Starobinsky Inflation: From non-SUSY to SUGRA Realizations

Starobinsky Inflation: From non-SUSY to SUGRA Realizations

Constantinos Pallis & Nicolaos Toumbas Department of Physics, University of Cyprus,
P.O. Box 20537, Nicosia 1678, CYPRUS

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

98.80.Cq, 11.30.Qc, 12.60.Jv, 04.65.+e


Cosmology, Modified Gravity, Supersymmetric models, Supergravity



[\fancyplain 1]\fancyplainStarobinsky Inflation: From non-SUSY to SUGRA Realizations \lhead[\fancyplain ]\fancyplain1 \cfoot

1 Introduction

The idea that the universe underwent a period of exponential expansion, called inflation [1], has proven useful not only for solving the horizon and flatness problems of standard cosmology, but also for providing an explanation for the scale invariant perturbations, which are responsible for generating the observed anisotropies in the Cosmic Microwave Background (CMB). One of the first incarnations of inflation is due to Starobinsky. To date, this attractive scenario remains predictive, since it passes successfully all the observational tests [4, 3]. Starobinsky considered adding an term, where is the Ricci scalar, to the standard Einstein action in order to source inflation. Recall that gravity theories based on higher powers of are equivalent to standard gravity theories with one additional scalar degree of freedom – see e.g. [5]. As a result, Starobinsky inflation is equivalent to inflation driven by a scalar field with a suitable potential and so, it admits several interesting realizations [6, 7, 15, 8, 9, 11, 14, 10, 12, 13].

Following this route, we show in this work that induced-gravity inflation (IGI) [16, 17, 18, 19, 20] is effectively Starobinsky-like, reproducing the structure and the predictions of the original model. Within IGI, the inflaton exhibits a strong coupling to and the reduced Planck scale is dynamically generated through the vacuum expectation value (v.e.v.) of the inflaton at the end of inflation. Therefore, the inflaton acquires a higgs-like behavior as in theories of induced gravity [22, 20, 21]. Apart from being compatible with data, the resulting theory respects perturbative unitarity up to the Planck scale [16, 15, 17]. Therefore, no concerns about the validity of the corresponding effective theory arise. This is to be contrasted with models of non-minimal inflation (nMI) [23, 24, 26, 25, 27, 28] based on a potential with negligible v.e.v. for the inflaton . Although these models yield similar observational predictions with the Starobinsky model, they admit an ultraviolet (UV) scale well below for , leading to complications with naturalness [30, 29].

Nonetheless, IGI allows us to embed Starobinsky inflation within Supergravity (SUGRA) in an elegant way. The embedding is achieved by incorporating two chiral superfields, a modulus-like field and a matter-like field appearing in the superpotential, , as well as various Kähler potentials, , consistent with an and discrete symmetries [15, 17, 31] – see also Refs. [10, 9, 11, 14, 18]. In some cases [9, 15, 17, 31], the employed ’s parameterize specific Kähler manifolds, which appear in no-scale models [32, 33]. Moreover, this scheme ensures naturally a low enough reheating temperature, potentially consistent with the gravitino constraint [15, 35, 34] if connected with a version of the Minimal SUSY Standard Model (MSSM).

An important issue in embedding IGI in SUGRA is the stabilization of the matter-like field . Indeed, when parameterizes the Kähler manifold [9, 10], the inflationary trajectory turns out to be unstable with respect to (w.r.t.) the fluctuations of . This difficulty can be overcome by adding a sufficiently large term , with and , in the logarithmic function appearing in , as suggested in \creflee for models of non-minimal (chaotic) inflation [26] and applied in Refs. [27, 28, 38, 39]. This solution, however, deforms slightly the Kähler manifold [40]. More importantly, it violates the predictability of Starobinsky inflation, since mixed terms with , which can not be ignored (without tuning), have an estimable impact [41, 17, 37] on the dynamics and the observables. Moreover, this solution becomes complicated when more than two fields are considered, since all quartic terms allowed by symmetries have to be considered, and the analysis of the stabilization mechanism becomes tedious – see e.g. Refs. [17, 37, 41]. Alternatively, it was suggested to use a nilpotent superfield [42], or a charged field under a gauged R symmetry [40].

