### by **Joel Beckles** with extensive commentary provided by **Dr. Indranil Banik**

A recent publication in the Monthly Notices of the Royal Astronomical Society (MNRAS) featuring authors from the University of St Andrews has sent shockwaves through the astronomical community. One of General Relativity’s contenders for the theory of gravity, Milgromian Dynamics, has suffered a massive, and possibly decisive, blow.

The results of a very thorough test of the predictions of Milgromian Dynamics, also known as Modified Newtonian Dynamics (MOND), have been published^{1} by our very own Dr. Indranil Banik (lead author) and Dr. Hongsheng Zhao, along with international collaborators. The evidence presented suggests that MOND is in very strong conflict with observational data. In this article, we get exclusive insights from Dr. Banik, who is a Galaxy Dynamics Research Fellow at St Andrews.

## The Birth of MOND: Why Modified Gravity Laws were First Considered

Milgromian Dynamics, often referred to by the acronym MOND, was a theory put forward (in publication) by physicist Mordehai Milgrom in 1983. Interestingly, a conference was held in St Andrews in commemoration of the theory’s 40th anniversary in June 2023^{2}. Roughly speaking, the original theory aimed to remove the need for dark matter in our understanding of galaxy dynamics.

MOND initially sought to explain a popular mystery in astrophysics: why stars’ orbital velocities remain fairly constant with distance away from their galactic centre. Normally, one expects a body’s orbital velocity to be slower the further away it is from its centre of orbit. (Think of the effect of string length when rotating a ball on a string.) In this regard, the dynamics of galaxies presented an unexpected result to astrophysicists. The most popular proposal to resolve this issue is that there is something affecting galaxy dynamics other than the visible matter we observe: dark matter. MOND, however, suggests something different: that the laws of gravity differ from Newtonian expectations at extremely low accelerations in a manner which can explain the observed orbital velocities of stars.

Over the past 40 years, the theory and variants thereof have been developed by many scholars, including Dr. Banik. A good summary of these developments is provided by Banik and Zhao, 2022^{3}.

## Testing MOND

Many tests of MOND have been discussed previously and there has been debate over the years about different aspects of the theory’s validity. Dr. Banik explains, however, why the latest evidence presented by him and his collaborators can be considered so decisive:

MOND postulates that gravity departs from classical expectations below an acceleration scale 𝑎

_{0}. This means it predicts anomalous behaviour not just in galaxies but even on stellar scales. In particular, widely separated binary stars (wide binaries) in the Solar neighbourhood are expected to orbit 20% faster than expected in Newtonian gravity.Instead of rehashing the technicalities, I will take this opportunity to explain some aspects which are not often discussed. The wide binary test is a particularly critical decisive test of MOND. Since I have been working on MOND for almost a decade, it was always clear that emotions would run high with such a decisive test. This meant a very high risk of moral hazards. I therefore took extreme precautions to mitigate the moral hazards associated with the wide binary test. The main thing was to fix the analysis protocol as much as possible in advance of the actual test, so we were not tempted to alter modelling choices to try and change the result. For this purpose, the detailed plan was posted on https://arxiv.org/abs/2109.03827 at the start of my current postdoc

^{4}, whose main objective is to implement the wide binary test. This plan mostly focuses on the computational cost of the test and how much memory it will use, since the test is also rather numerically taxing. But certainly an important consideration was to fix the protocol as much as possible. In the paper, I have explained that some very minor changes and simplifications were made to this protocol because the actual Gaia DR3 worked a bit differently to what was expected, with some things working better than expected – making us use alternative protocols that are better in order to exploit this.

## Diving Deeper

To further understand the analysis of this test of MOND, a basic introduction to the concept of Bayesian statistics might be useful. When thinking about probability, one might think about the likelihood of an event occurring (e.g. rolling 4 on a 6-sided die) over a large number of samples. This is known as frequentist statistics. In this approach, probabilities are assigned to specific outcomes. In contrast, Bayesian statistics assigns probabilities to hypotheses. This framework allows us to consider how probable it is that a hypothesis is correct – very useful for testing the validity of MOND. Bayesian statistics assumes some prior probability distribution based on pre-existing knowledge. This prior distribution is then modified by compromising with real-world data to create a new probability distribution known as the posterior.^{5}

Dr. Banik explains to us the significance of the posterior in the tests of MOND when considering a key model parameter known as 𝛼_{𝑔𝑟av}:

The most important thing was that the criteria for various possible outcomes were clear in advance. If the posterior on the 𝛼

_{𝑔𝑟𝑎𝑣}parameter strongly clusters near 0, this means Newtonian gravity is preferred. If 𝛼_{𝑔𝑟𝑎𝑣}clusters near 1, then MOND is preferred. A different value would mean some other gravity law is in operation, or there are some other issues with the wide binary test. A priori, it was expected that the result would be either 0 or 1, even though values in the range -2 to +3.6 were allowed. It was assumed that the posterior would be tight enough to reliably distinguish between 0 and 1, so both gravity laws would not simultaneously be consistent with the wide binary test.

## The Big Result: The Meticulous work Behind Testing MOND

Conducting an analysis to such a level of rigour as outlined above was not a trivial task. Dr. Banik describes in detail his discovery of the fate of MOND and the impact it had within the astronomical community:

A clear Newtonian result became apparent in the evening of fifth November 2022, with Newtonian gravity preferred over MOND at about 20 sigma confidence. The posterior on 𝛼

_{𝑔𝑟𝑎𝑣}was consistent with 0 at about 1 to 2 sigma confidence. This result was reported to the other authors. Given the significant implications, the head of department was rapidly informed. It has taken a while to write up the results and get them peer reviewed, but the basic result has barely changed.

