Can their originators explain why?

A recent BigThink.com article suggested that modified gravity could not explain every observation that dark matter theories explain. Unfortunately, the article does not cite any of the papers it is criticizing. This is a problem because there are numerous papers on modified gravity and some theories have been unambiguously falsified while others agree well with observations. This article shows why the latest research into a particular version of modified gravity explains all observations attributed to dark matter, but its ad hoc nature prevents it from being taken as seriously as a more elegant solution might be.

There are strong reasons why dark matter has been theorized to be a type of massive particle that does not interact with anything electromagnetically rather than an indication that Newtonian gravity, and therefore Einsteinian gravity does not work. It plays a role in how galaxies rotate, the Cosmic Microwave Background, the origin of light elements (Big Bang nucleosynthesis), small scale formation, and how light bends. Dark matter, the theory that matter we cannot see fills the universe, accounts for all these observations. Accounting for them in a modified gravity theory is a considerable challenge.

Dark matter was first proposed to explain why the stars in the outer regions of galaxies rotate about their galactic centers much more quickly than Newtonian gravity predicts. In fact, many galaxies rotate more like rigid bodies, e.g., a bicycle tire, rather than clusters of stars, almost as if the outer stars were connected to the center by spokes.

Solar systems and smaller clusters of stars like globular clusters do not behave this way and follow Newtonian physics perfectly well. Hence, from galaxies on up to the universe as a whole dark matter plays an important role. What role it plays at a smaller scale is not well understood and definitely not observed.

We also know that dark matter cannot be ordinary matter, known as baryonic matter, because of how that matter formed at the beginning of the universe. There is a cap on how much baryonic matter can have been formed at about 5% of the total mass observed in the universe. The Bigthink article explains this. That does not in fact imply that dark matter must exist, only that modified gravity must explain observations of the universe as a whole going back to the beginning of time, not just galaxies.

One of the best known alternatives to dark matter is MOdified Newtonian Dynamics (MOND). MOND’s basic premise is that you can explain the rotational curves of galaxies by modifying the force between two gravitating bodies. At high acceleration, inside the Solar System all the way up to subgalactic star clusters, MOND predicts the same as Newtonian physics, which is that the force is proportional to the acceleration. At low acceleration, beginning at the scale of dwarf galaxies, however, the force is proportional to the square of the acceleration divided by a fixed parameter. This means that the force required to keep those stars orbiting is much smaller and falls off more quickly than standard dynamics predicts. This turns out to lead to the velocity of stars being independent of their distance from the center of their galaxy as you would expect from a rigid body like a wheel.

MOND is not strictly a modification of gravity at all. It modifies dynamics. In order to modify gravity, you have to turn it into a more general theory. One theory is called AQUAL or A QUAdratic Lagrangian proposed in 1984 by Bekenstein and Milgrom. (No one said physicists were good with acronyms.)

AQUAL is a step away from modified dynamics to a modification of the gravitational potential. Yet, it is not enough. In particular, it does not explain light bending, which is a relativistic prediction, nor does it address the propagation of gravitational waves (so-called tensor modes), which is even more problematic for modified gravity.

In order to explain phenomena such as light bending (particularly gravitational lensing which requires dark matter to explain) Bekenstein proposed Tensor-Vector-Scalar or TeVeS theory.

 

TeVeS introduces new fields to Einstein’s general relativity, a vector and scalar field. Electromagnetism also uses a vector field while the Higgs field, believed to be responsible for mass, is a scalar. Original TeVeS suggested that all matter couples to a different gravitational field than gravity itself couples. (Later theorists showed how you could unify them by modifying the vector field.) The matter coupled gravitational field is constructed from the gravitational tensor field, the scalar field, and the vector field. These are arranged in what is called a disformal relationship with respect to the vector field, meaning not just a rescaling of units dependent on position alone as in a conformal relationship but rescaling that depends on direction as well.

The scalar field is what is responsible for the MOND behavior since it leads to AQUAL under the assumption that bodies like stars are moving much slower than the speed of light and weak gravity that isn’t changing too quickly. An admittedly ad hoc restriction has to be placed on the scalar field so that it predicts MOND. There is no real explanation for why it would follow this rule rather than being unrestricted which would lead to Newtonian dynamics at all accelerations. The scalar field has other functions, however, in modified gravity because it drives the particle or dust-like behavior we attribute to dark matter without needing additional matter.

The vector field helps to explain the gravitational lensing and resembles the vector field in the Einstein Aether theory. It is responsible for giving the appearance that spacetime is warping without needing additional matter. Unfortunately, the vector field is forced, in general, to point in the time direction and so breaks Lorentz invariance.

 

Given all these restrictions, TeVeS leads to AQUAL which leads to MOND. It has the sense of a theory someone created as an exercise to show they could explain dark matter with a modified general relativity, but it is taken seriously by a number of researchers.

