arXiv:2408.07711v1 Announce Type: new

Abstract: In the non-metric gravity proposed by K. Krasnov, this is a theory in which the scalar constraint of the Ashtekar’s formalism is modified in such way that the cosmological constant is replaced by a cosmological function that depends of the canonical variables $A_{a}{}^{i}$ and $E_{i}{}^{a}$, the General Relativity (GR) is the particular case when the cosmological function is a constant. Some years ago inspired by this theory Rosas-Rodr’iguez proposed two cosmological functions, one for the Ashtekar’s formalism and the other for the ADM formalism. In this paper we show that this cosmological functions are related thought the three-dimensional Ricci scalar.

## Non-Metric Gravity and Modified Scalar Constraint

The paper discusses the non-metric gravity theory proposed by K. Krasnov, which modifies the scalar constraint in Ashtekar’s formalism. In this theory, the cosmological constant is replaced by a cosmological function that depends on the canonical variables $A_{a}{}^{i}$ and $E_{i}{}^{a}$. The theory encompasses General Relativity (GR) as a particular case when the cosmological function is a constant.

## Cosmological Functions in Ashtekar’s and ADM Formalism

Sometime earlier, Rosas-Rodr’iguez proposed two cosmological functions, each corresponding to a different formalism. One cosmological function is for the Ashtekar’s formalism, and the other is for the ADM formalism.

## Linking Cosmological Functions through Ricci Scalar

This paper shows that the two cosmological functions proposed by Rosas-Rodr’iguez are related through the three-dimensional Ricci scalar.

## Roadmap for Future Research

Based on the conclusions of this paper, there are several potential directions for future research in the field of non-metric gravity:

- Further Investigating Cosmological Functions: Researchers can delve deeper into the properties and implications of the cosmological functions proposed by Rosas-Rodr’iguez. Understanding the relationship between these functions and the three-dimensional Ricci scalar is a promising avenue for exploration.
- Extending the Theory: While this paper establishes the link between the cosmological functions and the Ricci scalar, there may be scope for extending the theory to include additional variables or constraints. Researchers can explore how modifying these components may further refine and expand the theory.
- Comparison with Observational Data: It is crucial to compare the predictions of non-metric gravity with observational data to assess its viability as an alternative to GR. Future research should focus on devising experiments or data analysis techniques that can test the predictions made by the theory.
- Applications in Cosmology: Non-metric gravity and the modified scalar constraint have the potential to offer new insights into the behavior of the universe at large scales. Researchers can investigate the implications of this theory for cosmological phenomena such as the expansion of the universe, dark matter, and dark energy.

## Challenges and Opportunities

While the study of non-metric gravity and the modified scalar constraint opens exciting possibilities, there are also challenges to consider:

- Mathematical Complexity: Non-metric gravity involves complex mathematical calculations and equations. Researchers need to have strong mathematical and computational skills to navigate and analyze the theory effectively.
- Data Limitations: Validating the predictions of non-metric gravity requires accurate and precise observational data. However, obtaining such data may be challenging, particularly for phenomena that occur at large scales or involve extreme conditions.
- Peer Review and Collaboration: To establish non-metric gravity as a credible and widely accepted theory, it is essential to engage in rigorous peer review and collaborate with experts in the field. Researchers should actively seek opportunities to present their work, publish in reputable journals, and participate in conferences and workshops.

In conclusion, the paper establishes the relationship between cosmological functions in non-metric gravity and the three-dimensional Ricci scalar. The roadmap for future research includes further investigating cosmological functions, extending the theory, comparing with observational data, and exploring applications in cosmology. While there are challenges in terms of mathematical complexity, data limitations, and peer review, the opportunities for advancing our understanding of gravity and cosmology make this an exciting field of study.