Cosmological expansion and the uniqueness of gravitational action

TL;DR

This paper probes whether the gravitational action in cosmology is uniquely determined by the observed expansion history. By analyzing $f(R)$ gravity, it shows that for any barotropic fluid, there exists a two-parameter family of $f(R)$ functions that reproduce the same background expansion as the Einstein–Hilbert action, establishing non-uniqueness of the action at the level of background dynamics. The authors construct explicit forms of equivalent actions in several cosmologically relevant cases, and perform a stability analysis indicating that stable $f(R)$ extensions exist. They argue that to discriminate among viable theories, one must study perturbations and their observational consequences, since the background expansion alone cannot fix the correct gravitational action. The results also illuminate the classical equivalence between $f(R)$ gravity and scalar-tensor theories, guiding future work on dark energy and modified gravity.

Abstract

Modified theories of gravity have recently been studied by several authors as possibly viable alternatives to the cosmological concordance model. Such theories attempt to explain the accelerating expansion of the universe by changing the theory of gravity, instead of introducing dark energy. In particular, a class of models based on higher order curvature invariants, so-called $f(R)$ gravity models, has drawn attention. In this letter we show that within this framework, the expansion history of the universe does not uniquely determine the form of the gravitational action and it can be radically different from the standard Einstein-Hilbert action. We demonstrate that for any barotropic fluid, there always exists a class of $f(R)$ models that will have exactly the same expansion history as that arising from the Einstein-Hilbert action. We explicitly show how one can extend the Einstein-Hilbert action by constructing a $f(R)$ theory that is equivalent on the classical level. Due to the classical equivalence between $f(R)$ theories and Einstein-Hilbert gravity with an extra scalar field, one can also hence construct equivalent scalar-tensor theories with standard expansion.