The contradiction in our dark matter

It is straightforward to falsify theories that make clear predictions, should they happen to be wrong. It is impossible to falsify ideas that involve invisible components.

Specific theories of dark matter (e.g., WIMPs) can be seen to be increasingly unlikely, but can they be falsified outright? If we decide that WIMPs have been practically falsified – to all intents and purposes – that doesn’t mean dark matter is wrong, just that we’ve spent the past four decades building dozens of giant experiments costing billions of dollars and frustrating thousands of careers barking up the wrong tree. The unseen forest that is the ‘dark sector’ is vast; there are limitless opportunities to bark up other wrong trees.

How could we know if the entire dark sector is a non-entity? On many occasions, I’ve had colleagues say to me that they would “only consider MOND as a last resort.” OK, so when do we know we’ve reached that point? After an eternity of searching for unseen, undetected non-entities seems a little late.

This is the bind we’re in. Most of us (scientists working in the field) are unwilling to consider something as radical-seeming as MOND until dark matter has been falsified. But dark matter cannot be falsified. So we patch up our hypotheses accordingly, rinse and repeat with every new crisis^#. This has been going on for nearly my entire career, to the point that I now see junior scientists who seem to think this is how science is supposed to work. Why wouldn’t they? They’ve known nothing else.

I empathize. I started from the same place: there has to be dark matter, it has to be non-baryonic, it is almost certainly a new particle and that new particle is almost certainly a WIMP. It was the hardest thing to realize and accept that we could be wrong, wrong, wrong, and wrong on all counts. I worked incredibly hard to avoid that conclusion. I take solace only in the fact that at every epoch in human history we’ve had a cosmology that we were absolutely sure^$ was right but that later turned out not to be. Maybe we’re a blessed generation who finally got it right. Or maybe we’re just the latest in a long line of brilliant savants who fooled themselves into thinking we understand more than we actually do.

I remain unwilling to say that dark matter has been falsified because I don’t think it is falsifiable. That, in itself, is usually considered to be a bad thing for a scientific theory. Perhaps at some future point it will be portrayed that way for dark matter, but in the present I’ve heard plenty of scientists pretend like it is somehow a good thing. Certainly it makes a fertile playground for theorists, so I guess good/bad is a matter of perspective. I suppose an infinite forest of trees in a conveniently unobservable dark sector is an irresistible opportunity for a dog^! with an insatiable bladder.

I do, however, think that dark matter has been practically falsified. If I’m wrong about that, it is possible to demonstrate. To be clear, I don’t just mean WIMPs. I mean dark matter as an explanation for the mass discrepancies observed in extragalactic systems. No kind of dark matter* can suffice. I came to this conclusion before I really knew anything about MOND, which is why I was receptive to it. Others haven’t had that experience, so aren’t.

I don’t expect other scientists to accept that an entire paradigm is wrong because I say so. I do, however, expect them to acknowledge that the dark matter hypothesis should be falsifiable. It is incumbent on each scientist to establish for themselves criteria by which that conclusion could be reached, should it be appropriate.

For me, the problem is the contradiction present in the dynamical data for galaxies. We simultaneously require galaxy disks to be maximal and also for them not to be maximal. The only way to avoid this contradiction is to engage in fine-tuning: one must build a model in which everything works out just so. Since dark matter is not otherwise falsifiable, a requirement for fine-tuning^& is pretty much the worst thing we can say about it.

The contradiction as framed by others

The contradiction that concerns me has certainly been noticed by others. Usually, they choose to come down on one side or the other. The problem is that they’re both right.

On the one hand, it is clear observationally that the luminous mass matters to the dynamics of galaxies. For example, Swaters et al. note that

the luminous mass dominates the gravitational potential in the central regions, even in low surface brightness dwarf galaxies

which is to say, disk galaxies are maximal. We’ll explore what that means below, but you can see the effect by eye:

*Rotation curves color coded by galaxy surface brightness*. There is an almost perfect rainbow variation that is plain to see. The gravitational potential traced by the rotation curve correlates well with the distribution of stellar mass. *(Adapted from McGaugh 2020.)*

On the other hand, there are no residuals from the Tully-Fisher relation.

