Advances in High Energy Physics

Volume 2017 (2017), Article ID 6759267, 16 pages

https://doi.org/10.1155/2017/6759267

## Starobinsky Inflation: From Non-SUSY to SUGRA Realizations

Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus

Correspondence should be addressed to Constantinos Pallis; rg.htua.neg@sillapk

Received 29 December 2016; Revised 23 March 2017; Accepted 10 April 2017; Published 18 June 2017

Academic Editor: Elias C. Vagenas

Copyright © 2017 Constantinos Pallis and Nicolaos Toumbas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

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 sub-Planckian values of the inflation. The corresponding effective theories retain perturbative unitarity up to the Planck scale and the inflation mass is predicted to be . The supergravity embedding of these models is achieved by employing two gauge singlet chiral superfields, a superpotential that is uniquely determined by a continuous and a discrete symmetry and several (semi)logarithmic Kähler potentials that respect these symmetries. Checking various functional forms for the noninflation accompanying field in the Kähler potentials, we identify four cases which stabilize it without invoking higher order terms.

#### 1. Introduction

The idea that the universe underwent a period of exponential expansion, called inflation [1–3], 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, 5]. 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., [6, 7]). As a result, Starobinsky inflation is equivalent to inflation driven by a scalar field with a suitable potential, so it admits several interesting realizations [8–29].

Following this route, we show in this work that* induced-gravity inflation* (IGI) [30–38] is effectively Starobinsky-like, reproducing the structure and the predictions of the original model. Within IGI, the inflation exhibits a strong coupling to and the reduced Planck scale is dynamically generated through the* vacuum expectation value* (v.e.v.) of the inflation at the end of inflation. Therefore, the inflation acquires a Higgs-like behavior as in theories of induced gravity [36–42]. Apart from being compatible with data, the resulting theory respects perturbative unitarity up to the Planck scale [29–31]. Therefore, no concerns about the validity of the corresponding effective theory arise. This is to be contrasted with models of* nonminimal inflation* (nMI) [43–54] based on a potential with negligible v.e.v. for the inflation . 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 [55–57].

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 [29, 31, 58]; see also [20–22, 28, 32]. In some cases [20, 29, 31, 58], the employed ’s parameterize specific Kähler manifolds, which appear in no-scale models [59–61]. Moreover, this scheme ensures naturally a low enough reheating temperature, potentially consistent with the gravitino constraint [29, 62, 63] 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 [20, 21], the inflationary trajectory turns out to be unstable* with respect to* 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 [64] for models of nonminimal (chaotic) inflation [47–49] and applied in [50–54, 65–70]. This solution, however, deforms slightly the Kähler manifold [71]. More importantly, it violates the predictability of Starobinsky inflation, since mixed terms with , which cannot be ignored (without tuning), have an estimable impact [31, 72–74] 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., [31, 72–74]). Alternatively, it was suggested to use a nilpotent superfield [75] or a charged field under a gauged symmetry [71].

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 scenarios are almost indistinguishable. The case considered in [58] is included as one of the viable choices in class. Contrary to [58], 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 [58]. As in the non-SUSY case, IGI may be realized using sub-Planckian values for the initial (noncanonically normalized) inflation field. The radiative corrections remain under control and perturbative unitarity is not violated up to [31, 58, 76], consistently with the consideration of SUGRA as an effective theory.

Throughout this review we focus on the standard CDM cosmological model [4]. An alternative framework is provided by the running vacuum models [77–84] 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 [18].

The plan of this paper is as follows. In Section 2, we establish the realization of Starobinsky inflation as IGI in a non-SUSY framework. In Section 3 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 Section 4. Our conclusions are summarized in Section 5. Throughout, charge conjugation is denoted by a star (), the symbol as subscript denotes derivation with respect to , 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 Section 2.1 and then proceed to describe the inflationary analysis in Section 2.2. Armed with these prerequisites, we present inflation as a type of nMI in Section 2.3 and exhibit its connection with IGI in Section 2.4.

##### 2.1. Coupling Nonminimally the Inflation to Gravity

We consider an inflation that is nonminimally coupled to the Ricci scalar , via a coupling function . We denote the inflation potential by and allow for a general kinetic function —in the cases of pure nMI [33–35, 45, 46] . The* Jordan Frame* (JF) action takes the formwhere 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 [45] to the* Einstein frame* (EF), we write the actionwhere 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 the 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 [33–35] 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. [85–88]).

(1) The number of e-foldings that the scale experiences during inflation must be large enough for the resolution of the horizon and flatness problems of the standard hot Big Bang model; that is, [4, 45]where is the value of when crosses the inflationary horizon. In deriving the formula above (cf. [65–67]), we take into account an equation-of-state with parameter [89], since can be well approximated by a quadratic potential for low values of ; see(20b), (32b), and (71b) below. Also is the reheating temperature after nMI. We take a representative value throughout, which results in . 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 nontrivial modifications of , and thus of , may have a significant effect on the parameters above, modifying the inflationary observables.

(2) The amplitude of the power spectrum of the curvature perturbation generated by at has to be consistent with the data [90]; that is,As shown in Section 3.4, the remaining scalars in the SUGRA versions of nMI may be rendered heavy enough, so they do not contribute to .

(3) 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 [4, 5] with the CDM model; that is,at the 95%* confidence level* (c.l.) with . Although compatible with (7)(b), all data taken by the BICEP2/*Keck Array* CMB polarization experiments, up to the 2014 observational season (BK14) [5], seem to favor ’s of the order of 0.01, 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 .

(4) 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; that is,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 (2) about . Although these expansions are not strictly valid [57] during inflation, we take extracted this way to be the overall UV cutoff scale, since the reheating phase, realized via oscillations about , is a necessary stage of the inflationary dynamics.

##### 2.3. From Nonminimal to Inflation

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., [71]):As we can see from (10), 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 (10) and (3), we obtain the EF quantities:For , the plot of versus is depicted in Figure 1(a). An inflationary era can be supported since becomes flat enough. To examine further this possibility, we calculate the slow-roll parameters. Plugging (12) into (5) yieldsNotice that since is slightly concave downwards, as shown in Figure 1(a). The value of at the end of nMI is determined via (5), giving