{2 b}{3}}e^{\frac{1}{3} \kappa ^2\rho ^2}\right) ^2. \end{aligned}$$

(67)

Note that we set $b= \xi _1$ and rescaled the second FI parameter by $\xi = \xi _2/q$. We are now ready to study the role of the new FI term in inflation.

For $F=0$, one finds that for $\xi < -1$ and $b= 3$ the potential has a maximum at the origin, and a supersymmetric minimum. Since we set the superpotential to zero, the SUSY breaking is measured by the D-term order parameter, i.e. the Killing potential associated with the gauged $\textrm{U}(1)$, which is given by

$$\begin{aligned} \mathcal {D} &= i\kappa ^{-2}\frac{-iqX}{ W}\bigg (\frac{\partial W}{\partial X} +\kappa ^2 \frac{\partial \mathcal {K}}{\partial X} W \bigg )+ \kappa ^{-2} q \xi (\kappa \rho )^2 e^{(\kappa \rho )^2/3}. \end{aligned}$$

(68)

This enters the scalar potential as $\mathcal {V}_D=\mathcal {D}^2/2$. So, at the local maximum and during inflation $\mathcal {D}$ is of order q and supersymmetry is broken. On the other hand, supersymmetry is conserved and the potential vanishes at the global minimum. Since the D-auxiliary vanishes, the new FI term becomes singular, making the supersymetric minimum invalid. So in any case, a small F is needed.

For $F\ne 0$, the potential has still a local maximum at $\rho =0$ for $b=3$ and $\xi < -1$. For this choice, the derivatives of the potential have the following properties,

$$\begin{aligned} \mathcal {V}'(0) = 0, \quad \mathcal {V}''(0) = 6\kappa ^{-4} q^2(\xi +1). \end{aligned}$$

(69)

For $\xi < -1$, the extremum is indeed a local maximum as required.

After activating the F-term contribution, let us discuss about the global minimum. The change in the global minimum $\kappa \rho _v$ is relatively modest, of order $\mathcal O (F^2/q^2)$, as long as $F^2/q^2 \ll 1$. The plot of this change is shown in Fig. 2.

Fig. 2

This plot presents the scalar potentials for $F=0$ and $F\ne 0$ cases. For $F=0$, we have a local maximum at $\rho = 0$ and the global minimum has zero cosmological constant. For $F \ne 0$, the origin $\rho = 0$ is still the maximum but the global minimum now has a positive cosmological constant

A remark on super symmetry breaking in the case $F \ne 0$. The Killing potential $\mathcal {D}$ and the F-term contribution $\mathcal {F}_X$ are the order parameters, which read

$$\begin{aligned} \mathcal {D} &\propto q [3 + (\kappa \rho )^2 (1+ \xi e^{(\kappa \rho )^2/3})],\quad \mathcal {F}_X \propto F (\kappa \rho )^2 (3+(\kappa \rho )^2) e^{(\kappa \rho )^2/2}, \end{aligned}$$

(70)

where the F-term order parameter $\mathcal {F}_X$ is defined by

$$\begin{aligned} \mathcal {F}_X &= -\frac{1}{\sqrt{2}} e^{\kappa ^2 \mathcal {K}/2} \bigg (\frac{\partial ^2 \mathcal {K}}{\partial X\partial \bar{X}}\bigg )^{-1/2} \bigg (\frac{\partial \bar{ W}}{\partial \bar{X}}+\kappa ^2 \frac{\partial \mathcal {K}}{\partial \bar{X}}\bar{ W} \bigg ). \end{aligned}$$

(71)

Therefore, near the local maximum, $\mathcal {F}_X/\mathcal {D} \sim \frac{F}{q}\rho ^2$. On the other hand, at the global minimum, both $\mathcal {D}$ and $\mathcal {F}_X$ are of the same order i.e. $\mathcal {F}_X/\mathcal {D} \sim \frac{F}{q}$, assuming that $\rho$ at the minimum is of order $\mathcal {O}(1)$, which is true in our models below. This makes tuning of the vacuum energy between the F- and D-contribution in principle possible.