Residuals from the Tully-Fisher relation as a function of size at a given mass. Blue points are star-dominated galaxies; cyan points are gas-dominated. Compact galaxies are to the left, diffuse ones to the right. The red dashed line [δlog(V)/δlog(R) = -1/2] is what should happen with maximal disks: more compact galaxies should rotate faster at a given mass. This is what Newton predicts, but galaxies didn’t get the memo. (Adapted from McGaugh 2005.)

The inference from this observation is that galaxy disks cannot be maximal. As Courteau & Rix put it,

The case of δlog(V)/δlog(R) = -0.5 expected^% for a maximal disk is ruled out

which is to say, disk galaxies are not maximal – even the high surface brightness (HSB) disks that dominated their sample.

They say this because maximal disks – or even those that contribute noticeably to the mass budget at small radii – predict deviations from the Tully-Fisher relation. They did this by intentionally focusing on the point where the stars should contribute the most. I’ve shown similar things many times, so here let’s let LCDM advocate Frank van den Bosch demonstrate it:

Model galaxies (left) of the same mass but with different spin parameters (van den Bosch 2001). In this model, spin determines size and surface brightness: lower spin galaxies are more compact and have higher surface brightness. This causes them to evince higher velocities that **should** cause residuals from Tully-Fisher, but these are not observed (right, McGaugh 2021).

So galaxies cannot be maximal. Only they have to be to have the observed correlation between the luminous mass distribution and the kinematics, as seen in the first figure above. Well, which is it? Are all galaxies maximal? Or none? Or can they both^ be right?

Maximal disks

Maximal disk is a technical term that is well known to people who work on the subject but not outside that relatively narrow^&& field. So what do we mean by this term?

The rotation curve of NGC 6946. The solid blue line shows the expected rotation curve for the visible mass including both stars (dashed line: stellar disk; dot-dashed line: bulge) and atomic gas (dotted line). The stars suffice to describe the inner rotation curve, illustrating the case of **maximum disk**. The red points are the rotation curve measured optically (Daigle et al. 2006; Epinat et al. 2008); the orange points that measured with radio interferometric observations of the gas (Boomsma et al. 2008; I’ve suppressed the overlap for clarity). The thin line is the prediction of MOND; the black line is the implied dark matter distribution.

In essence, a maximal disk is one in which the stars provide practically all the mass at small radii. The depiction of NGC 6946 above illustrates maximum disk. Sure, the rotation curve flattens out at large radii and we need to invoke dark matter. But the observed stars explain the amplitude and shape of the inner rotation curve quite well. In this case there is a compact bulge at the center of the galaxy that causes a sharp rise in the rotation curve right from R = 0. (This is an example of Renzo’s Rule.) The disk (plus bulge) is maximal in the sense that we cannot attribute any more mass to them without exceeding the observed rotation curve.

The amplitude of the portion of the rotation curve due to the stars depends on their mass-to-light ratio. While this cannot exceed maximum disk, it could be lower. So conceivably, the good match to the shape of the observed rotation curve is a chimera, and really this galaxy is dark matter dominated. That is a possibility many seem to have embraced, but while it might work for the disk, it does not work for the bulge. As we suppress the contribution of the stars as Courteau & Rix argue we must, then the rotation curve looks more and more like that of the dark matter halo alone. That goes up and flattens out (and ultimately must turn over again somewhere beyond the edge of the data) but it has no features.

*The rotation curve of NGC 6946* as above but with the stellar mass reduced by a factor of two. This disk is submaximal, contributing less than the implied dark matter.

One thing the rotation curve due to the dark matter halo cannot do is go up then down then up again. Yet that is exactly what it needs to do to explain the inner peak inside 1 kpc if the bulge component is not maximal. I suppose we could have a smaller dark matter halo inside the main dark matter halo that does this, but its mass distribution would have to be practically identical to that of the bulge. That’s insane. Why would we invoke a second dark matter halo when the stars are right there?

In this case, the stars have the right mass for what we expect from stellar populations. There was a long debate historically about whether the optical band mass-to-light ratios for maximum disk were consistent with those expected from stellar population synthesis models. For a long time they looked close but a bit high. This difference has pretty much gone away now that we have access to near-infrared data: the two are consistent, and having a stellar mass-to-light ratio much below the maximum disk value for high surface brightness disks becomes problematic from a population perspective.