In this review, we revisit the issue of stabilizing , disallowing terms of the form , , without caring much about the structure of the Kähler manifold. Namely, we investigate systematically several functions (with ) that appear in the choices for , and we find four acceptable forms that lead to the stabilization of during and after IGI. The output of this analysis is new, providing results that did not appear in the literature before. More specifically, we consider two principal classes of ’s, and , distinguished by whether and appear in the same logarithmic function. The resulting inflationary scenaria are almost indistinguishable. The case considered in \crefsu11 is included as one of the viable choices in the class. Contrary to \crefsu11, though, we impose here the same symmetry on and . Consequently, the relevant expressions for the mass spectrum and the inflationary observables get simplified considerably compared to those displayed in \crefsu11. As in the non-SUSY case, IGI may be realized using subplanckian values for the initial (non-canonically normalized) inflaton field. The radiative corrections remain under control and perturbative unitarity is not violated up to [2, 17, 31], consistently with the consideration of SUGRA as an effective theory.

Throughout this review we focus on the standard CDM cosmological model [3]. An alternative framework is provided by the running vacuum models [43] which turn out to yield a quality fit to observations, significantly better than that of CDM. In this case, the acceleration of the universe, either during inflation or at late times, is not attributed to a scalar field but rather arises from the modification of the vacuum itself, which is dynamical. A SUGRA realization of Starobinsky inflation within this setting is obtained in the last paper of \crefketov1.

The plan of this paper is as follows. In Sec. 2, we establish the realization of Starobinsky inflation as IGI in a non-SUSY framework. In \Srefsec:sugra we introduce the formulation of IGI in SUGRA and revisit the issue of stabilizing the matter-like field . The emerging inflationary models are analyzed in Sec. 4. Our conclusions are summarized in Sec. 5. Throughout, charge conjugation is denoted by a star (), the symbol as subscript denotes derivation w.r.t. , and we use units where the reduced Planck scale, , is set equal to unity.

2 Starobinsky Inflation From Induced Gravity

We begin our presentation demonstrating the connection between inflation and IGI. We first review the formulation of nMI in \Sreffhi, and then proceed to describe the inflationary analysis in \Srefobs. Armed with these prerequisites, we present inflation as a type of nMI in \Srefr2nmi, and exhibit its connection with IGI in \Srefigi.

2.1 Coupling non-Minimally the Inflaton to Gravity

We consider an inflaton that is non-minimally coupled to the Ricci scalar , via a coupling function . We denote the inflaton potential by and allow for a general kinetic function – in the cases of pure nMI [19, 24, 25] . The Jordan Frame (JF) action takes the form


where is the determinant of the Friedmann-Robertson-Walker metric, , with signature . We require to ensure ordinary Einstein gravity at low energies.

By performing a conformal transformation [24] to the Einstein frame (EF), we write the action


where a hat denotes an EF quantity. The EF metric is given by , and the canonically normalized field, , and its potential, , are defined as follows:


For , the coupling function acquires a twofold role. On one hand, it determines the relation between and . On the other hand, it controls the shape of , thus affecting the observational predictions – see below. The analysis of nMI can be performed in the EF, using the standard slow-roll approximation. It is [19] completely equivalent with the analysis in the JF. We just have to keep track the relation between and .

2.2 Observational and Theoretical Constraints

A viable model of nMI must be compatible with a number of observational and theoretical requirements summarized in the following – cf. \crefreview.


The number of e-foldings that the scale experiences during inflation must to be large enough for the resolution of the horizon and flatness problems of the standard hot Big Bang model, i.e. [24, 3],


where is the value of when crosses the inflationary horizon. In deriving the formula above – cf. \crefnMkin – we take into account an equation-of-state with parameter [45], since can be well approximated by a quadratic potential for low values of – see Eqs. (2.\theparentequationb), (2.\theparentequationb) and (4.\theparentequationb) below. Also is the reheating temperature after nMI. We take a representative value throughout, which results to . The effective number of relativistic degrees of freedom at temperature is taken in accordance with the Standard model spectrum. Lastly, is the value of at the end of nMI, which in the slow-roll approximation can be obtained via the condition


Evidently non trivial modifications of , and thus of , may have a significant effect on the parameters above, modifying the inflationary observables.


The amplitude of the power spectrum of the curvature perturbation generated by at has to be consistent with the data [46], i.e.,


As shown in \Srefstab, the remaining scalars in the SUGRA versions of nMI may be rendered heavy enough and so, they do not contribute to .


The remaining inflationary observables (the spectral index , its running , and the tensor-to-scalar ratio ) must be in agreement with the fitting of the Planck, Baryon Acoustic Oscillations (BAO) and Bicep2/Keck Array data [3, 4] with the CDM model, i.e.,


at the 95 confidence level (c.l.) with . Although compatible with Eq. (2.7b), all data taken by the Bicep2/Keck Array CMB polarization experiments, up to the 2014 observational season (BK14) [4], seem to favor ’s of the order of , as the reported value is at the 68 c.l.. These inflationary observables are estimated through the relations:


where and the variables with subscript are evaluated at .


