in a second phase. Likewise the velocity of the disk varies as $\frac{\kappa_E}{\kappa}$ (increasing in a first phase and decreasing in a second phase). In fact the motion perceived by the terrestrial observer is fictitious, the real velocity is constant (free particle).

# 5 Conclusion

We have presented in this paper the formalism of the $\kappa$-model. Astonishingly this formalism is fairly straightforward when compared to other formalisms which try to modify the standard general relativity [5, 6, 7]. The $\kappa$-model is thus characterized by its relative simplicity. Furthermore any type of modified gravity necessarily leads to the introduction of new fields and surreptitiously to new particles. Thus all the various versions of modified gravity, with their lot of new unknown particles, even seems much more complex compared to the dark matter paradigm (Newtonian in essence), apparently for a very limited gain in the understanding. However the number of parameters in the modified gravity models is generally restricted to two or three. It is a very good point, also regardless of the type of modified gravity model, it can be considered as a predictive one whereas dark matter, with also two or three parameters but different for each galaxy, is not at all predictive, but merely postdictive. By contrast, it is indisputable that the MOND model [8, 9], at least in its original form, and characterized by an unique universal parameter, $a_0$, still constitutes even today the better way to predictively generate the rotational curves of individual galaxies. Unfortunately some issues persist for MOND at the scale of the galactic clusters, even though some solutions have been proposed to circumvent the problem [10]. The $\kappa$-model has the same level of efficiency as MOND to fairly predict the profiles of the rotational curves of individual spiral galaxies [4], but moreover can also help to understand the physics of galactic clusters [3]. Other interesting ways have also been suggested to circumvent the dark matter conundrum [11, 12, 13, 14, 15]. We have already noticed that the latter theories share some common points with the $\kappa$-model [4]. Eventually we can report that, in parallel, a study focusing on the cosmological implications of the $\kappa$-model is in progress.

**Data availability statement:** The authors confirm that the data supporting the findings of this study are available within the article and the reference list.

**Conflicts of Interest:** The authors declare no conflict of interest.

# 6 References

[1] Pascoli, G., and Pernas, L., 2020, hal.archives-ouvertes.fr/hal-02530737

[2] Pascoli, G., 2022, Astrophys. Space Sci., 367, 121

[3] Pascoli, G., 2023, Can. J. Phys., 101, 11

[4] Pascoli, G., 2024, Universe, 10(3), 151

[5] Moffat, J. W., 2006, Journal of Cosmology and Astroparticle Physics, 3, 4

[6] Mannheim, P.D., Prog.Part.Nucl.Phys. 2006, 56, 340

[7] Li, Q., and Modesto, L., Grav.Cosmol., 2020, 26, 99

[8] Milgrom, M., Astrophysical Journal, 1983, 270, 365
in a second phase. Likewise the velocity of the disk varies as $\frac{\kappa_E}{\kappa}$ (increasing in a first phase and decreasing in a second phase). In fact the motion perceived by the terrestrial observer is fictitious, the real velocity is constant (free particle).

# 5 Conclusion

We have presented in this paper the formalism of the $\kappa$-model. Astonishingly this formalism is fairly straightforward when compared to other formalisms which try to modify the standard general relativity [5, 6, 7]. The $\kappa$-model is thus characterized by its relative simplicity. Furthermore any type of modified gravity necessarily leads to the introduction of new fields and surreptitiously to new particles. Thus all the various versions of modified gravity, with their lot of new unknown particles, even seems much more complex compared to the dark matter paradigm (Newtonian in essence), apparently for a very limited gain in the understanding. However the number of parameters in the modified gravity models is generally restricted to two or three. It is a very good point, also regardless of the type of modified gravity model, it can be considered as a predictive one whereas dark matter, with also two or three parameters but different for each galaxy, is not at all predictive, but merely postdictive. By contrast, it is indisputable that the MOND model [8, 9], at least in its original form, and characterized by an unique universal parameter, $a_0$, still constitutes even today the better way to predictively generate the rotational curves of individual galaxies. Unfortunately some issues persist for MOND at the scale of the galactic clusters, even though some solutions have been proposed to circumvent the problem [10]. The $\kappa$-model has the same level of efficiency as MOND to fairly predict the profiles of the rotational curves of individual spiral galaxies [4], but moreover can also help to understand the physics of galactic clusters [3]. Other interesting ways have also been suggested to circumvent the dark matter conundrum [11, 12, 13, 14, 15]. We have already noticed that the latter theories share some common points with the $\kappa$-model [4]. Eventually we can report that, in parallel, a study focusing on the cosmological implications of the $\kappa$-model is in progress.

**Data availability statement:** The authors confirm that the data supporting the findings of this study are available within the article and the reference list.

**Conflicts of Interest:** The authors declare no conflict of interest.

# 6 References

[1] Pascoli, G., and Pernas, L., 2020, hal.archives-ouvertes.fr/hal-02530737

[2] Pascoli, G., 2022, Astrophys. Space Sci., 367, 121

[3] Pascoli, G., 2023, Can. J. Phys., 101, 11

[4] Pascoli, G., 2024, Universe, 10(3), 151

[5] Moffat, J. W., 2006, Journal of Cosmology and Astroparticle Physics, 3, 4

[6] Mannheim, P.D., Prog.Part.Nucl.Phys. 2006, 56, 340

[7] Li, Q., and Modesto, L., Grav.Cosmol., 2020, 26, 99

[8] Milgrom, M., Astrophysical Journal, 1983, 270, 365
[9] Milgrom, M., Studies in History and Philosophy of Modern Physics, 2020, 7,170

[10] Hodson, A.O., and Zhao, H., A&A, 2017, 598, A127

[11] Varieschi, G.U., J.Appl.Math.Phys., 2018 06, 1247

[12] Varieschi, G.U., Found.Phys., 2020, 50 11, 1608

[13] Cesare V., Diaferio A., Matsakos T., and Angus, G., 2020, A&A, 637, A70

[14] Varieschi, G.U., Universe, 2021, 10, 387

[15] Varieschi, G.U., 2021, Eur. Phys. J. Plus, 136, 183
[9] Milgrom, M., Studies in History and Philosophy of Modern Physics, 2020, 7,170

[10] Hodson, A.O., and Zhao, H., A&A, 2017, 598, A127

[11] Varieschi, G.U., J.Appl.Math.Phys., 2018 06, 1247

[12] Varieschi, G.U., Found.Phys., 2020, 50 11, 1608

[13] Cesare V., Diaferio A., Matsakos T., and Angus, G., 2020, A&A, 637, A70

[14] Varieschi, G.U., Universe, 2021, 10, 387

[15] Varieschi, G.U., 2021, Eur. Phys. J. Plus, 136, 183