Indeed, the 3.6 micron M*/L = 0.37 M_☉/L_☉ in the case depicted for NGC 6946 with a maximum disk. That’s reasonable but on the low side for what is plausible for the stars in a mature spiral galaxy like this. Halving that strains credulity, so there is no room for a second inner halo, or even for the cusp predicted** for the primary cold dark matter halo. Stellar mass really does seem to dominate in the inner parts, just as Swaters et al. said.

LSB galaxies

The issue that confounded me was whether the low surface brightness (LSB) galaxies I was working on were maximal or not. My inital expectation was that LSB galaxies would be stretched out versions of HSB galaxies. I expected them to shift off of the Tully-Fisher relation and follow the line δlog(V)/δlog(R) = -0.5. They did not do that. If I didn’t have them be maximal, I found that I could explain pretty much any slope other than the one observed (δlog(V)/δlog(R) = 0). That required fine-tuning to perfectly balance the lesser contribution of stars in LSB galaxies which we had to back fill with dark matter just so. I spent ages running around in circles trying to make that work. Every time I thought I had succeeded, I realized I had assumed something that made it so: tautologies abound.

If we want to explain the shapes of rotation curves as seen up top, we need the stars to contribute to the gravitational potential. For that to work for LSB galaxies, we have to turn maximum disk up to eleven:

*The low surface brightness, gas rich dwarf DDO 154. The lines have the same meaning as above, with the case of maximum disk in the left panel and that natural for stellar populations at right.* The difference in disk mass is a factor of ten.

A crazy-high stellar mass-to-light ratio is what happens if we just ignore what we know about stars and just focus on the kinematics. But we do know a lot about stars. Population models indicate stellar masses that are very submaximal. Even boosting the mass-to-light ratio doesn’t get us very far. LSB galaxies aren’t really maximal in the same sense as HSB galaxies, and there is even less room for the expected cuspy halos that are already problematic when the stellar contribution is small.

Fine tuning is unavoidable

Even if we ignore what we know about stars, we still have a fine-tuning problem. The lack of a shift in the Tully-Fisher relation with either surface brightness or radial size implies that disks are all the same mass surface density. So we observe a wide range of surface brightness, but the surface mass density is always the same. That makes no sense, and is just another example of squeezing the toothpaste tube: we can make a model look OK from one perspective as long as we don’t look from another.

Worse, we still need to explain the role of the luminous mass in LSB galaxies. These are dark matter dominated at almost all radii, and yet the distribution of the observed stars and gas is predictive of the kinematics. This is a contradiction to Newtonian dynamics. The only theory that does this right – and predicted it a priori – is MOND. But that’s too horrible to contemplate, so we shield our eyes and ignore^%% one or the other set of inconvenient facts. As a result, the field has become moribund, and will remain so until we free ourselves of our invisible demons.

^#We’ve experienced so many crises that we seem no longer able to recognize new ones. JWST observations of high redshift galaxies follows a well-worn trajectory: an observation that contradicts the standard model is made, much huffing and puffing ensues, the theorists get to work constructing implausible models, these are accepted as patching up the hypothesis (whether satisfactory or not), and the field moves on as if nothing happened.

^$To give one historical example, prior to Hubble’s discoveries in the 1920s, it was thought that the Milky Way was the entire universe. Certainly there were no other galaxies comparable to the Milky Way:

“No competent thinker, with the whole of the available evidence before him, can now, it is safe to say, maintain any single nebula to be a star system of coordinate rank with the Milky Way. A practical certainty has been attained that the entire contents, stellar and nebular, of the sphere belong to one mighty aggregation.” [i.e., the Milky Way]

-Agnes Mary Clerke in The System of the Stars (1890)

^!It used to be that one would not claim a detection of dark matter until all astrophysical alternatives had been exhausted. Now it seems to be the fad to claim a detection first on the off-chance it works out later. I already peed on that tree! It’s mine!

