# “Mass Discrepancy-Acceleration Relation” in ΛCDM Paradigm

Rotation curves of galaxies seem to require more mass than observed - the conventional solution is the introduction of Dark Matter.

Reading some articles on this topic (in particular a recent philosophical investigation of cosmology) showed a remarkable relation between the apparent missing mass of galaxies and the acceleration at that position of the galaxy (Fig 2 in the article):

The x-axis is the acceleration, and the y-axis stands for the apparent mass-discrepancy. The data-points stem from many different galaxies and positions within those galaxies As it is known, the further out [=smaller acceleration] one looks, the larger the apparent requirement for dark matter.

Discovery of this particular plot go back to a paper in 1990 by Sanders. Unfortunatly I was not able to find any theoretical explanation from the standard-model of cosmology ΛCDM. Why would dark matter behave exactly like that - as far as I understand, predictable from visibile matter?

Concretely, my two questions:

1. How can this phenomenon be described with ΛCDM?
2. Is it related to a recent article on rotational velocities in PRL, which finds also very simple empirical laws on galaxy rotations without the need of dark matter?

The mass-discrepancy-acceleration relation is one of the many ways people have been observing that there is an acceleration scale in the rotation curves of galaxies. To see this more clearly, you can realise that the velocity and acceleration are related as $V^2/r = a$, so that $V^2/V_{N}^2 = a/a_N$, where $a$ is the effective radial acceleration. Your plot is then talking about the relation between the observed and expected radial acceleration. This is also the subject of the paper of McGaugh, Lelli, and Schombert from (2016) which you link, so I will treat both these results as the same issue.

One of the historically earliest observations of these kind of relations would be the baryonic Tully-Fisher relation discovered in 1977. What one essentially sees in the rotation curves is that for gravitational acceleration well above a certain $g_0 \approx 10^{-10} m s^{-2}$, the rotation curves agree with the one computed from Newtonian gravity and the presence of matter, and for accelerations where Newtonian theory would predict acceleration well below $g_0$, there is a discrepancy.

This lead M. Milgrom to publish a paper in 1983 proposing a possible modification of Newtonian gravity which implements this acceleration scale in a framework called MOND. By fitting this one parameter $g_0$, MOND is then able to reproduce any of the mass-discrepancy-acceleration or Tully-Fisher relations almost perfectly. The paper of McGaugh, Lelli, and Schombert from (2016) does a fit closely analogous to what Milgrom did, just on new data.

In the framework of $\Lambda$CDM, one can explain this tight acceleration relation across galaxies as emergent from a combination of non-linear and dissipative effects. Specifically, Keller & Wadsley (2017) included dissipative collapse of baryons in their cosmological simulation and obtained exactly the relations you link to. I would not say that this issue is entirely settled, and surely more work has to be invested to fully understand this universal acceleration scale in cosmology.

• @NiceDean Well, MOND is exactly taking the acceleration scale identifiable in the Mass-Dicrepancy-Acceleration relation and implementing it in some kind of theory. Additionally the PRL paper you link essentially uses a MOND prescription of this sort. I thought that this was obvious, but perhaps I should clarify this. – Void Dec 10 '17 at 19:10
• @NiceDean I see, you can then understand the comments on MOND as redundant, the paper of Keller & Wadsley I link and shortly comment on still provides an answer to your question. – Void Dec 10 '17 at 19:51