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

Abstract

We discuss a modified gravity theory defined by $f(R) = \sum_{n}^{l} α_n M^{2(1-n)} R^n$. 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.