*Excepting some sort of hybrid “dark matter” that is invented to do what ordinary dark matter cannot. By ordinary I mean CDM, WDM, SIDM, and every other variation on particle physics that simply invents new mass with no consideration of how the observed galaxy dynamics comes about. That would include primordial black holes and various macroscopic DM ideas (e.g., MACHOS, strange nuggets). Coming up with half-baked ideas for new particle dark matter is big business these days, but any idea not informed by observed astrophysics (which are most of them) is doomed to fail.

Examples of hybrid dark matter that are informed by observed astrophysics include dipolar dark matter and superfluid dark matter. Regardless of whether these specific cases are viable, the point is that the observed dynamics are a fundamental aspect if nature and require a commiserate explanation. Simply throwing in some extra mass with some fine-tuned feedback models can never provide a satisfactory explanation. Note that coming up with extra mass is mostly done by particle phenomenologists while feedback models are built by numerical astrophysicists. There is very little overlap between these communities; they pretty much just take it on faith that since dark matter has to exist, the part they don’t know about will magically work out.

^&The classic example of fine-tuning in the sense that I mean is the Ptolemaic model of epicycles and deferents. If one adds enough of these and tunes them just so, anything can be fit. Note that epicycles are not explicitly falsifiable for this reason; we rejected them because they got ridiculously complicated and there turns out to be a more parsimonious explanation. The same thing holds now for dark matter and MOND.

^%This slope is expected because Newton teaches us that V² = GM/R. δlog(V)/δlog(R) = -0.5 follows from taking the logarithm of this at fixed mass. Galaxies are observed to span a large range of radius at a given mass, but not a corresponding range in circular velocity.

^Yes, they can both be right, but not with dark matter. Only MOND naturally explains both observations simultaneously.

Also, for the hyper vigilant, Courteu and Rix (1999) use a slightly different definition of velocity than I do in this Tully-Fisher residual plot. I went through all that in McGaugh & de Blok (1998) and in McGaugh (2005) and it makes no difference to the discussion here.

^&&A vote we held at a conference on disk dynamics in Rome in 2000. The question of whether disks were maximal was posed; most people voted no based on the statistical lack of residuals from Tully-Fisher. After the vote, one of the dissenters noted that those who voted in favor of maximal disks were the people who actually worked in the subject. Those of us with other concerns were persuaded by the statistical evidence because we didn’t engage with the details of real, individual galaxies in the same way.

**The NFW halo famously gets the inner shape of the rotation curve wrong (the cusp-core problem), but it is also wrong at intermediate radii and at large radii. Other than that it’s great.

^%%A common excuse I here for this behavior is that galaxies are “small” and nonlinear – complicated entities that we can never hope to understand, so whatever they do can be ignored as irrelevant. As a scientific argument, that’s pathetic. Galaxies should be complicated in LCDM, but in observational reality they’re kinematics are sufficiently simple that they obey a single effective force law. That’s one thing they should not do, just as a complicated set of epicycles and deferents shouldn’t always add up to the inverse square law.

The blinding influence of the Bullet Cluster

A few posts back, the issue of the Bullet Cluster came up in the comments. I was still working my way up to addressing that in a full post, so I limited myself to making a sociological observation based on my experience:

People who invoke the bullet cluster in response to queries about MOND are usually doing so to deflect from the need to engage with it.

That is a general observation that was not aimed at anyone in particular, but it was made in response to a comment citing Don Lincoln saying this, so Dr. Lincoln hopped into the comments to reply:

Or…and bear with me here…some people actually find the Bullet Cluster (and the DF2/DF4 situation) to be persuasive.
FWIW, I’m a physicist, working in the field. And my views have changed over the years. In the early 1990s, I was pretty sure that dark matter was MACHOs. When MACHO, OGLE and all them disproved that conjecture, I then strongly favored MOND (broadly defined…not necessarily Milgrom’s conjecture, but rather the much vaguer paradigm that inertia or gravity needed some improved understanding). However, in the modern day, Bullet and Dragon Fly have again changed my leaning.
Yes, yes, the Bullet Cluster is moving awfully fast. Yes, yes, it’s a problem for LCDM. But this isn’t the same level of problem as it poses for modified physics. For LCDM, it is simply an unusual entry on the tail of a known velocity distribution, while the problems it poses for MOND-ish issues is more central.
Now, this is your page and you are allowed to have an echo chamber and sycophants…no problem. But to characterize those who disagree with you as somehow not being thoughtful is just…sloppy. And dismissive. And patronizing.
The broad community might be wrong. But, you know what? So could you.
FWIW, it will be difficult to convince me of >>ANY<< solution without DM being produced in particle accelerators. Indirect measurements are background-prone. And direct measurements, while super helpful, will tell us where to look in accelerators. (Think the DAMA debacle.)
My fear is that dark matter is real, but it only interacts gravitationally or far weaker than the weak force. If that’s the case, our grandkids will be having this argument.
Don Lincoln, posting as Science Guy, 4 June 2026