It is possible that like many theories TeVeS is just an effective theory and all the supposedly ad hoc additions have good explanations in a more unified theory.

With that preamble, let’s look at the various phenomena where dark matter is implied and how TeVeS fairs.

Galactic Rotation Curves

MOND is surprisingly successful in explaining the rotational curves of galaxies but does not explain any other aspects of dark matter.

MOND, in fact, explains these rotational curves better than dark matter halo models in many cases including a recent one done on Fornax cluster dwarf galaxies. Halo models have their own problems because they predict the formation of halos of many sizes (hierarchical formation) and that these must be intimately connected to the formation of galaxies or we would see them on their own, which we don’t.

MOND depends on a fixed parameter that needs to apply to all galaxies, and some galaxies fit with one parameter and others fit with another. Because there are other variable parameters like mass to light ratio and distances in addition to the MOND interpolation function that explains how we go from Newtonian to MOND-ian physics for moderate accelerations, MOND hasn’t been ruled out on account of parameter fitting.

MOND still doesn’t explain everything observed in galaxies, i.e., there has to be some additional unseen mass. Some of this missing mass was discovered after MOND was proposed but much of it still is unexplained based on the distribution of light emitted from galaxies.

Gravitational Lensing

Gravitational lensing happens because gravity bends light. The more matter you have the more light bends. Since galaxies are very large, as light travels across their lengths, the light from objects behind them bends a lot both around the top and bottom. Because of this phenomenon, galaxies can act like lenses and magnify more distant objects. Astronomers can use this phenomenon to see further than they might ordinarily be able to.

Dark matter models of galaxies fit well with lensing, but MOND has nothing to say about it. TeVeS was developed to explain lensing using the vector field which explains how light can be warped as if matter were there when it is not. The vector field does this through the disformal relationship so that the additional warping becomes directional.

Cosmic Microwave Background and Acoustic Oscillations

Damning evidence against TeVeS came from the earliest light we can see, the Cosmic Microwave Background (CMB). It was believed, as this 2013 article claims, based on the short paper by Scott Donaldson, that the smooth power spectrum of the CMB rules out TeVeS which predicts large oscillations in the spectrum called Baryonic Acoustic Oscillations.

A recent 2021 paper by Skordis et al. addressed this problem showing that their version of TeVeS which they call a Relativistic MOND or RMOND theory resolved the issue. This solution is to parameterize the model so that the large acoustic oscillations are suppressed by MOND, and the scalar field behaves at the cosmological scale like a massive dust-like field while the vector field decouples.

This demonstrates one feature of TeVeS which is that there just isn’t one version of it and little to back up one over another except empirical data. In this regard, TeVeS is like the anti-string theory. It sacrifices elegance for empiricism.

The BigThink article reproduces the damning graphic from Donaldson’s paper without mentioning the work by Skordis from last year which was published in the prestigious Physical Review Letters.

Gravitational Wave Propagation

Perhaps one of the strongest blows against TeVeS and MOND was the discovery that gravitational waves travel at the speed of light. This discovery was thanks to the detection of gravitational waves at LIGO and VIRGO detectors from an event called GW170817 in 2017. It was the first time a gravitational wave event was observed by gamma ray detectors. Caused by the merger of two neutron stars, this event signaled the beginning of a new era in astronomy and nearly killed TeVeS. While it seems like a no brainer that gravitational and electromagnetic waves should both travel at the speed of light, TeVeS actually predicts that they propagate at different rates.

Skordis et al. in a 2019 paper in Phys. Rev. D fixed the problem by modifying one of the constants in TeVeS so it became dependent on the scalar field, canceling out the difference in propagation speed and thereby rescuing TeVeS. This also had the impression of an ad hoc fix but could be the result of some deeper theory.

Field or Particle

Another criticism is that modified gravity replaces dark matter with a gravitational field. It’s hard to see how this is a problem since the scalar field drives gravitational dust-like behavior in the dynamics of general relativity as Skordis et al. 2021 showed. We know one can obtain both clustered and smooth behavior from the same scalar field, so there is really no difference (Padmanabhan 2002). When it comes to explaining small scale observations, a lot of work remains but some is already being done and showing some differences between dark matter and MOND-based models (Renaud 2016).

Stability and Singularity Formation

The BigThink article didn’t mention this problem, but Bekenstein’s original TeVeS has been shown to be both unstable (Seifert 2007) and prone to forming singularities in the absence of matter (Contaldi 2008). That is pretty damning. Luckily, the modified gravity community have almost universally moved on from that version to a much more general framework. These general frameworks, such as the one Skordis et al. work with, avoid these problems.

Should we abandon MOND?

