Globally Stable Dark Energy in $F(R)$ Gravity

TL;DR

This work addresses the instability of many $F(R)$ dark-energy models at negative curvature and proposes globally stable constructions by enforcing $F_R(R)>0$ and $F_{RR}(R)>0$ for all $R$ via sigmoid-inspired forms of $F_R$. Building on the Appleby–Battye framework, the authors introduce both a revisited AB model and a new sigmoid-based model with a square-root form, yielding analytic $F(R)$ functions that approach a high-curvature $\, ext{ΛCDM}$ limit while maintaining stability. They analyze cosmological evolution, showing a slightly enhanced $H(z)$ and phantom crossing in the dark-energy EoS, and perform a thorough set of local-gravity tests (chameleon mechanism, solar-system, equivalence-principle) to constrain model parameters. A unified description of inflation and dark energy is developed by incorporating an $R^2$ term with a tunable $j$-factor, enabling a smooth transition across cosmic histories and preserving global stability. The results provide a concrete pathway to a single $F(R)$ Lagrangian capable of describing both early- and late-time acceleration while remaining compatible with local tests, though further cosmological tests and reheating analyses remain for future work.

Abstract

$F(R)$ models for dark energy generally exhibit a weak curvature singularity, which can be cured by adding an $R^2$ term. This correction allows for a unified description of primordial and late-time accelerated expansions. However, most existing models struggle to achieve this, as they become unstable over certain negative ranges of the Ricci scalar, where either the first or second derivative of $F(R)$ turns negative. These instabilities may disrupt the post-inflationary evolution when the Ricci scalar oscillates about the vacuum state after the $R^2$ inflation. In this work, we introduce a new model-building to guarantee global stability, i.e., the first and second derivatives are positive for all real Ricci scalars. By extending the idea from Appleby and Battye, we demonstrate that viable models can be constructed by imposing a positive, bounded first derivative of $F(R)$ with a sigmoid shape. As examples, we first reformulate and generalize the original Appleby-Battye model. Then, we propose a new dark energy model, which successfully explains the acceleration of cosmic expansion and passes local gravity tests.

Figures (5)

• Figure 1: Comparison of $F(R)/\Lambda$ versus $R/\Lambda$ for existing models, with all the parameters set to unity. The Starobinsky model is not shown here, as it coincides with the Hu-Sawicki model for $n=1$.
• Figure 2: Comparison of $F_{R}(R)$ (left panel) and $\Lambda F_{RR}(R)$ (right panel) versus $R/\Lambda$ for existing models, with all the parameters set to unity.
• Figure 3: Comparison of the Jordan-frame field (upper panel) and potential (lower panel) between the Hu-Sawicki model (left panel with $n=1$ and $\xi=0.1$) and our model (right panel with $n=1$ and $\xi=0.1$). Here, $\rho_\Lambda=\frac{\Lambda}{\kappa^2}$.
• Figure 4: Hubble parameter $H$ as a function of $1+z$, with $n=2.5$ and $\xi=0.01$. The lower panel shows the relative difference $\frac{H(z)-H^{\Lambda\mathrm{CDM}}(z)}{H^{\Lambda\mathrm{CDM}}(z)}$.
• Figure 5: Comparison of the EoS parameter between our model and the $\Lambda$CDM model, with $n=2.5$ and $\xi=0.01$. The upper and lower panels show the effective EoS of dark energy, $w_{\mathrm{de}}$, and the effective EoS of the entire system, $w_{\mathrm{eff}}$, respectively.