In my experience, Dr. Lincoln is one of the more reasonable people connected to this debate. He says some things that are fair, and also some things that are revealing of the current sociology, so it is worth exploring point by point.

Or…and bear with me here…some people actually find the Bullet Cluster (and the DF2/DF4 situation) to be persuasive.

The sentence starts with sociology: the “bear with me” trope is an assertion of reasonableness before it is demonstrated, which it may or may not be – often it is employed by people who think they’re being reasonable when really not so much. But in this case, yes: lots of people find the Bullet Cluster to be persuasive. My complaint is not that. It is that they cite the Bullet Cluster as an excuse to not think further about MOND and stifle debate. That has been my lived experience.

The Bullet Cluster achieved the status of a totem object long ago:

https://www.tumblr.com/calamitiesofnature-blog/16164456869/dark-matter

I myself find clusters persuasive. As I’ve written repeatedly, they pose a real problem for MOND. The Bullet Cluster is just one example, and an extremely weird case at that. The universe is big, so there’s always a unicorn somewhere: astronomers long ago learned not rely too much on the weirdest object in a category. So I’m more impressed when people cite clusters in general as a problem for MOND, both because that’s true, and because it evinces awareness of the subject beyond the totem that has become the go-to code word for dismissing a predictively successful paradigm without understanding it.

Dr. Lincoln also cites “the DF2/DF4 situation.” I’m not sure why that comes up if the Bullet Cluster by itself is entirely persuasive. This is a topic very much in my expertise, and I do not find it all that persuasive. I spent some time reading up on this situation – the data keep changing – in hopes of saying something quantitative here, and that left me feeling it was even less persuasive than I had given it credit for.

The DF2/DF4 (and now DF9) galaxies have become prominent examples of galaxies that appear to lack dark matter. Such objects are a problem for MOND, because you can imagine stripping away dark matter (though it is hard to do) but you can’t switch off the force law. However, the prominence of these particular objects has more to do with advertising (a ridiculous amount of attention has been devoted to these few weird objects) than with how convincing they are. That’s not to say they aren’t important, just that their importance is exaggerated.

These DF galaxies are a good example of cognitive dissonance in action. People give more weight to evidence that supports what they already believe, and less to that which supports things that don’t. In this case, the other evidence is all the other galaxies in the universe. This is a classic case of missing the forest for a few outlying trees.

I used to spend a lot of time fact-checking claims^# to falsify MOND. So when DF2 was first announced as a problem for MOND with great fanfare, I went to check. It was not. Indeed, had I known of this object’s in advance, I could have used MOND to correctly predict its velocity dispersion as I did with Crater 2 and the 30+ dwarf satellites of Andromeda. This rather quenched my enthusiasm for fact-checking every claim, so I vowed* not to spend more time doing so.

This is a damned-if-you-do, damned-if-you-don’t situation. If incorrect claims are left uncontested, the community seems to assume they are correct. If one spends the time to write a paper, you are diverted from your own research. Once published, much of the community remains unaware of the rebuttal, or chooses to believe their preferred narrative (another example of cognitive dissonance).

So before writing this post, I found myself breaking my vow. These are interesting objects, and there is now a new one (DF9). All three of these dwarfs have unusually large diameters and low velocity dispersions, about 8 km/s. That’s pretty much what we expect for the stars we see. No dark matter, no MOND. Even though it is really weird to find galaxies without dark matter in a universe made of dark matter that requires dark matter to make galaxies, it is even worse for MOND, if true.

