In the last post, I noted some of the sociological overtones underpinning attitudes about dark matter and modified gravity theories. I didn’t get as far as the more scientifically  interesting part, which  illustrates a common form of reasoning in physics.

About modified gravity theories, Bertone & Tait state

“the only way these theories can be reconciled with observations is by effectively, and very precisely, mimicking the behavior of cold dark matter on cosmological scales.”

Leaving aside just which observations need to be mimicked so precisely (I expect they mean power spectrum; perhaps they consider this to be so obvious that it need not be stated), this kind of reasoning is both common and powerful – and frequently correct. Indeed, this is exactly the attitude I expressed in my review a few years ago for the Canadian Journal of Physics, quoted in the image above. I get it. There are lots of positive things to be said for the standard cosmology.

This upshot of this reasoning is, in effect, that “cosmology works so well that non-baryonic dark matter must exist.” I have sympathy for this attitude, but I also remember many examples in the history of cosmology where it has gone badly wrong. There was a time, not so long ago, that the matter density had to be the critical value, and the Hubble constant had to be 50 km/s/Mpc. By and large, it is the same community that insisted on those falsehoods with great intensity that continues to insist on conventionally conceived cold dark matter with similarly fundamentalist insistence.

I think it is an overstatement to say that the successes of cosmology (as we presently perceive them) prove the existence of dark matter. A more conservative statement is that the ΛCDM cosmology is correct if, and only if, dark matter exists. But does it? That’s a separate question, which is why laboratory searches are so important – including null results. It was, after all, the null result of Michelson & Morley that ultimately put an end to the previous version of an invisible aetherial medium, and sparked a revolution in physics.

Here I point out that the same reasoning asserted by Bertone & Tait as a slam dunk in favor of dark matter can just as accurately be asserted in favor of MOND. To directly paraphrase the above statement:

“the only way ΛCDM can be reconciled with observations is by effectively, and very precisely, mimicking the behavior of MOND on galactic scales.”

This is a terrible problem for dark matter. Even if it were true, as is often asserted, that MOND only fits rotation curves, this would still be tantamount to a falsification of dark matter by the same reasoning applied by Bertone & Tait.

Lets look at just one example, NGC 1560:


The rotation curve of NGC 1560 (points) together with the Newtonian expectation (black line) and the MOND fit (blue line). Data from Begeman et al. (1991) and Gentile et al. (2010).

MOND fits the details of this rotation curve in excruciating detail. It provides just the right amount of boost over the Newtonian expectation, which varies from galaxy to galaxy. Features in the baryon distribution are reflected in the rotation curve. That is required in MOND, but makes no sense in dark matter, where the excess velocity over the Newtonian expectation is attributed to a dynamically hot, dominant, quasi-spherical dark matter halo. Such entities cannot support the features commonly seen in thin, dynamically cold disks. Even if they could, there is no reason that features in the dominant dark matter halo should align with those in the disk: a sphere isn’t a disk. In short, it is impossible to explain this with dark matter – to the extent that anything is ever impossible for the invisible.

NGC 1560 is a famous case because it has such an obvious feature. It is common to dismiss this as some non-equilibrium fluke that should simply be ignored. That is always a dodgy path to tread, but might be OK if it were only this galaxy. But similar effects are seen over and over again, to the point that they earned an empirical moniker: Renzo’s Rule. Renzo’s rule is known to every serious student of rotation curves, but has not informed the development of most dark matter theory. Ignoring this information is like leaving money on the table.

MOND fits not just NGC 1560, but very nearly* every galaxy we measure. It does so with excruciatingly little freedom. The only physical fit parameter is the stellar mass-to-light ratio. The gas fraction of NGC 1560 is 75%, so M*/L plays little role. We understand enough about stellar populations to have an idea what to expect; MOND fits return mass-to-light ratios that compare well with the normalization, color dependence, and band-pass dependent scatter expected from stellar population synthesis models.

The mass-to-light ratio from MOND fits (points) in the blue (left panel) and near-infrared (right panel) pass-bands plotted against galaxy color (blue to the left, red to the right). From the perspective of stellar populations, one expects more scatter and a steeper color dependence in the blue band, as observed. The lines are stellar population models from Bell et al. (2003). These are completely independent, and have not been fit to the data in any way. One could hardly hope for better astrophysical agreement.