The wide binary test is the most complicated statistical hypothesis test I have ever conducted, by a very large margin. The main stage is an approximately eight hundred line block of Fortran code that determines the absolute binomial likelihood of each model. This is done at a marginal cost of about two seconds, so a full length one hundred thousand element Markov Chain Monte Carlo analysis takes a bit over two days. The calculations were done on a computing cluster in Bonn, which is well suited to the test. Parallel programming is used, assuming about sixty cores are available, as was typically the case (this is half the cluster, letting other users work on the cluster as well).

It has recently become clear that the failure of MOND on the sub-galactic scales of wide binary stars is also apparent in Cassini radio tracking data of the Earth-Saturn distance. MOND predicts some anomalies here, for much the same reasons as it does for wide binaries. This is observationally excluded at about eight sigma confidence thanks to timing of signals between Earth and the Cassini spacecraft, getting the relative distance very precisely. Therefore, MOND does not work on scales smaller than about one parsec. MOND also fails significantly in galaxy clusters, where it predicts too little gravity in the inner regions and too much gravity in the outskirts.

## The Death of MOND (?)

The paper by Banik et. al. revealed a conflict between MOND and observational data amounting to 16 times the uncertainty^{1}. Evidence to this extent seems to strongly suggest that this alternative theory of gravity is fundamentally flawed. Dr. Banik shares his thoughts on the matter:

In the future, I will not be working further on MOND given these insurmountable difficulties. Once the Solar System results

^{6}are published in about six months to a year by another group I am not involved with, MOND will not be an investable proposition. Theories that fail Solar System constraints are automatically uninvestable.

While this is not the outcome that I envisaged when I first started working on MOND a decade ago, I am proud to have conducted such a detailed test of what at the time was a very plausible hypothesis. The failure of the only viable alternative to dark matter on galaxy scales significantly strengthens the case for galaxies having their own dark matter halos.

## Moving Forward: Modified Gravity and the Hubble Tension

General Relativity (GR) still appears to be our strongest model of gravity. Although modifying the standard model of gravity has proven problematic in the realm of galaxy dynamics, the possibility for modifications to gravity at far larger scales has not been ruled out. Dr. Banik alerted us to another of his recent publications^{7} where he and collaborators propose such a modification to resolve another mystery in astrophysics known as the Hubble Tension. Simply put, the Hubble tension is a disagreement within the astrophysics community on the rate at which the Universe is expanding. Different, seemingly valid, methods for its calculation give conflicting results.

Dr. Banik believes that modifying gravity on scales of the order of 10 megaparsecs (Mpc) may be the key to resolving this conflict:

I think there is a very good chance of deviations from GR on those scales. Obviously, this is a huge extrapolation of GR from the Solar System scales for which it was designed. And this extrapolation does cause very real problems like the Hubble tension. Modified gravity on those scales is quite reasonable. I got into it in one particular way, but one does not have to believe that galaxies are purely baryonic to think that gravity might be modified.

## Further Reading and Resources

For a further breakdown on these potentially decisive tests of MOND, Dr. Banik recommends this YouTube video, made independently by Dr. Rebecca Smethurst, as an excellent resource: https://www.youtube.com/watch?v=HlNSvrYygRc.

Dr. Banik’s recent article for the Institute of Arts and Ideas is also a great read on this topic: https://iai.tv/articles/the-challenge-to-dark-matter-mond-is-wrong-auid-2676?_auid=2020

See here as well for a recent article in The Conversation by Dr. Banik on the Hubble Tension: https://theconversation.com/do-we-live-in-a-giant-void-it-could-solve-the-puzzle-of-the-universes-expansion-216687.

A department talk by Dr. Banik about the Hubble Tension can be found here: https://www.youtube.com/watch?v=fsBykeVXHG0.

## References and Footnotes

- Banik I, Pittordis C, Sutherland W, Famaey B, Ibata R, Mieske S, et al. Strong constraints on the gravitational law from gaia DR3 wide binaries. Monthly Notices of the Royal Astronomical Society. 2023;527(3):4573–4615. doi:10.1093/mnras/stad3393 ↩︎
- Celebrating 40 years of Milgromian Dynamics and charting the road ahead: [Internet]. [cited 2023 Dec 31]. Available from: http://star-www.st-and.ac.uk/~hz4/MOND40Conf2023/index.html ↩︎
- Banik I, Zhao H. From galactic bars to the Hubble tension: Weighing up the astrophysical evidence for Milgromian Gravity. Symmetry. 2022;14(7):1331. doi:10.3390/sym14071331 ↩︎
- Banik I, Pittordis C, Sutherland W. Detailed numerical implementation of the wide binary test [Internet]. 2021 [cited 2023 Dec 31]. Available from: https://doi.org/10.48550/arXiv.2109.03827 ↩︎
**Footnote:**For a relatively gentle introduction to Bayesian statistics, see these articles: Understanding the Differences Between Bayesian and Frequentist Statistics – International Journal of Radiation Oncology, Biology, Physics (redjournal.org) and A Gentle Introduction to Bayesian Analysis: Applications to Developmental Research – Schoot – 2014 – Child Development – Wiley Online Library. ↩︎- Desmond H, Hees A, Famaey B. On the incompatibility of the radial acceleration relation and solar system quadrupole in modified Gravity Mond [Internet]. 2024 [cited 2024 Jan 13]. Available from: https://arxiv.org/abs/2401.04796 ↩︎
- Mazurenko S, Banik I, Kroupa P, Haslbauer M. A simultaneous solution to the Hubble tension and observed bulk flow within 250
*h*^{-1}Mpc. Monthly Notices of the Royal Astronomical Society. 2023;527(3):4388–96. doi:10.1093/mnras/stad3357 ↩︎