F(R) Gravity To Explain Dark Matter Phenomena

One of the most well studied variants of conventional general relativity is f(R) gravity. As the introduction to the paper below explains:

f(R) gravity is a straightforward extension of General Relativity (GR) where, instead of the Hilbert-Einstein action, linear in the Ricci scalar R, one considers a power-law f(R) = f(0n)R^n in the gravity Lagrangian. In the weak field limit, a gravitational potential is of the form:

This paper argues that this modification to general relativity can recover the baryonic Tully-Fisher relation which is also produced by MOND in the context of galaxy dynamics, but in the naturally relativistic and mathematically consistent framework of f(R) gravity (it is not the first paper to do so). The money chart is this one:

The conclusion of the paper explains that:

In this paper we use f(R) theories of gravity, particularly power-law Rn gravity, and demonstrate that the missing matter problem in galaxies can be addressed by power-law Rn gravity. Using this approach, it is possible to explain the Fundamental Plane of elliptical galaxies and the baryonic Tully-Fisher relation of spiral galaxies without the DM hypothesis. Also, we can claim that the effective radius is led by gravity and the whole galactic dynamics can be addressed by f(R) theories. Also, f(R) gravity can give a theoretical foundation for rotation curve of galaxies. We have to stress that obtained value for parameter β from galactic rotation curves or BTF differs from parameter β obtained using observational data at planetary or star orbit scales. The reason for this result is that gravity is not a scale-invariant interaction and then it differs at galactic scales with respect to local scales. 

Also, we investigated some forms of TFR in the light of f(R) gravities. These investigations are leading to the following conclusions: 

- f(R) gravity can give a theoretical foundation for the empirical BTFR, 

- MOND is a particular case of f(R) gravity in the weak field limit, 

- ΛCDM is not in satisfactory agreement with observations, 

- FP [i.e. the Fundamental Plane of elliptical galaxies] can be recovered by Rn gravity.

The paper and its abstract are as follows:

Here we use the samples of spiral and elliptical galaxies, in order to investigate theoretically some of their properties and to test the empirical relations, in the light of modified gravities. We show that the baryonic Tully-Fisher relation can be described in the light of f(R) gravity, without introducing the dark matter. Also, it is possible to explain the features of fundamental plane of elliptical galaxies without the dark matter hypothesis.

V. Borka Jovanović, D. Borka, P. Jovanović, "The baryonic Tully-Fisher relation and Fundamental Plane in the light of f(R) gravity" (April 15, 2025) arXiv:2504.11135 (Accepted for publication in Contrib. Astron. Obs. Skalnate Pleso https://doi.org/10.31577/caosp.2025.55.2.24).