On the Starobinsky Model of Inflation from Supergravity (1307.1137v2)

Abstract: We discuss how the higher-derivative Starobinsky model of inflation originates from N=1 supergravity. It is known that, in the old-minimal supergravity description written by employing a chiral compensator in the superconformal framework, the Starobinsky model is equivalent to a no-scale model with F-term potential. We show that the Starobinsky model can also be originated within the so-called new-minimal supergravity, where a linear compensator superfield is employed. In this formulation, the Starobinsky model is equivalent to standard supergravity coupled to a massive vector multiplet whose lowest scalar component plays the role of the inflaton and the vacuum energy is provided by a D-term potential. We also point out that higher-order corrections to the supergravity Lagrangian represent a threat to the Starobinsky model as they can destroy the flatness of the inflaton potential in its scalar field equivalent description.

• The paper demonstrates how the Starobinsky R² inflation model is embedded in supergravity using old-minimal (chiral superfield) and new-minimal (real linear compensator) formulations.
• It establishes that the old-minimal formulation recasts the model as a no-scale supergravity theory with an F-term potential linked to a scalar inflaton dynamics.
• The paper highlights that higher-order corrections may compromise the flat inflaton potential, drawing parallels to the η-problem in supersymmetric inflation models.

The Starobinsky Model of Inflation from Supergravity

The paper under consideration explores the intricate relationship between the Starobinsky model of inflation and its embedding within ${\cal N}=1$ supergravity. The authors focus on illustrating how the renowned higher-derivative $R^2$ Starobinsky model can be derived using two different formulations of supergravity: the old-minimal supergravity and the new-minimal supergravity. This analysis not only deepens the understanding of theoretical frameworks within which inflationary models can be analyzed but also sheds light on potential pitfalls, especially concerning higher-order corrections in supergravity.

Background and Motivation

The Starobinsky model of inflation, characterized by its $R^2$ term in the gravitational Lagrangian, remains a significant model within inflationary cosmology due to its success in explaining various observational constraints, notably the spectral index and the tensor-to-scalar ratio. This model is pivotal in demonstrating a viable inflationary scenario without requiring additional scalar fields beyond those introduced by the theory of gravity itself.

The seamless integration of the Starobinsky model within the context of supergravity stems from its necessity to incorporate quantum corrections naturally through higher-order terms. Furthermore, supergravity, as a natural extension to general relativity in the presence of supersymmetry, offers a compelling framework for exploring the dynamics of inflation at high energies.

Old-Minimal Supergravity Formulation

In the old-minimal supergravity formulation, the embedding of the Starobinsky model uses a chiral superfield $S_0$ as a compensator in a superconformal approach. The analysis demonstrates that the Starobinsky Lagrangian can be equivalently framed as a no-scale supergravity model with an $F$-term potential, which enables the investigation into the dynamics associated with complex superfields and Kähler potentials. The theoretical formalism translates the $R^2$ correction into a scalar degree of freedom (inflaton), thereby linking gravitational dynamics directly with field-theoretic representations in supergravity.

New-Minimal Supergravity Formulation

An alternative, novel embedding is achieved through the new-minimal supergravity formulation, which employs a real linear compensator superfield. Here, the Starobinsky model is equivalently coupled to a massive vector multiplet, with a $D$-term providing the necessary vacuum energy and inflaton potential. This formulation shifts the focus towards gauge fields as central players in the inflationary dynamics and ties the scalar degrees of freedom to vector multiplet configurations.

Higher-Order Corrections and Their Implications

A critical concern discussed in the paper is the impact of higher-order corrections on the robustness of the inflaton potential. In both formulations, non-renormalizable terms become permissible and can distort the otherwise flat potential across the inflationary plateau. Such distortions may severely affect the anticipative power of the Starobinsky model, especially regarding the requisite flatness for sustaining inflation. The authors emphasize the similarity between this higher-order correction challenge and the $\eta$-problem previously faced in supersymmetric inflation models, highlighting supergravity's subtleties in maintaining potential flatness.

Conclusion and Future Prospects

The paper concludes with an acknowledgment of the intricate challenges and opportunities presented by embedding inflationary models like the Starobinsky model within a supergravity framework. Further paper into higher-order corrections and alternative formulations could provide deeper insights into both the theoretical foundations of inflation and the practical implications for cosmological observations.

The research offers promising directions for future exploration in AI and computational physics to simulate complex supergravity scenarios and account for quantum-level corrections, potentially leading to a more comprehensive understanding of the early universe's dynamics.