The vast majority of astrophysicists and the preponderance of funding has gone to the theory that dark matter is some kind of particle. Which particle and how it fits into the Standard Model of particle physics (sterile neutrinos, WIMPS, axions, etc.) is open to debate. Most motivations based on string theory have vanished, but there is hope that dark matter detectors will shed light on the phenomenon. If, however, these are all caused by gravity, we will never discover that particle, at least not until we discover the graviton.

TeVeS has gone through considerable modifications in the last 20 years, becoming a much more general theory and largely merging with Einstein Aether theory. Most observations can be explained by either dark matter or modified gravity models, and each has its own problems.

The main criticism I levy at TeVeS is that it is so ad hoc. There is no fundamental underlying theory that would explain why all the parameters are what they are. Einstein’s elegant theory of general relativity becomes a mess of fields, parameters, and constraints in a general TeVeS formalism. All that needs explanation.

One such explanation is Verlinde’s Entropic Gravity (EG) theory which reproduces MOND from fundamentals of quantum entanglement entropy (Verlinde 2017). In his theory, which reproduces both Einstein’s theory and Newton’s, MOND arises because competing sources of entropy cause spacetime to behave as an elastic medium. So far there is no relativistic version of EG comparable to TeVeS, but it is a compelling theory.

Dark matter models of course suffer from their own theoretical gaps. It is not as if you could just add extra matter to general relativity and that’s it. A lot goes into a dark matter model to make it explain galactic rotation curves and the research into their formation goes back to the 1960s. Self-interaction, for example, is often ignored but may be critical to predicting the correct substructures in dark matter halos (Weinberg 2015).

Potentially, if we observed a galaxy made entirely of dark matter, that might settle the problem. No matter, no MOND, surely. The problem with that is how would we see it? There must be some lensing at least. In 2016, astronomers claimed to have found a galaxy, Dragonfly 44 (DF44), that was nearly all dark matter, 98% or so. This was splashed all over the news at the time, and it could have been bad news for MOND. It turned out, however, to be false. The original researchers had made a faulty assumption that led to a much higher mass estimate. The galaxy was not mostly dark matter at all, and it was no different than any other dwarf galaxy.

Thus, until we either find evidence for dark matter in a detector or TeVeS becomes unambiguously explained, we cannot abandon MOND, and the reports of its death have been greatly exaggerated.

Bekenstein, Jacob, and Mordehai Milgrom. “Does the missing mass problem signal the breakdown of Newtonian gravity?.” The Astrophysical Journal 286 (1984): 7–14.

Bekenstein, Jacob D. “Relativistic gravitation theory for the modified Newtonian dynamics paradigm.” Physical Review D 70.8 (2004): 083509.

Begeman, K. G., A. H. Broeils, and R. H. Sanders. “Extended rotation curves of spiral galaxies: Dark haloes and modified dynamics.” Monthly Notices of the Royal Astronomical Society 249.3 (1991): 523–537.

Gentile, Gianfranco, B. Famaey, and W. J. G. De Blok. “THINGS about MoND.” Astronomy & Astrophysics 527 (2011): A76.

Dodelson, Scott. “The real problem with MOND.” International Journal of Modern Physics D 20.14 (2011): 2749–2753.

Skordis, Constantinos, and Tom Złośnik. “New relativistic theory for modified Newtonian dynamics.” Physical review letters 127.16 (2021): 161302.

Skordis, Constantinos, and Tom Złośnik. “Gravitational alternatives to dark matter with tensor mode speed equaling the speed of light.” Physical Review D 100.10 (2019): 104013.

Padmanabhan, T., and T. Roy Choudhury. “Can the clustered dark matter and the smooth dark energy arise from the same scalar field?.” Physical Review D 66.8 (2002): 081301.

Elena Asencio et al, The distribution and morphologies of Fornax Cluster dwarf galaxies suggest they lack dark matter, Monthly Notices of the Royal Astronomical Society (2022).

Contaldi, Carlo R., Toby Wiseman, and Benjamin Withers. “TeVeS gets caught on caustics.” Physical Review D 78.4 (2008): 044034.

Seifert, Michael D. “Stability of spherically symmetric solutions in modified theories of gravity.” Physical Review D 76.6 (2007): 064002.

Renaud, Florent, Benoit Famaey, and Pavel Kroupa. “Star formation triggered by galaxy interactions in modified gravity.” Monthly Notices of the Royal Astronomical Society 463.4 (2016): 3637–3652.

Weinberg, David H., et al. “Cold dark matter: controversies on small scales.” Proceedings of the National Academy of Sciences 112.40 (2015): 12249–12255.

Skordis, Constantinos, et al. “Large scale structure in Bekenstein’s theory of relativistic modified Newtonian dynamics.” Physical Review Letters 96.1 (2006): 011301.

Verlinde, Erik. “On the origin of gravity and the laws of Newton.” Journal of High Energy Physics 2011.4 (2011): 1–27.

Verlinde, Erik. “Emergent gravity and the dark universe.” SciPost Physics 2.3 (2017): 016.