arXiv:2407.16787v1 Announce Type: new

Abstract: In a series of recent papers we put forward a “fractional gravity” framework striking an intermediate course between a modified gravity theory and an exotic dark matter (DM) scenario, which envisages the DM component in virialized halos to feel a non-local interaction mediated by gravity. The remarkable success of this model in reproducing several aspects of DM phenomenology motivates us to look for a general relativistic extension. Specifically, we propose a theory, dubbed Relativistic Scalar Fractional Gravity or RSFG, in which the trace of the DM stress-energy tensor couples to the scalar curvature via a non-local operator constructed with a fractional power of the d’Alembertian. We derive the field equations starting from an action principle, and then we investigate their weak field limit, demonstrating that in the Newtonian approximation the fractional gravity setup of our previous works is recovered. We compute the first-order post-Newtonian parameter $gamma$ and its relation with weak lensing, showing that although in RSFG the former deviates from its GR values of unity, the latter is unaffected. We also perform a standard scalar-vector-tensor-decomposition of RSFG in the weak field limit, to highlight that gravitational waves propagate at the speed of light, though also an additional scalar mode becomes dynamical. Finally, we derive the modified conservation laws of the DM stress energy tensor in RSFG, showing that a new non-local force emerges, and hence that the DM fluid deviates from the geodesic solutions of the field equations.

Summary:

This article discusses a new framework called “fractional gravity” that is an intermediate approach between modified gravity theories and exotic dark matter scenarios. The framework proposes a non-local interaction between dark matter and gravity, and has been successful in reproducing aspects of dark matter phenomenology. The authors then introduce a relativistic extension called Relativistic Scalar Fractional Gravity or RSFG, where the trace of the dark matter stress-energy tensor couples to the scalar curvature via a non-local operator. The article explores the weak field limit of RSFG, computes the first-order post-Newtonian parameter $gamma$, discusses weak lensing and gravitational waves in RSFG, and highlights the modified conservation laws of the dark matter stress-energy tensor in RSFG.

## Future Roadmap:

### 1. General Relativistic Extension

The first step in the future roadmap is to develop a general relativistic extension of the fractional gravity framework proposed. This will involve a deeper investigation and understanding of the coupling between the trace of the dark matter stress-energy tensor and the scalar curvature using a non-local operator constructed with a fractional power of the d’Alembertian operator.

### 2. Weak Field Limit Analysis

Once the general relativistic extension is established, it is important to study its weak field limit. The article mentions that in the Newtonian approximation, the fractional gravity setup of previous works is recovered. This analysis will involve deriving the first-order post-Newtonian parameter $gamma$ and comparing it to its values in General Relativity (GR). Additionally, the effect of weak lensing in RSFG needs to be explored to understand if it differs from GR predictions.

### 3. Scalar-Vector-Tensor Decomposition

The scalar-vector-tensor decomposition in the weak field limit of RSFG needs to be performed to examine the propagation of gravitational waves. The article suggests that gravitational waves still propagate at the speed of light in RSFG but also introduces an additional scalar mode that becomes dynamical. Understanding the behavior of gravitational waves in this framework is essential for its further development.

### 4. Modified Conservation Laws

The next step is to derive and investigate the modified conservation laws of the dark matter stress-energy tensor in RSFG. The article hints that a new non-local force emerges, suggesting that the dark matter fluid deviates from the geodesic solutions of the field equations. Understanding the implications of these modified conservation laws is crucial for evaluating the overall consistency of the RSFG framework.

## Challenges and Opportunities:

**Challenges:**Developing a general relativistic extension of the fractional gravity framework may involve complex mathematical calculations and require deeper insights into the non-local interaction between dark matter and gravity. Additionally, understanding the implications of the modified conservation laws and the behavior of gravitational waves in RSFG may pose theoretical challenges.**Opportunities:**The success of the fractional gravity framework in reproducing aspects of dark matter phenomenology presents an opportunity to explore new avenues for understanding the nature of dark matter. The RSFG extension provides a theoretical framework to investigate the weak field limit, gravitational waves, and modified conservation laws, which can lead to new insights and potentially explain observed phenomena that cannot be accounted for by traditional modified gravity or dark matter scenarios.

In summary, the future roadmap for readers includes developing a general relativistic extension of the fractional gravity framework, analyzing the weak field limit and weak lensing effects in RSFG, performing a scalar-vector-tensor decomposition to understand gravitational wave propagation, and investigating the modified conservation laws of the dark matter stress-energy tensor. While there are challenges involved, these steps present opportunities to deepen our understanding of dark matter and gravity.