A Note on de Sitter Embedding of $f(R)$ Theories

TL;DR

This paper examines the constraints on $f(R)$ gravity theories imposed by de Sitter embedding, analyzing stability, viability of models, and implications for cosmological phenomena like inflation and dark energy.

Contribution

It provides a detailed analysis of de Sitter embedding constraints on $f(R)$ models, identifying which Lagrangians are viable and exploring their cosmological implications.

Findings

Many common $f(R)$ Lagrangians do not admit de Sitter embedding.

Stable embeddings are only possible with anti-de Sitter spaces.

A class of $f(R)$ models with positive and negative powers of $R$ could unify inflation and late-time acceleration.

Abstract

The consequences of the constraints which de Sitter embedding of $f(R)$ theories imposes on the Lagrangian's parameters, are investigated within the metric formalism. It is shown, in particular, that several common $f(R)$ Lagrangians do not actually admit such an embedding. Otherwise, asymptotic matching of local solutions of the corresponding models with background (maximally symmetric) spaces of constant curvature is either unstable or, anti-de Sitter embedding is the only stable embedding. Additional arguments are given in favour of a previous claim that a class of $f(R)$ models comprising both positive and negative powers of $R$ (two different mass scales), could be a nice scenario where to address, in a united picture, both early-time inflation and late-time accelerated expansion of the universe. The approach undertaken here is used, also, to check ghost-freedom of a Dirac-Born-Infeld modification of general relativity previously studied in the bibliography.