This article summarizes my work published in Chinese Physics C.¹

Research background

The accelerated expansion of the Universe can be interpreted either as the existence of an unknown dark energy component or as a modification of gravity itself. Among various modified gravity theories, $F(R)$ gravity provides an attractive framework in which cosmic acceleration emerges from the gravitational sector without introducing an additional dark energy field.

By replacing the Ricci scalar $R$ in the Einstein–Hilbert action with a general function $F(R)$, the theory introduces an additional scalar degree of freedom, known as the scalaron, which modifies the gravitational dynamics and can drive cosmic acceleration².

During the past decades, many viable $F(R)$ dark energy models have been proposed. A number of these models can reproduce a $\Lambda$CDM-like background evolution and satisfy current cosmological constraints. However, their theoretical consistency is usually examined only within the curvature range relevant to the standard cosmological evolution. Extending the analysis to a broader Ricci scalar range can expose stability problems that are not apparent along the standard cosmological trajectory^3-4.

In this work, I investigate how globally stable $F(R)$ dark energy models can be constructed, where the conditions $F_R>0$ and $F_{RR}>0$ are satisfied over the entire Ricci scalar space:

$$F_R(R)\equiv\frac{\mathrm dF}{\mathrm dR}>0,\qquad F_{RR}(R)\equiv\frac{\mathrm d^2F}{\mathrm dR^2}>0,$$

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Constructing $F(R)$ models from stability requirements

In metric $F(R)$ gravity, $F_R$ determines the effective gravitational coupling, while $F_{RR}$ is closely related to the stability of the scalaron. The standard viability conditions include

$$F_R>0,\qquad F_{RR}>0,$$

where the first condition ensures a positive effective gravitational coupling and avoids ghost-like instabilities in the gravitational sector².

Many commonly studied $F(R)$ dark-energy models are constructed by first proposing a functional form with the desired cosmological behavior and then examining whether it satisfies the stability conditions. In this work, I take the stability requirements as the starting point and construct models from the properties of $F_R(R)$:

$$F_R(R)\rightarrow F(R).$$

Specifically, I first construct $F_R(R)$ satisfying the required stability properties and then derive the corresponding $F(R)$ function through integration.

In addition to the stability conditions, a viable dark-energy model should recover general relativity at high curvature:

$$F_R(R)\rightarrow 1.$$

Therefore, $F_R(R)$ should be positive, bounded, and monotonically increasing. Sigmoid-like functions provide a convenient class of bounded monotonic functions with these properties.

It should be emphasized that a sigmoid form itself does not guarantee global stability. Rather, it provides a useful parameterization for constructing functions that satisfy the required monotonic and bounded behavior.

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The sigmoid structure of the Appleby–Battye model

The Appleby–Battye model is an important example of an $F(R)$ dark-energy construction designed to maintain stability beyond the curvature range directly relevant to the late-time cosmological evolution⁵.

Its first derivative can be written as

$$F_R(R)=\frac12\left(1+\tanh(aR-b)\right).$$

Using the relation between hyperbolic and exponential functions, this expression can be rewritten as a Logistic sigmoid:

$$F_R(R)=\frac{1}{1+b e^{-R/R_c}}.$$

Here $b$ is a newly defined positive parameter in the Logistic parameterization, not the shift parameter denoted by the same letter in the preceding hyperbolic-tangent form.

This representation provides a more transparent interpretation of the model: the Appleby–Battye construction can be viewed as an $F(R)$ model corresponding to a Logistic-type sigmoid transition in $F_R(R)$.

This viewpoint allows the construction to be generalized by considering other sigmoid-like functions with different asymptotic behaviors.

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A new family of globally stable $F(R)$ models

Based on this construction approach, I further investigate alternative sigmoid-like forms of $F_R(R)$ and construct a new family of $F(R)$ models satisfying global stability conditions.

A representative example is

$$F_R(R) = \frac12 \left( 1+ \frac{x}{\sqrt{x^2+c}} \right),$$

where $x$ is a shifted and normalized curvature variable.

After integration, the corresponding model becomes

$$F(R) = \frac{R}{2} + \frac{R_c}{2} \sqrt{ \left(\frac{R}{R_c}-b\right)^2+c } +C.$$

Unlike the Appleby–Battye model, whose high-curvature corrections are exponentially suppressed, this model exhibits power-law corrections in the high-curvature regime:

$$F(R) \rightarrow R-2\Lambda+\text{power-law correction}.$$

The resulting models satisfy

$$F_R>0,\qquad F_{RR}>0$$

over the full Ricci scalar range while maintaining desirable high-curvature behavior similar to widely studied viable models such as the Hu–Sawicki model.

The construction can be further generalized to a broader family of globally stable modified gravity models.

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Toward dark energy–inflation unification

The global stability discussed above refers to the dark-energy $F(R)$ sector before adding the inflationary correction. A further question is whether such models can be consistently combined with the $R^2$ inflationary mechanism.

A possible unified description of primordial and late-time cosmic acceleration can be written as

$$F(R)=F_{\rm DE}(R)+\frac{R^2}{6M^2},$$

where $F_{\rm DE}(R)$ describes late-time acceleration and the $R^2$ term provides the Starobinsky inflationary mechanism⁶.

For the combined inflation-plus-dark-energy model,

$$F_R^{\rm total} = F_R^{\rm DE} + \frac{R}{3M^2}.$$

After inflation, the Ricci scalar oscillates around the vacuum state and can enter the negative-curvature regime:

$$R<0.$$

In this region, the contribution from the $R^2$ term becomes negative. Therefore, global stability of the dark-energy sector alone is not sufficient to guarantee the stability of the full inflation-plus-dark-energy model.

This places additional constraints on the asymptotic behavior of $F_R(R)$ at negative curvature. Generalized sigmoid-like constructions with a non-vanishing lower bound provide one possible direction to compensate for the contribution from the $R^2$ term.

This approach provides a stability-based perspective for exploring unified dark energy–inflation models within the $F(R)$ framework.

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Research perspective

The main purpose of this work is not to construct a phenomenological model that achieves a better background fit than $\Lambda$CDM. Instead, it addresses a more fundamental model-building question:

If $F(R)$ gravity is considered as a candidate explanation for cosmic acceleration, what functional properties are required for theoretical consistency?

By constructing models from stability requirements rather than from purely phenomenological parameterizations, this approach provides an alternative perspective on viable modified gravity model building.

Future studies will investigate the observational consequences of these globally stable models through:

• cosmic microwave background observations;
• large-scale structure formation;
• weak gravitational lensing;
• inflationary dynamics and reheating.

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References

1. H. Chen, “Globally Stable Dark Energy in $F(R)$ Gravity”, Chinese Physics C 50, 015106 (2026).
   arXiv: 2411.08617 ↩

2. A. De Felice and S. Tsujikawa, “f(R) theories”, Living Reviews in Relativity 13, 3 (2010).
   arXiv: 1002.4928 ↩¹ ↩²

3. A. A. Starobinsky, “Disappearing cosmological constant in $f(R)$ gravity”, JETP Letters 86, 157–163 (2007).
   arXiv: 0706.2041 ↩

4. W. Hu and I. Sawicki, “Models of $f(R)$ Cosmic Acceleration that Evade Solar-System Tests”, Physical Review D 76, 064004 (2007).
   arXiv: 0705.1158 ↩

5. S. A. Appleby and R. A. Battye, “Do consistent $F(R)$ models mimic General Relativity plus $\Lambda$?”, Physics Letters B 654, 7–12 (2007).
   arXiv: 0705.3199 ↩

6. A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singularity”, Physics Letters B 91, 99–102 (1980). ↩

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