Dark Matter versus MOND

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1. Introduction

Dark matter is generally believed to account for the approximately flat velocity curves characteristic of spiral galaxies. Observations of the colliding “bullet cluster” galaxies 1E0657-56 provide further evidence for the existence of dark matter. However, some physicists believe the observed flat velocity curves indicate the law of gravity must be modified at large distances according to MOdified Newtonian Dynamics (MOND). Recently Merritt [1] argued for MOND by claiming dark matter models cannot account for the acceleration threshold and the () relation emerging from the MOND approach [2] .

2. Purpose

The purpose of this paper is to evaluate the validity of Merritt’s claim by considering a specific model based on dark matter, the holographic large scale structure (HLSS) model [3] . The HLSS model was developed within the LCDM paradigm and employs the holographic principle based on thermodynamics and general relativity [4] . This note shows the HLSS model can account for both the MOND acceleration threshold and the () relation.

3. Analysis

In the HLSS model, galaxies with total mass inhabit spherical holographic

screens with radius if the Hubble constant

. The HLSS model considers galactic matter density

distributions, where r is the distance from the galactic center.

The spherical isothermal halo of dark matter, with radius and mass

, has density distribution so the dark matter mass within radius R is. There is no singularity in the galactic matter density distribution because mass inside a core volume of

radius at the galactic center is concentrated in a central black hole with

mass [3] . Radial acceleration at radius R due to dark matter is then. At radii R sufficiently distant from the galactic

center that total baryonic mass of the galaxy can be treated as concentrated at the galactic center, Newtonian radial acceleration resulting from

baryonic matter is. The radius where is found from

Since and, , and at that radius

consistent with the MOND estimate.

Another indication that the MOND acceleration is a natural scale in the dark matter based HLSS model involves the situation at the radius of the spherical holographic screen. Then the Newtonian assumption, that total galactic mass can be considered as concentrated at the galactic center, is certainly mathematically justified. There, the sum of radial acceleration from dark matter and radial acceleration from baryonic matter is

Using

then yields

equal to the estimated MOND acceleration.

The tangential velocity V at radius R is related to radial acceleration by. So, the ratio () is approximately

resulting in

Then, when,

as noted by Merritt [1] . Next, using

and

results in

When and

again as noted by Merritt [1] . Since when, using

and gives

also known as the baryonic Tully-Fisher relation.

Finally, if the Hubble constant, the cosmological constant, and the accelerations

and are both consistent with the acceleration

estimated above.

4. Conclusion

Contrary to Merritt’s claim [1] , this note demonstrates that the HLSS model [3] , based on dark matter, can account for the MOND acceleration threshold, the () relation, and the baryonic Tully-Fisher relation. After this paper was accepted for publication, I learned that Man Ho Chan previously reached the same conclusion [5] using a dark matter based analysis independent of the holographic approach used in this paper.

Acknowledgements

I thank the reviewer for important suggestions about how to improve the presentation of these results.

References

[1] Merritt, D. Cosmology and Convention. arXiv:1703.02389

[2] Famaey, B. and Mc Gaugh, S. (2012) Living Reviews in Relativity, 15, 10. arXiv:1112:3960

[3] Mongan, T.R. (2011) JMP, 2, 1544, and (2013) JMP, 4, 50.

[4] Bousso, R. (2002) Reviews of Modern Physics, 74, 825.

https://doi.org/10.1103/RevModPhys.74.825

[5] Chan, M.H. (2013) Physical Review D, 88, 103501. arXiv:1310.6801

https://doi.org/10.1103/PhysRevD.88.103501