One can also fit rotation curve data with dark matter halos. These require a minimum of three parameters to the one of MOND. In addition to M*/L, one also needs at least two parameters to describe the dark matter halo of each galaxy – typically some characteristic mass and radius. In practice, one finds that such fits are horribly degenerate: one can not cleanly constrain all three parameters, much less recover a sensible distribution of M*/L. One cannot construct the plot above simply by asking the data what it wants as one can with MOND.

The “disk-halo degeneracy” in dark matter halo fits to rotation curves has been much discussed in the literature. Obsessed over, dismissed, revived, and ultimately ignored without satisfactory understanding. Well, duh. This approach uses three parameters per galaxy when it takes only one to describe the data. Degeneracy between the excess fit parameters is inevitable.

From a probabilistic perspective, there is a huge volume of viable parameter space that could (and should) be occupied by galaxies composed of dark matter halos plus luminous galaxies. Two identical dark matter halos might host very different luminous galaxies, so would have rotation curves that differed with the baryonic component. Two similar looking galaxies might reside in rather different dark matter halos, again having rotation curves that differ.

The probabilistic volume in MOND is much smaller. Absolutely tiny by comparison. There is exactly one and only one thing each rotation curve can do: what the particular distribution of baryons in each galaxy says it should do. This is what we observe in Nature.

The only way ΛCDM can be reconciled with observations is by effectively, and very precisely, mimicking the behavior of MOND on galactic scales. There is a vast volume of parameter space that the rotation curves of galaxies could, in principle, inhabit. The naive expectation was exponential disks in NFW halos. Real galaxies don’t look like that. They look like MOND. Magically, out of the vast parameter space available to galaxies in the dark matter picture, they only ever pick the tiny sub-volume that very precisely mimics MOND.

The ratio of probabilities is huge. So many dark matter models are possible (and have been mooted over the years) that it is indefinably huge. The odds of observing MOND-like phenomenology in a ΛCDM universe is practically zero. This amounts to a practical falsification of dark matter.

I’ve never said dark matter is falsified, because I don’t think it is a falsifiable concept. It is like epicycles – you can always fudge it in some way. But at a practical level, it was falsified a long time ago.

That is not to say MOND has to be right. That would be falling into the same logical trap that says ΛCDM has to be right. Obviously, both have virtues that must be incorporated into whatever the final answer may be. There are some efforts in this direction, but by and large this is not how science is being conducted at present. The standard script is to privilege those data that conform most closely to our confirmation bias, and pour scorn on any contradictory narrative.

In my assessment, the probability of ultimate success through ignoring inconvenient data is practically zero. Unfortunately, that is the course upon which much of the field is currently set.

*There are of course exceptions: no data are perfect, so even the right theory will get it wrong once in a while. The goof rate for MOND fits is about what I expect: rare, but  more frequent for lower quality data. Misfits are sufficiently rare that to obsess over them is to refuse to see the forest for a few outlying trees.

Here’s a residual plot of MOND fits. See the peak at right? That’s the forest. See the tiny tail to one side? That’s an outlying tree.

Residuals of MOND rotation curve fits from Famaey & McGaugh (2012).

73 thoughts on “It Must Be So. But which Must?

  1. @gdp
    Thank you for your lucid explanation. (Why not use your real name Dr. S?) Unfortunately, “dynamical friction” is also a misnomer. In classical mechanics, dynamical friction requires physical contact between solid bodies and/or fluids. In fluids, dynamical friction is called drag. What Zwicky and Chandrasekhar are talking about is deceleration due to gravitational field. This is quite different from dynamical friction.

    Dr. McGaugh,
    You said:
    “if dark matter exists in the way is “must” for cosmology, I don’t see how one keeps it out of galaxies. Similarly, if the phenomena in galaxies indicates a change in the force law, this ought to have consequences for cosmology.”

