We have explained MOND in our previous post, and how impressive this simple modification to the Newtonian laws without free parameters can explain the rotation curves of galaxies without the need for any dark matter. But perhaps the most well-known critic against MOND is that it falls short to explain the velocities or mass discrepancies found in galaxy cluster dynamics. In particular, one finds mass discrepancies of about 50-10:1 ratio (that is, that one needs even 50 times more mass that observed to explain the motion of galaxies) at the cores and 7:1 ratio at 1 Megaparsec in clusters, and with MOND, the discrepancies are reduced to 3-2:1 at 1 Megaparsec and even smaller at higher distance from the core.
This means that, although MOND helps to reduce the problem of dark matter in clusters (because most of part of the clusters are below the acceleration scale of MOND, that is, in the MOND regime), it does not solve entirely the need for dark matter.
One of the first tentative thoughts is to conclude that MOND is incomplete, and we are missing some stronger correction to Newtonian laws. This correction can’t be done in terms of accelerations, because that would change how good MOND explains galaxy rotation curves. Perhaps the modification to solve the clusters kicks in at a larger scale than galaxies? In the same way Milgrom identified that the mass discrepancy in galaxies occur at a certain acceleration scale, Jakob Bekenstein identified that the mass discrepancy in clusters when using MOND occurs at a certain potential scale, and decreases with increasing distance from the cores. Follow me in trying to extend MOND on potentials.
The first thing we need to notice is that we can’t simply modify Newton’s laws in high potentials. One could be tempted to boost the gravitational constant in terms of potentials, due to the Machian coincidence of the gravitational constant being equal to the speed of light squared divided by the potential of the universe. But this would change the strength of gravity at neutron stars (which are the highest gravitational potential objects we understand), and a variation of the gravitational constant in them is highly constrained by their internal dynamics.
So, our goal must be modifying MOND in the low accelerations and at the same time, high potential regimes, because clusters, although having higher potentials at their cores than the galaxies themselves, are around the low acceleration regime of MOND. In particular, as pointed out by Bekenstein, we need to boost the acceleration scale constant of MOND in clusters by a factor of 2 or 3 to solve the majority of the problem in clusters when using MOND.
We start with our Machian approach to MOND, which comes from equating the MOND’s acceleration scale to the gravitational potential of the universe. Now we introduce an effective boost to , so that at high accelerations, the interpolating function still reduces to unity to recover the usual Newtonian laws. We can easily do this by multiplying a function of potentials, which will be a fraction with the potential of the universe as a potential scale to make it dimensionless.
And now we can test if our extension to MOND still reduces to Newton in the solar system, to MOND inside galaxies, to no correction to high-potential high-acceleration regimes such as neutrons stars, and boosts gravity in high-potential low-acceleration regimes such as clusters by a factor equivalent to a few times a greater MOND’s acceleration scale. The resulting possibilities are the following. And to fit the data more precisely, we would expect to be a small number coming from missing integers and Pi in our equation. It turns out, that needs to be a rather big number, which doesn’t look natural.
Because of this, the potential scale of the universe doesn’t look like to be behind the extension of MOND to explain clusters. One could think that comes from another dimensionless factor, such as this one which matches the velocities of galaxies in clusters with respect to the CMB or the mass of the universe. But I think this dependence on velocities is probably ruled out, as it would result in very different predictions depending on the velocities of galaxies in the clusters.
You can already see how messy and unnatural this fix looks, and together with the fact that the potential scale of the universe is not the one behind the needed to fit clusters in this way, I did not publish it. But it turns out I wasn’t the only one who came up with fix! Bekenstein did basically the same, although he used an exponential function, instead of my function, which behaves very similar. This is due to the fact that the Lagrangian should not depend on the zero point of the field, and an exponential function achieves this. He also uses a new free parameter of a speed squared instead of the Machian approach of using the potential of the universe. Curiously, with this approach, one breaks with Newton’s shell theorem, and parts of the sources of a gravitational field distribution could be outside the region considered, which what you need to explain the bullet cluster by modifying gravity.
If you want to read more about extending MOND to fit clusters, Hongsheng Zhao is probably the researcher who has devoted the most time to it.
After all this attempt, one is tempted to abandon the idea to simply modify MOND to account for clusters, and indeed I recall Milgrom saying he doesn’t like this approach (can’t find the reference for this). The alternative is to conclude that there is some missing mass in clusters, while MOND explains galaxy rotation curves without dark matter in them. And this is reasonable for a couple of reasons. As I stated earlier, the discrepancies found in clusters are found at the cores (with high potentials), while in galaxies are found far away from the cores! This makes sense for dark matter in clusters, since dark matter would tend to clump at the cores. But it doesn’t make sense in galaxies. Also, dark matter was first observed in clusters with mass discrepancies of around 70:1, which were reduced to 7:1 after finding a lot of missing mass in the form of hot gas. Perhaps we are not done with finding more missing mass. Some MOND researchers even speculate that the dark matter in clusters doesn’t even need to be dark matter, but could be explained by massive neutrinos, which we already have in the standard model.
One of my lasts thoughts is that perhaps, for some unknown reason, the field of clusters sums to that of the field of the universe (the Machian acceleration scale constant) when studying clusters. This is effectively the boost of the acceleration scale constant in MOND needed to fit clusters, but it doesn’t work if you apply it to galaxies. And this is due to the question of why the total field intensity of the universe defines the acceleration scale constant, but only the local field intensities appear in the denominator. How does the universe differentiate between the local and global field intensities in MOND? Maybe the global field intensity is the one defining the inertial frame, and absolute accelerations are measured with respect to it á la Mach?
References:
Machian MOND: https://arxiv.org/abs/2410.19007
Jakob Bekenstein’s presentation and suggestions with clusters (scale dependent critical acceleration): • TWEAKING GENERAL RELATIVITY: MOND RELATIVI…
Hongsheng Zhao’s EMOND (extended MOND): https://arxiv.org/abs/1701.03369
Stacy McGaugh blog post about MOND and clusters:
Clusters of galaxies ruin everything