Let us make a comment on the case when $b = 0$ only the new FI parameter $\xi$ is contributing to the potential. For $-3<\xi < 0$, the condition for the scalar potential’s local maximum at $\rho = 0$ can be satisfied. In the case where F is set to zero, the scalar potential (67) has a minimum at $\kappa \rho _{\text {min}}^2 = 3\ln \left( -\frac{3}{\xi } \right)$. In order to have $\mathcal {V}_{\text {min}} = 0$, we can choose $\xi = - \frac{3}{ e}$. We find that this parameter xi choice does not, however, allow slow-roll inflation close to the scalar potential maximum. Similar to the previous section, it may be possible to achieve both the scalar potential satisfying slow-roll conditions and a small cosmological constant at the minimum by adding correction terms to the Kähler potential and turning on a parameter F. However, in the next section, we will focus on $b = 3$ case where less parameters are required to satisfy the observational constraints.

4.3 An Example for D-Term Inflation Model

Let’s concentrate on the scenario when $b = 3$ and we assume that the scalar potential is D-term dominated by fixing $F = 0$. In this scenario, the model has just two free parameters, q and xi. The amplitude $A_s$ of the CMB data will determine the value of the first parameter, which regulates the potential’s overall scale. The second parameter $\xi$ is the sole remaining free parameter. We identify the criterion for slow-roll inflation scenarios, where the horizon crossing occurs close to the potential’s maximum at $\rho = 0$.

Since we assume inflation to start near the origin $\rho = 0$, the expansion of slow-roll parameters for small $\kappa \rho$ can be written as

$$\begin{aligned} \epsilon &= \frac{4}{9} (\xi +1)^2 (\kappa \rho )^2 + \mathcal {O}(\rho ^3), \nonumber \\ \eta &= \frac{2(1+ \xi )}{3} + \mathcal {O}(\rho ^2) . \end{aligned}$$

(72)

Note also that $\eta$ is negative when $\xi <-1$. We can therefore tune the parameter $\xi$ to avoid the $\eta$-problem. The observation is that at $\xi =-1$, the effective charge of X vanishes and thus the $\rho$-dependence in the D-term contribution (67) becomes of quartic order.

Note that we obtain the same relation between $\epsilon$ and $\eta$ as in the model of inflation from supersymmetry breaking driven by an F-term from a linear superpotential and $b=1$ (see Eqs. (17) and (18)). As a result, it is possible to have a scalar potential in which the flat plateau close to the maximum satisfies the slow rotation conditions and at the same time yields a negligible cosmological constant at a nearby minimum.

The number of e-folds N during inflation is determined by using Eq. (16) where we choose $|\epsilon (\chi _\textrm{end})| =1$. It is worth mentioning that the slow-roll parameters for small $\rho ^2$ satisfy the simple relation $\epsilon =\eta (0)^2\rho ^2+O(\rho ^4)$ by Eq. (72). As a result, the number of e-folds between $\rho _1$ and $\rho _2$ ($\rho _1 < \rho _2$) has the simple approximation form shown in [2],

$$\begin{aligned} N \simeq \frac{1}{|\eta (0)|} \ln \left( \frac{\rho _2}{\rho _1} \right) = \frac{3}{2|\xi +1|} \ln \left( \frac{\rho _2}{\rho _1} \right) . \end{aligned}$$

(73)

as long as the expansions in (72) are valid in the region $\rho _1 \le \rho \le \rho _2$. It should also be noted that we made the approximation $\eta (0) \simeq \eta _*$, which is valid in this case.

Next, we compare the theoretical predictions from this model with observational data via the power spectrum of scalar perturbations of the CMB, namely amplitude, As, tilt $n_s$ and the tensor-to-scalar ratio r. From the spectral indices relations described above, we should obtain $\eta _*\simeq -0.02$, where we find that Eq. (73) gives approximately the required number of e-folds when the logarithm is of order 1.

