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.
Highlights
Cosmic inflation [1,2,3] is a well established, consistent with the data [4], theory of the early universe which predicts cosmic acceleration and generation of seeds of the large scale structure of the present universe
In this paper we extend this analysis as follows: we assume that the Einstein frame potential of the f (R) theory with higher order terms up to Rl has a flat area around any of its stationary points
We investigate the issue of stability of the general relativity (GR) vacuum of the model
Summary
Cosmic inflation [1,2,3] is a well established, consistent with the data [4], theory of the early universe which predicts cosmic acceleration and generation of seeds of the large scale structure of the present universe. In order to obtain quasi de Sitter evolution of space–time one needs a wide range of energies for which the R2 M−2 term dominates the Lagrangian density This would require all higher order corrections (such as R3, R4 etc.) [8,9] to be suppressed by a mass scale much bigger than M. The influence of higher order corrections on the Starobinsky model may spoil the flatness of the Einstein frame potential and prevent the early universe from inflating. In this paper we extend this analysis as follows: we assume that the Einstein frame potential of the f (R) theory with higher order terms up to Rl has a flat area around any of its stationary points.
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