Do consistent $F(R)$ models mimic General Relativity plus $Λ$?

TL;DR

Do consistent $F(R)$ models mimic General Relativity plus a cosmological constant? The paper investigates whether a broad class of consistency-constrained $F(R)$ gravity theories can reproduce GR with a cosmological constant rather than yielding large deviations. By constructing a concrete trial function that respects the standard constraints, and by analyzing both the Newtonian limit and the full cosmological evolution, the authors show that satisfying the constraints drives the Friedmann equation toward GR with an effective cosmological term $\Lambda \approx b/(2a)$ and with corrections that are exponentially suppressed in $\alpha = aR - b$. They argue that a viable late-time acceleration still requires fine-tuning of the free parameters, and claim this is a generic feature within this constrained class. The result suggests that such constrained $F(R)$ models may be effectively indistinguishable from GR over the cosmic history, although potential local-gravity signatures could provide tests if the constraints are kept strict.

Abstract

Modified gravity models are subject to a number of consistency requirements which restrict the form that the function $F(R)$ can take. We study a particular class of $F(R)$ functions which satisfy various constraints that have been found in the literature. These models have a late time accelerating epoch, and an acceptable matter era. We calculate the Friedmann equation for our models, and show that in order to satisfy the constraints we impose, they must mimic General Relativity plus $Λ$ throughout the cosmic history, with exponentially suppressed corrections. We also find that the free parameters in our model must be fine tuned to obtain an acceptable late time accelerating phase. We discuss the generality of this conclusion.

Figures (2)

• Figure 1: $(a)$ is a plot of the function Q(R). This function has three zeros, which correspond to three vacuum states of our model. $(b)$ are the functions $m$ (solid), $r$ (dotted) and $dm/dr$ (dashed), plotted parametrically as functions of $R$. We can clearly see that there exists an $R$ such that $r=-2$ and $m<1$, suggesting a late-time accelerating phase. In addition, we have $m \sim 0$ as $r \to -1$, which is the condition required for an acceptable matter era am1. We have set $a=2$, $b=1.5$.
• Figure 2: $(a)$ The Einstein frame potential. Note the presence of two minima; one at $V=0$, and one at some $V = V_{0} > 0$. It is this minima (shown in $(b)$) that corresponds to a stable de Sitter solution in the Jordan frame. $(b)$ is $V(\sigma)$ in the small $\sigma$ regime. We have set $a=2$, $b=1.5$.