The effective theory describing nMI remains valid up to a UV cutoff scale , which has to be large enough to ensure the stability of our inflationary solutions, i.e.,


As we show below, for the models analyzed in this work, contrary to the cases of pure nMI with large , where . The determination of is achieved expanding in \Erefaction about . Although these expansions are not strictly valid [30] during inflation, we take the extracted this way to be the overall UV cut-off scale, since the reheating phase – realized via oscillations about – is a necessary stage of the inflationary dynamics.

2.3 From Non-Minimal to Inflation

The inflation can be viewed as a type of nMI, if we employ an auxiliary field with the following input ingredients


Using the equation of motion for the auxiliary field, , we obtain the action of the original Starobinsky model (see e.g. \crefnick):


As we can see from \Erefstr1, the model has only one free parameter (), enough to render it consistent with the observational data, ensuring at the same time perturbative unitarity up to the Planck scale. Using \Erefstr1 and \ErefVJe, we obtain the EF quantities


For , the plot of versus is depicted in Fig. 1-(a). An inflationary era can be supported since becomes flat enough. To examine further this possibility, we calculate the slow-roll parameters. Plugging \ErefVJstr into \Erefsr yields


Notice that since is slightly concave downwards, as shown in Fig. 1-(a). The value of at the end of nMI is determined via \Erefsr, giving


Under the assumption that , we can obtain a relation between and via \ErefNhi


The precise value of can be determined enforcing \ErefProb. Recalling that , we get


The resulting value of is large enough so that


consistently with Eq. (2.9b) – see Fig. 1-(a). Impressively, the remaining observables turn out to be compatible with the observational data of \Erefnswmap. Indeed, inserting the above value of into \Erefns (), we get


Without the simplification of \Erefsgxstr, we obtain numerically , and . We see that turns out to be appreciably lower than unity thanks to the negative values of – see \Erefsrstr. The mass of the inflaton at the vacuum is


As we show below this value is a salient future in all models of Starobinsky inflation.

Furthermore, the model provides an elegant solution to the unitarity problem [30, 29], which plagues models of nMI with , and . This stems from the fact that and do not coincide at the vacuum, as Eq. (2.12a) implies . In fact, if we expand the second term in the right-hand side (r.h.s.) of \Erefaction about , we find


Similarly, expanding in Eq. (2.12b), we obtain


Since the coefficients of the above series are of order unity, independent of , we infer that the model does not face any problem with perturbative unitarity up to the Planck scale.

Figure 1: The inflationary potential as a function of for inflation () and IGI with (). Values corresponding to and are also indicated.

2.4 Induced-Gravity Inflation

It would be certainly beneficial to realize the structure and the predictions of inflation in a framework that deviates minimally from Einstein gravity, at least in the present cosmological era. To this extent, we incorporate the idea of induced gravity, according to which is generated dynamically [22] via the v.e.v. of a scalar field , driving a phase transition in the early universe. The simplest way to implement this scheme is to employ a double-well potential for – for scale invariant realizations of this idea see \crefjones. On the other hand, an inflationary stage requires a sufficiently flat potential, as in \Erefstr1. This can be achieved at large field values if we introduce a quadratic [19, 20]. More explicitly, IGI may be defined as a nMI with the following input ingredients:


Given that , we recover Einstein gravity at the vacuum if


We see that in this model there is one additional free parameter, namely appearing in the potential, as compared to the model.


VJe and \Erefig1 imply


For , the plot of versus is shown in Fig. 1-(b). As in the model, develops a plateau and so, an inflationary stage can be realized. To check its robustness, we compute the slow-roll parameters. \Erefsr and \ErefVJig give


IGI is terminated when , determined by the condition


Under the assumption that , \ErefNhi implies the following relation between and


Imposing Eq. (2.9b) and setting , we derive a lower bound on :


Contrary to inflation, does not control exclusively the normalization of \ErefProb, thanks to the presence of an extra factor of . This is constrained to scale with . Indeed, we have


If, in addition, we impose the perturbative bound , we end-up with following ranges:


where the lower bounds on and correspond to – see Fig. 1-(b). Within the allowed ranges, remains constant, by virtue of \ErefProbig. The mass turns out to be


essentially equal to that estimated in \Erefmsstr. Moreover, using \Erefsgxig and \Erefns, we extract the remaining observables


Without making the approximation of \Erefsgxig, we obtain numerically . These results practically coincide with those of inflation, given in Eqs. (2.\theparentequationa) – (2.\theparentequationc), and they are in excellent agreement with the observational data presented in \Erefnswmap.