    I agree. Assuming dark matter is not in the Standard Model and not sparticles, MOND and dark matter are two sides of one coin. A new particle will have a corresponding new field that could interact with gravity causing a deviation from Newtonian dynamics. This DM field combines with general relativity for a complete theory of gravity. The unified theory of MOND and dark matter:

    On galactic scale:
    MOND + DM particles = DM field
    On cosmic scale:
    GR + DM field = Modified gravity


  2. I thought it might be useful to clarify just what I mean by viscosity in the context of this discussion. The following quote comes from a definition of viscosity at

    “Because part of a fluid that is forced to move carries along to some extent adjacent parts, viscosity may be thought of as internal friction between the molecules; such friction opposes the development of velocity differences within a fluid.”

    It is precisely in the sense of opposing “the development of velocity differences” that the viscosity concept seems to fit observations.of galactic rotation curves. Those curves do not develop the velocity differences expected.

    By contrast, the general analytical approach seems well characterized by Chandrasekhar:

    “It is now generally agreed that the forces influencing the motions
    of the individual stars in a stellar system are essentially gravitational in character. In a
    general way it is clear that these forces arise, first, from the smoothed-out distribution
    of matter in the system and, secondly, from the effect of chance stellar encounters. The
    forces of the first kind are derivable from a gravitational potential S3 representing the
    smoothed-out distribution of matter in the system. This gravitational potential is a
    function of the space and time co-ordinates only. On the other hand, the forces of the
    second kind arise from the accidental encounters with the other stars which happen to
    be in the neighborhood of the star we are considering.”

    The first case is just Newtonian dynamics which only accounts for the centripetal gravitational force and the second simply considers random stellar encounters. Neither approach can possibly capture the viscosity-like behavior that galaxy rotation curves exhibit. This is an analytical problem


  3. Off topic, but is anyone else laughing out load about the “Dark Matter Hurricane”?
    Popular news has always been poor at reporting science, but this seems like an all time low point.


  4. Hadn’t heard about the dark matter hurricane, but it is a good example of why I get my science from refereed journals and not the popular press.

    1810.00374 talks about MOND, but doesn’t appear to do MOND. Their argument against it appears to hinge on the conventional dynamical mass exceeding the luminous mass in the inner parts of an elliptical galaxy. The accelerations are typically above the MOND scale there, so there should be no such discrepancy. However, this is more a test of stellar mass estimates than of MOND. If M*/L is exactly what they population model they quote says it should be, then this might be interesting. In practice that is only true on average; individual galaxies vary. So you’d have to build a mass model and ask what M*/L would be necessary to fit the data. They don’t do that. What they do show is qualitatively consistent with what I’ve seen in other ellipticals: the data can probably be explained with stars only, needing neither dark matter nor MOND at small radii. Then one can ask if the implied M*/L is reasonable; I don’t see the information necessary to evaluate that.
    I have fact-checked literally hundreds of claims to falsify MOND. This one doesn’t even meet the threshold that deserves checking.


  5. @budrap
    “Viscosity” is just a metaphor. Fluid viscosity is due to Van der Waals forces not gravitational force.

    Dr. McGaugh,
    What do you think of Ant 2?

    “Ant 2’s low density could mean that a modification to the dark matter properties is needed. The currently favoured theory predicts dark matter to pack tightly in the centres of galaxies. Given how fluffy the new dwarf appears to be, a dark matter particle which is less keen to cluster may be required.”

    “The gap between Ant 2 and the rest of the Galactic dwarfs is so wide that this may well be an indication that some important physics is missing in the models of dwarf galaxy formation.”


  6. @Enrico, it is more properly an analogy. That galactic systems exhibit viscosity-like behavior as in, opposing “the development of velocity differences”, seems straightforward qualitative analysis. The fact that the forces involved are different does not necessitate that similar effects cannot be produced by those forces, especially on very different scales. Therefore it would seem to be an appropriate modelling avenue of exploration.

    My complaint against the current failed model is that gravitational viscosity has been ignored as a conceptual framework for understanding and modelling galactic rotation curves on the basis of a rather flawed mathematical analysis written 77 years ago.


  7. @budrap
    Better to use a different term to replace “viscosity” in galaxy dynamics. Gravity and electromagnetic force are more analogous. Newton’s law and Coulomb’s law even have the same mathematical form but we don’t call the electromagnetic force “gravity.”

    I don’t think Chandrasekhar’s paper is flawed. If you do, write a paper showing its flaws and modelling galactic rotations using “viscosity.” If you can pass peer review and publish in a journal, it would be a notable contribution to astronomy.


