In this work we present a new framework of the gravity sector by considering

the extension $F(R,w)$, in which $R$ is the Ricci scalar and $w$ is the

equation of state. Three different choices of function $F(R,w)$ are

investigated under the Palatini formalism. The models appear equivalent to

$F(R)$ models of gravity with effective momentum-energy tensors. For linear

dependence of Ricci scalar in which $F(R,w)=k(w)R$, the model appears

equivalent to Einstein-Hilbert action with effective momentum-energy tensor.

Recovering the minimal coupling case of the last choice does not face

Jordan-Einstein frame ambiguities and exhibits natural alignments with general

relativity results in the mattertext{/} radiation dominated eras. We discuss

some astrophysical implications of the model by considering scalar fields as

dominant matter forms. We show that the Higgs inflation could be saved within

the $F(R,w)$ model. We suggest some future investigations exemplified by

constant-roll inflation and universe evolution for $F(R)=f(R)k(w)$ where $f(R)$

represents the Starobinsky gravitational form. Using the model and comparing it

with pure $F(R)$ gravity, we provide preliminary indications of $F(R,w)$’s

impact. As a final note, we suggest using the Polytropic equation of state in

future works to investigate $F(R,w)$.

**Conclusion:**

- A new framework of the gravity sector, considering the extension $F(R,w)$, has been presented in this work.
- Three different choices of function $F(R,w)$ have been investigated under the Palatini formalism.
- These models appear equivalent to $F(R)$ models of gravity with effective momentum-energy tensors.
- For the linear dependence of Ricci scalar ($F(R,w)=k(w)R$), the model is equivalent to the Einstein-Hilbert action with an effective momentum-energy tensor.
- The minimal coupling case of the last choice does not face Jordan-Einstein frame ambiguities and aligns naturally with general relativity results in the matter/radiation dominated eras.
- Astrophysical implications of the model have been discussed, considering scalar fields as dominant matter forms.
- The Higgs inflation can be saved within the $F(R,w)$ model.
- Potential future investigations include constant-roll inflation and universe evolution for $F(R)=f(R)k(w)$, where $f(R)$ represents the Starobinsky gravitational form.
- Preliminary indications of $F(R,w)$’s impact have been provided by comparing it with pure $F(R)$ gravity using the model.
- The use of the Polytropic equation of state is suggested for future works to investigate $F(R,w)$.

**Future Roadmap:**

Readers who are interested in exploring further research in the gravity sector with the extension $F(R,w)$ can consider the following potential roadmap:

## Potential Challenges:

- Understanding the implications and limitations of the different choices of function $F(R,w)$ under the Palatini formalism.
- Investigating how the models in the gravity sector with $F(R,w)$ relate to existing $F(R)$ models of gravity.
- Addressing the challenges and ambiguities in the minimal coupling case of the $F(R,w)$ model.
- Exploring the astrophysical implications of the $F(R,w)$ model with dominant scalar fields as matter forms.
- Validating and further studying the potential impact of the $F(R,w)$ model on the Higgs inflation.

## Potential Opportunities:

- Investigating the constant-roll inflation and universe evolution for $F(R)=f(R)k(w)$, with a focus on the Starobinsky gravitational form.
- Comparing and analyzing the effects of the $F(R,w)$ model with pure $F(R)$ gravity to understand its potential advantages or disadvantages.
- Exploring the use of the Polytropic equation of state in future works to investigate the behavior and properties of $F(R,w)$.

By following these potential pathways, researchers can contribute to a deeper understanding of the gravity sector and its extensions through the $F(R,w)$ framework, uncovering new insights and discoveries.

**Reference:** The conclusions and roadmap outlined in this article are based on the work “A New Framework for Modeling Gravity: F(R,w)” by the authors (please provide full citation details).