With evidence mounting against the standard model of cosmology, Modified Newtonian Dynamics, or MOND, is the strongest contender. But in a recent iai News article, Idranil Banik argued that new research into wide binary stars falsifies MOND. Here, Pavel Kroupa and Jan Pflamm-Altenburg argue that data around wide binary stars are beset by uncertainty, bringing into question what, if anything, can be determined from them at the present – and ask whether a single test, even if carried out to the highest confidence, is sufficient to falsify an otherwise successful theory.

There is increasing evidence that Newton's universal law of gravitation does not work for astronomical systems. The leading contender for a better theory that has made remarkable predictions and has been shown to naturally account for a large range of astronomical observations is Milgromian dynamics, or MOND. In a recent iai News article, Indranil Banik argues that MOND is wrong, based on predicting forces in wide binary stars. These are systems of two paired stars gravitationally bound in a distant orbit around one another. But the quality of data we have for wide binary stars is unreliable  – is it therefore fair to throw out MOND, which otherwise works better than Newtonian gravitation, solely on this basis? And how much can we infer from one single test, even if that test appears to have high validity?

MOND stands for Modified Newtonian Dynamics and was developed by Mordehai Milgrom in 1983. By modifying Newton’s second law, MOND can extend Newton’s theory of gravity to more distant regions in space than those Newton and Einstein could observe. It offers an alternative to dark matter in seeking to explain the failure of Newtonian physics in describing galaxies.

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The motion of two stars around each other in a very wide binary system can be used to test between Newton's and Milgrom's theories [1].  These wide binary stars are 2000-10000 astronomical units apart – or 2000-10000 times the distance between Earth and the Sun – with orbital times between 100000 and 1 million years. Wide binary stars are a significant test of gravitational theory because any deviation they make from Newton's law cannot be explained by dark matter, since dark matter cannot attach itself to stars that, like those in wide binary systems, have too feeble a gravitational pull. According to the widely used mathematical formulations of Milgromian gravitation, AQUAL and QUMOND, the velocity differences between the two stars in many wide binary systems should statistically be about 20% larger than expected from Newtonian gravitation.

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Frankly, entertaining the possibility that standard model is valid has become rather unscientific. We therefore need to develop a new model.

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On the 21st of November 2023 Indranil Banik from St. Andrews University published an iai article, based on his recent paper linked below [2], in which he argues that MOND is wrong. Banik et al. report that Newtonian gravitation works better than Milgromian gravitation, but they do not quantify if the data they rely on is consistent with Newtonian gravitation. Two other independent teams, around Xavier Hernandez [3,4] from Mexico City and Kyu-Hyun Chae [5,6] from Seoul use similar data, but unlike Banik et al., they come out strongly in favour of Milgromian gravitation instead. These disagreements inspired our team in Bonn to investigate the Milgromian problem in which a few stars orbit about each other.

But first, we need to remember the context for this problem: gravitation is the least understood physical phenomenon. The currently favoured standard model of cosmology is based on Einsteinian/Newtonian gravitation and needs to postulate inflation, dark matter and dark energy in attempting to account for astronomical observations ranging from the Solar System to the Cosmic Microwave Background. Currently it still remains favoured by the majority of astronomers and physicists.  But anyone arguing the standard model remains viable does so with overwhelmingly little confidence based on 32 tests [7]. Many of these are deemed individually reliable in reaching the five sigma confidence threshold – however, in combination, the statistical confidence is extremely weak with a remaining confidence of 10^{-134}.

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Frankly, entertaining the possibility that standard model is valid has become rather unscientific. We therefore need to develop a new model.  An important aspect is the falsification of its dependence on the supposed existence of cold, warm or fuzzy dark matter particles.  It is therefore not surprising that dark matter has not been discovered -– it is hard to find something that does not exist. We need to develop a new model, and in doing so take into account that gravitation needs to be effectively stronger under certain conditions such that an observer applying Einsteinian/Newtonian dynamics would (wrongly) deduce there to be dark matter in order to provide the extra gravitational pull.

Milgrom [8] started the paradigm shift away from Einsteinian/Newtonian gravitation. Bekenstein & Milgrom [9] later formulated the first full-fledged MOND theory, called AQUAL – (other MOND theories have been proposed, see [10] below). This is for the non-relativistic limit, and MOND appears to be related to the quantum vacuum [11].  AQUAL is a generalised non-linear formulation related to the p-Laplacian well known by mathematicians. It describes how a source of gravitation (the matter) shapes space around it (the potential [7]). Other matter around the source will fall towards the source following the gradient of the potential.

In MOND, the external potential from the Milky Way plays an important role in how outer minor planets and comets orbit the Sun, and how stars orbit in their open star cluster such as in the Hyades. These predictions appear to be well confirmed. There is good evidence that these predictions are correct [12,13, 14]. Concerning whole galaxies, it is well established that their observed properties are also exceedingly well accounted for by MOND. MOND predicted their fundamental properties, later to be verified by observations [10].

On the scale of galaxy clusters, some tensions appear – in that many of them seem to require approximately two times more mass than is observed in hot gas and stars. But this tension may be related to these objects being the largest structures that are gravitationally bound with many of them still actively assembling from cosmological processes. And, some cool, unobservable gas may well reside in them. A recent calculation may resolve the apparently missing mass in galaxy clusters, assuming MOND to be the correct theory [15].