  8. @Enrico, Semantics and career advice aside, saying, “I don’t think Chandrasekhar’s paper is flawed”, without explaining why you think that, is simply a way of making a content-less argument.


  9. @budrap
    You claim Chandrasekhar’s paper is flawed. The burden of proof is on you. It passed peer review and published in a reputable journal. Chandrasekhar is a Nobel laureate in physics. Zwicky himself did not publish any paper claiming Chandrasekhar is wrong. You want us to accept Chandrasekhar is wrong without making any sensible argument.


  10. I was wondering the other day if it would be possible to derive a test for MoND in the solar system by looking at near-parabolic / hyperbolic comets. Given the high eccentricity exhibited by these comets, they will, for sure, enter the MoND regime and some (many) of them would turn out to be, in fact, bounded to the sun. A test that I could think of is to make predictions about the chemical properties for the next claimed interstellar comets that will be discovered – i.e. if, in MoND, they are really on “elliptical” orbits, then they should be nearly indistinguishable from the other solar system comets. However, if they are on unbounded trajectories in MoND, then they will have a high chance of having some special characteristics that differentiate them from the solar system comets.
    Would a lower fraction of observed comets with parabolic / hyperbolic trajectories (as would be the case in MoND) mean anything?
    As a note, I used quote marks for elliptical as I’m not really sure how to call a closed orbit (but with very high eccentricity) in MoND – my expectation is the shape of the orbit is no longer an ellipse.
    Anyway – what would be the excess velocity at infinite distance, calculated using ND, for which a comet’s trajectory in MoND would become “parabolic”? (For the present interloper – Borisov, I highly doubt that it could be on a bounded trajectory since its excess velocity is in the range of 30km/s, but say, at 3km/s, would that make a comet bounded in MoND?)


    1. Yes, people have thought about this. There should be some excess precession, for example. It is very hard to tell in practice, but constraints on MOND in the solar system are nevertheless pretty tight (e.g.,
      The asymptotic velocity of a solar mass object in MOND is a hair under 0.4 km/s, so that’s the scale to have in mind. Also bear in mind that the sun is in a relative high acceleration location – about 1.8*a0. So the immediate environment is more Newtonian than MONDian.


      1. Thank you for the answer.
        I would have expected an excess velocity in the order of 1 – 2 km/s so the figure you gave is quite low.
        With this value, I’d say that it’s not possible to distinguish between solar and interstellar comets because, especially with comets which are prone to outgasing events – the uncertainty in the actual orbit / escape velocity is too large to be able to make a clear cut between ND and MoND.
        That would leave only orbit precession – but good luck with waiting enough time to observe it…. after all, we’d be talking about very long period objects/comets.
        What I find interesting though is that if MoND has an underlying physical explanation, there will be an excess orbit precession both for very strong (i.e. GR) and very weak fields (i.e. MoND).


        1. Yes, excess precession in these very different regimes is intriguing. But also very hard to detect for long period comets – if I thought I could do it, I would be.


          1. What about accretion disks sporting spiral or some other traceable patterns, around young stars?
            For instance this disk around AB Aurigae ( seems to have discernible spiral patterns, at the star’s estimated distance, up to ~1600AU.
            It is true that the star is more massive than the Sun and, given that it is relatively close to us on galactic scales, the dominating regime for that 1600AU distance is clearly ND. But this is just for illustration – maybe there are other (smaller) stars that have disks with spiral patterns extending in the 10 000AU range.
            Would those allow for a direct test for MoND? Would the stellar light pressure on the accretion disk at that distance skew the results?


            1. I doubt that we could get precise enough data to apply the test to an object like that. However, there are serious proposals to look for MOND effects in binary stars with large separations.


              1. Thank you again for the reply.
                I knew about the proposal with wide binaries, however, I always thought that it will be difficult to get data with enough precision to make a clear cut if MoND manifests or not. I assume that orbital velocities are measured using spectral methods as direct observations offer only a very limited precision given the orbital period. That’s why I though that maybe a clear pattern in the accretion disk would be better as models with MoND or just ND will diverge in the pattern different at large distances.
                But since you’re saying that there are serious proposals regarding wide binaries, I understand that the data (GAIA?) can provide already good enough resolution.


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