Implications of extreme flatness in a general f(R) theory

TL;DR

This paper explores a class of f(R) gravity theories with flat Einstein frame potentials, showing that only Starobinsky-like models can have an extremely flat area around stationary points, with stable GR vacuum and AdS vacuum not reachable.

Contribution

It demonstrates that f(R) theories with maximally flat stationary points necessarily lead to Starobinsky-like models with stable GR vacuum and an unreachable AdS vacuum.

Findings

Starobinsky model emerges as the limit for infinite series.

The Einstein frame potential has a stable GR vacuum and an inaccessible AdS vacuum.

Maximally flat stationary points imply a saddle point or local maximum depending on the series parity.

Abstract

We discuss a modified gravity theory defined by $f(R) = \sum_{n}^{l} \alpha_n M^{2(1-n)} R^n$. We consider both finite and infinite number of terms in the series while requiring that the Einstein frame potential of the theory has a flat area around any of its stationary points. We show that the requirement of maximally flat stationary point leads to the existence of the saddle point (local maximum) for even (odd) $l$. In both cases for $l\to\infty$ one obtains the Starobinsky model with small, exponentially suppressed corrections. Besides the GR minimum the Einstein frame potential has an anti de Sitter vacuum. However we argue that the GR vacuum is absolutely stable and AdS cannot be reached neither via classical evolution nor via quantum tunnelling. Our results show that a Starobinsky-like model is the only possible realisation of $f(R)$ theory with an extremely flat area in the Einstein frame potential.