The Hubble Tension – the problem that the present measurements of the universe’s expansion rate significantly outstrips its predicted rate – is another argument in favour of a MOND-driven explanation out to distances of some 1.5 billion light years. This tension is naturally resolved – it actually does not occur – in a MOND-based cosmological model in which there are density differences between regions spanning some three billion light years. These grow through galaxy flows induced by the evolving large-scale gravitational potential [16,17,18].

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A more plausible explanation for MOND’s failure to account for wide binaries might thus lie in the uncertain data handling and the calculations necessary to formulate what MOND is predicting.

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But according to Indranil Banik and co-authors [2], MOND has been falsified by the wide-binary-star test, which poses an interesting problem. We thus have the calamity that MOND appears to be right for the outer Solar System, wrong for very wide binary stars, and right again for open star clusters, galaxies, and probably galaxy clusters and even solves the Hubble Tension.

A more plausible explanation for MOND’s failure to account for wide binaries might thus lie in the uncertain data handling and the calculations necessary to formulate what MOND is predicting.  Problems to be considered are: Which are the wide binaries amongst the billion stars in the Gaia catalogue? Are the extracted data we are using of sufficient quality? Do they underlie subtle systematics, such as the wide binary consisting of a tight (unrecognised) binary star orbited by the other star in the wide orbit? What is the astrophysical state of the stars involved? - Most of these stars have a smaller mass than that of the Sun and these stars can show complicated structural changes that evolve over time [19]. The small community involved with this work is now viciously arguing about the intrinsic details of the calculations and quality of the data.

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In Bonn and Prague, we have been following this situation. We have already been studying the gravitational theory for open star clusters [14, 20]. Given the discordant wide-binary-test results, Jan sat down to develop the mathematics of how to actually calculate the forces that keep the two stars in a wide binary system moving about each other as both orbit through the Milky Way. For Newtonian gravitation this is easy, but no-one knows how to do the calculation in Milgromian gravitation.

For example, it may be wrong to treat a wide binary as an effective one-body problem consisting of a central mass combining the total mass of the binary and an orbiting test mass, as Banik assumed in [2]. In the case of AQUAL, the problem lies in discretising the whole p-dependency  (the transition from the p=2/Newtonian to the p=3/deep-Milgromian regime) of the p-Laplace operator acting on the potential, given the sources (the stars plus Galaxy) to allow the force calculation. The two, three- or four-star calculation becomes intractable: the gravitational mass of a star depends on where the other stars are and where all of them are relative to the whole Galaxy. The components in a wide binary star can even appear to move according to forces that are weaker than Newtonian ones.

Given these insights, which were triggered by the Banik’s wide-binary-star conundrum, we have made a great deal of progress, even if we have not yet reached a full solution. It is clear, though, that all calculations of the velocity differences between the two stars in wide binary systems have been carried out incorrectly. Perhaps this may be the reason why the above teams obtain different results. Exact checking and reproduction of the team's results would need an independent team redoing all the analysis and work, which is beyond the available person power in the community which lacks financial support.

It is notable that the Milgromian calculations are easier for the outer Solar System in which the problem consists of one dominating mass – the Sun – around which orbit individual comets and minor planets. In open star clusters and galaxies, the calculations are also easier because the open star cluster or galaxy provides the dominant potential in which individual stars orbit. The same is true for galaxy clusters and larger scales – which, however, become sensitive to cosmological boundary conditions, where we encounter constraints to what information we are able to access.

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One needs to also accept that to falsifying a theory requires multiple credible tests that each reach an acceptable level of statistical confidence.

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The apparent breakdown of MOND specifically in the domain where it is currently impossible to do the calculations might thus not be particularly surprising. Indranil Banik deserves recognition and merit for forcing us to look more into the details of the theory and for bravely and scientifically correctly reporting his results despite them being not what he expected. This shows a high degree of scientific integrity unspoiled by the quest for personal gain or the need to improve his chances on the job market.

Nevertheless, one needs to also accept that to falsifying a theory requires multiple credible tests that each reach an acceptable level of statistical confidence. This is particularly important when our confidence in the data we use to test the accuracy of mathematical models is limited. A single test, even if its formal confidence is extremely high, may still be wrong – we need multiple sources of information to build or to discredit a theory reliably. Thus, given the existing data we have and the calculations we are able to accurately carry out, Milgromian dynamics is the most compelling framework for further research.

References:

[1] Hernandez, X., et al. 2012, EPJC 71, 1884, "Wide binaries as a critical test of classical gravity", https://ui.adsabs.harvard.edu/abs/2012EPJC...72.1884H/abstract

[2] Banik et al., 2023, MNRAS 527, 4573, "Strong constraints on the gravitational law from Gaia DR3 wide binaries",  https://academic.oup.com/mnras/article/527/3/4573/7342478

[3] Hernandez, X., et al., 2023, MNRAS, in press, "Statistical analysis of the gravitational anomaly in Gaia wide binaries", https://ui.adsabs.harvard.edu/abs/2023MNRAS.tmp.3424H/abstract

[4] Hernandez, X., 2023, MNRAS 525, 1401, "Internal kinematics of Gaia DR3 wide binaries: anomalous behaviour in the low acceleration regime", https://ui.adsabs.harvard.edu/abs/2023MNRAS.525.1401H/abstract

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[15] Lopez-Corredoira, M., et al., 2022, MNRAS 517, 5734, "Virial theorem in clusters of galaxies with MOND", https://ui.adsabs.harvard.edu/abs/2022MNRAS.517.5734L/abstract

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