Actually, using this formula, we can estimate the upper bound of the tensor-to-scalar ratio r and the Hubble scale $H_*$ following the same argument given in [2]; the upper bounds are given by computing the parameters $r,H_*$ assuming that the expansions (72) hold until the end of inflation. By using $\eta _*= -0.02$, $N \simeq 50\,-\,60$ and $\kappa \rho _\textrm{end} \lesssim 0.5$, we obtain the bound

$$\begin{aligned} r \lesssim 16(|\eta _*|\kappa \rho _\textrm{end} e^{-|\eta _*|N})^2 \simeq 10^{-4}, \quad H_* \lesssim 10^{12} \, \textrm{GeV}. \end{aligned}$$

(74)

Note that the value of r mentioned above, despite being consistent with the observational data, is too small to be detected.

4.4 A Small Field Inflation Model from Supergravity with Observable Tensor-to-Scalar Ratio

In this section, we demonstrate how our model can achieve high r at the cost of adding a few extra terms to the Kähler potential. Take into account the prior model with additional cubic and quadratic terms in $X\bar{X}$:

$$\begin{aligned} K = X\bar{X} + A \kappa ^2(X\bar{X})^2 + B \kappa ^4 (X\bar{X})^3 + \kappa ^{-2} b \ln (\kappa ^2 X\bar{X}), \end{aligned}$$

(75)

while keeping the superpotential constant as in Eq. (65). We assume that inflation is driven by the D-term by setting the parameter $F = 0$. The scalar potential in terms of the field variable $\rho$ can be written as:

$$\begin{aligned} \mathcal {V} = \frac{q^2}{\kappa ^4}\left( b+(\kappa \rho )^2 + 2A (\kappa \rho )^4 + 3B (\kappa \rho )^6 + \xi (\kappa \rho )^{\frac{2 b}{3}} e^{\frac{1}{3} \left( A (\kappa \rho )^4+B (\kappa \rho )^6+(\kappa \rho )^2\right) }\right) ^2. \end{aligned}$$

(76)

A and B are two new parameters, but since they appear in higher orders in $\rho$ in the scalar potential, these parameters have no influence on our previous discussion about the possibility of parameter b. As a result, we can continue to consider the case $b = 3$. The formula (73) for the number of e-folds also holds for small $(\kappa \rho )^2$ even when A, B are not zero because the new parameters appear at order $\rho ^4$ and higher. However these two parameters can increase the value of the tensor-to-scalar ratio r. In order to obtain $r \approx 0.01$, we choose the parameter set as

$$\begin{aligned} q = 8.68 \times 10^{-6}, \quad \xi = -1.101, \quad A = 0.176, \quad B = 0.091. \end{aligned}$$

(77)

By choosing the initial condition $\kappa \rho _* = 0.445$ and $\kappa \rho _\textrm{end} = 1.155$, we get the results $N = 58$, $n_s = 0.96$, $r = 0.01$ and $\mathcal {A}_s =2.2 \times 10^{-9}$ , which is in agreement with the CMB data. Note that an application of the new FI term in no-scale supergravity model for inflation can be found for example in [19,20,21].

5 Conclusions

In this chapter, we addressed the possibility that supersymmetry breaking may lead to inflation, with the scalar component of the goldstino superfield serving as the inflaton. An interesting class of small field inflation models is generated by imposing gauged R-symmetry. These models have an inflationary plateau around the maximum of the scalar potential close to the origin, where R-symmetry is restored. The inflaton rolling down to a minimum with an infinitesimal tuneable positive vacuum energy. Inflation can be driven by either an F- or a new FI D-term. The above models are in agreement with cosmological observations and in the simplest case predict a rather small tensor-to-scalar ratio of primordial perturbations.

We also described the MSSM-inflaton couplings, and estimated the reheating temperature, generally $T_\textrm{reh}\sim 10^{8}$ GeV, from perturbative decay channels of the inflaton. The inflaton can decay into any MSSM sparticle in our model. Because the inflaton mass is less than double that of the gravitino mass, the former cannot perturbatively decay into two gravitini. The full picture of reheating, however, requires further investigation after taking into account non-perturbative effects such as Bose condensation and possible resonant production of fermions. Finally, as explained in [5], our minimal models do not allow for thermal LSP dark matter, but superheavy LSP dark matter (e.g,. neutralino) is possible depending on the parameter choice.