As in the previous section, the model retains perturbative unitarity up to . To verify this, we first expand the second term in the r.h.s. of \Erefaction1 about , with given by Eq. (2.3a). We find


Expanding given by Eq. (2.3b), we get


Therefore, as for inflation. Practically identical results can be obtained if we replace the quadratic exponents in \Erefig1 with as first pointed out in \crefgian. This generalization can be elegantly performed [18, 17] within SUGRA, as we review below.

3 Induced-Gravity Inflation in SUGRA

In \Srefsec:sugra1, we present the general SUGRA setting, where IGI is embedded. Then, in \Srefsec:ks, we examine a variety of Kähler potentials, which lead to the desired inflationary potential – see \Srefsec:ks2. We check the stability of the inflationary trajectory in \Srefstab.

3.1 The General Set-Up

To realize IGI within SUGRA [15, 17, 18, 31], we must use of two gauge singlet chiral superfields , with and being the inflaton and a “stabilizer” superfield respectively. Throughout this work, the complex scalar fields are denoted by the same superfield symbol. The EF effective action is written as follows [26]


where is the Kähler metric and its inverse (). is the EF F–term SUGRA potential, given in terms of the Kähler potential  and the superpotential by the following expression


Conformally transforming to the JF with , where is a dimensionless positive parameter, takes the form


Note that reproduces the standard set-up [26]. Let us also relate and by


Then taking into account the definition [26] of the purely bosonic part of the auxiliary field when on shell,


we arrive at the following action


By virtue of \ErefOmg1, takes the form


with and . As can be seen from \ErefSfinal, introduces a non-minimal coupling of the scalar fields to gravity. Ordinary Einstein gravity is recovered at the vacuum when


Starting with the JF action in \ErefSfinal, we seek to realize IGI, postulating the invariance of under the action of a global discrete symmetry. With stabilized at the origin, we write


where is a positive integer. If during IGI and assuming that ’s are relatively small, the contributions of the higher powers of in the expression above are small, and these can be dropped. As we verify later, this can be achieved when the coefficient is large enough. Equivalently, we may rescale the inflaton, setting . Then the coefficients of the higher powers in the expression of get suppressed by factors of . Thus, and the requirement determine the form of , avoiding a severe tuning of the coefficients . Under these assumptions, in \ErefOmg1 takes the form


where is assumed to be stabilized at the origin.

Eqs. (3.2) and (3.1) require that and acquire the following v.e.v.s


These v.e.v.s can be achieved, if we choose the following superpotential [17, 18]:


Indeed the corresponding F-term SUSY potential, , is found to be


and is minimized by the field configuration in \Erefig1.

As emphasized in Refs. [15, 17, 31], the forms of and can be uniquely determined if we limit ourselves to integer values for (with ) and , and impose two symmetries:

  • An R symmetry under which and have charges and respectively;

  • A discrete symmetry under which only is charged.

For simplicity we assume here that both and respect the same , contrary to the situation in \crefsu11. This assumption simplifies significantly the formulae in Secs. 3.3 and 3.4. Note, finally, that the selected in \ErefOmdef does not contribute in the kinetic term involving in \ErefSfinal. We expect that our findings are essentially unaltered even if we include in the r.h.s. of \ErefOmdef a term [18] or [17] which yields – the former choice, though, violates the symmetry above.

3.2 Proposed Kähler Potentials

It is obvious from the considerations above, that the stabilization of at zero during and after IGI is of crucial importance for the viability of our scenario. This key issue can be addressed if we specify the dependence of the Kähler potential on . We distinguish the following basic cases:


where the various choices , , are specified in \Treftab1, and is defined as follows


As shown in \Treftab1 we consider exponential, logarithmic, trigonometric and hyperbolic functions. Note that and parameterize the and Kähler manifolds respectively, whereas parameterizes the Kähler manifold – see \crefsu11.

Table 1: Definition of the various ’s, and masses squared of the fluctuations of and along the inflationary trajectory in \Erefinftr for and .

To show that the proposed ’s are suitable for IGI, we have to verify that they reproduce in Eq. (2.3b) when , and they ensure the stability of at zero. These requirements are checked in the following two sections.

3.3 Derivation of the Inflationary Potential

Substituting of \ErefWn and a choice for in \Erefk32 (with the ’s given in \Treftab1) into \ErefVsugra, we obtain a potential suitable for IGI. The inflationary trajectory is defined by the constraints


where we have expanded and in real and imaginary parts as follows


Along the path of \Erefinftr, reads


From \ErefWn we get . Also, \Erefk32 yields


where we take into account that along the path of \Erefinftr. Moreover,