# Starobinsky-Like Inflation and Running Vacuum in the Context of Supergravity

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## Abstract

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## 1. Introduction

## 2. Running Vacuum: A Natural Arena for Vacuum Decay in Cosmology

## 3. Generic Starobinsky Inflation

## 4. Starobinsky-Type Inflation in Dynamically-Broken SUGRA

- The effective equations$$\begin{array}{c}\hfill \frac{\partial \Gamma}{\partial \Lambda}=0\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{1.em}{0ex}}\frac{\partial \Gamma}{\partial \sigma}{|}_{\sigma ={\sigma}_{c}}=0\phantom{\rule{3.33333pt}{0ex}},\end{array}$$
- The cosmological constant Λ is small and positive, satisfying (29), and for phenomenological reasons, it should be of order:$$0<\Lambda \sim {10}^{-10}{M}_{\mathrm{Pl}}^{2}\phantom{\rule{3.33333pt}{0ex}},$$

## 5. “Decay” of Effective Vacuum Energy: Running Vacuum Model

#### 5.1. A Distinct Class of Running Vacuum Models

#### 5.2. Running Vacuum Evolution: From the Current to the Inflationary Era

#### 5.3. Geometrical Description: RVM versus Starobinsky

#### 5.4. Scalar Field Description: RVM versus Starobinsky

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The effective potential (8) of the collective scalar field ϕ that describes the one-loop quantum fluctuations of matter fields, leading to the higher-order scalar curvature corrections in the Starobinsky model for inflation (5). Notice that according to (8), we have $\beta =8\pi /3{\mathcal{M}}^{2}$. The potential is sufficiently flat for $\kappa \phi \gg 1$ to ensure slow-roll conditions for inflation are satisfied, in agreement with the Planck data, for appropriate values of the scale $1/\sqrt{\beta}\propto \mathcal{M}$ (which sets the overall scale of inflation in the model).

**Figure 2.**Generic shape of the one-loop effective potential (in dimensionless units ${\tilde{\kappa}}^{4}\phantom{\rule{0.166667em}{0ex}}{V}_{\mathrm{eff}}$, where $\tilde{\kappa}$ is the conformally-rescaled gravitational coupling; see the text) for the gravitino condensate field σ in dynamically-broken (conformal) supergravity models in the presence of a non-trivial de Sitter background with cosmological constant $\Lambda >0$ [9]. The Starobinsky inflationary phase is associated with fluctuations of the condensate and gravitational field modes near the non-trivial minimum of the potential, where the condensate ${\sigma}_{c}\ne 0$ and the potential assumes the value $\Lambda >0$, consistent with supersymmetry breaking. The dashed green lines denote “forbidden” areas of the condensate field values, violating the condition (17), for which imaginary parts appear in the effective potential, thereby destabilizing the broken symmetry phase.

**Figure 3.**The RVM effective potential $\alpha {\kappa}^{2}U/{H}_{I}^{2}$ (solid line) versus the scalaron field $\kappa \phi $. In order to produce the curve, we utilize $\mathcal{M}\sim {M}_{X}\sim {10}^{16}$ GeV, ${H}_{I}\sim 0.81\times {10}^{14}$ GeV (see Equation (3)) and $\alpha \sim {10}^{-4}$. The dashed line corresponds to the Starobinsky effective potential (see Figure 1).

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Basilakos, S.; Mavromatos, N.E.; Solà, J.
Starobinsky-Like Inflation and Running Vacuum in the Context of Supergravity. *Universe* **2016**, *2*, 14.
https://doi.org/10.3390/universe2030014

**AMA Style**

Basilakos S, Mavromatos NE, Solà J.
Starobinsky-Like Inflation and Running Vacuum in the Context of Supergravity. *Universe*. 2016; 2(3):14.
https://doi.org/10.3390/universe2030014

**Chicago/Turabian Style**

Basilakos, Spyros, Nick E. Mavromatos, and Joan Solà.
2016. "Starobinsky-Like Inflation and Running Vacuum in the Context of Supergravity" *Universe* 2, no. 3: 14.
https://doi.org/10.3390/universe2030014