There are devils in the details. The data for DF2 have changed repeatedly, both its distance and velocity dispersion. Which version to believe? Working my way through the literature, I found the statement “We find an instrumental resolution σ_inst = 0.375 Å (13.0 km s⁻¹)” and decided to stop right there. A rule of thumb in this business is that you shouldn’t try to measure a velocity dispersion smaller than your instrumental resolution because, well, you can’t resolve it. 8 km/s is smaller than 13 km/s. Now, in principle, you can tease more information out of the data, but that’s hard to do. In the best case, the two dispersions add in quadrature, so to infer 8 km/s, what you’ve really observed is √(8²+13²) = 15 km/s. That’s not much different from the instrumental resolution, and I’ve seen plenty of claims where that obscured a correct MOND prediction (e.g., Cetus). For DF2, we expect an intrinsic velocity dispersion of ~13 km/s, which is sensitive to the distance that keeps changing and to the EFE of the rough neighborhood in which these dwarfs find themselves. That corresponds to observing √(2*13²) = 18 km/s. So we have to be able to distinguish between 15 and 18 km/s. That can be done, but it is a lot less clean than the difference between 8 and 13 km/s sounds.

There are other corrections for broadening and binaries, so the above is the sanitized version. Binaries are hard to correct for in these unresolved objects. In nearby ultrafaint dwarfs where we measure velocity dispersions one star (or unresolved binary) at a time, the correction can be dramatic. Boötes III provides a rececnt example, having: “a velocity dispersion of σ_v=1.69^+1.03_−0.85 km s⁻¹, about six times smaller than the previously reported 10.7±3.5 km s⁻¹.” That takes it from having lots of dark matter, completely inconsistent with MOND, to bang on what MOND predicts. So you can perhaps appreciate my relutance to put too much credence in every claim made about measurements of these ultrafaint/ultradiffuse galaxies. This is the hardest place to work, and my experience has been that as the data improve, so too does agreement with MOND.

Is DF2 even a significant problem? Accepting the updated numbers as stated, corrected (and perhaps overcorrected) to reflect all the above effects, Keim et al. report two independent measurements for DF2 that give 6.3^+3.1_-3.7 and 9.2^+3.8_-4.5 km/s. In our paper we found that MOND predicts 13.4^+4.8_-3.7 km/s. Those are all consistent within the stated uncertainties: the one-sigma error bar of the lower measurement overlaps with that of the prediction,^& and the higher measurement is in as good agreement with MOND as could be expected given the uncertainty on both measurement and prediction. Dark matter advocates would be lauding such agreement to high heaven if they could^% make this prediction.

That’s a long segue for a parenthetical comment by Dr. Lincoln. It takes a lot of words to address five, and it isn’t for me to judge if the interpretation he seems to take for granted is better or worse than the alternative I describe. Even so, there’s a saying about this that might apply.

OK, let’s return to Dr. Lincoln’s comments. In case you’d forgotten – which I almost did, having spent so much time trying to track down DF2 data – this is a post about the sociology illustrated by Dr. Lincoln’s comments.

my views have changed over the years. In the early 1990s, I was pretty sure that dark matter was MACHOs. When MACHO, OGLE and all them disproved that conjecture, I then strongly favored MOND (broadly defined…)

My views have changed over the years too. In the early 1990s, I was completely sure that the dark matter was WIMPs. Had to be. I remember joking with other astronomers about how futile the search for MACHOs would be. As Rob Kennicutt put it in 1995 at IAU 171: “What are the MACHO people doing? Have they never heard of Big Bang Nucleosynthesis?” I recall shrugging and thinking that these projects wouldn’t detect dark matter, but they would provide a great variable star database. And so it came to pass.

We have here two different recollections of the same time period. Both are valid, but which is a fair representation of the community? Could be both. Gradually I’ve come to recognize that there were [at least] two distinct communities working on this issue, one in physics and one in astronomy. They don’t need to be echo chambers to develop mutually exclusive attitudes; the networks of communication can be broad and yet largely distinct. There is also a temporal aspect. I’m told by astronomers a few years older than myself that baryonic dark matter (particularly brown dwarfs) was favored in the late 1970s; indeed, that it seemed pretty obvious at that time. This changed quickly and was mostly (though never totally) supplanted by non-baryonic dark matter by the mid-1980s. Even that depended on the sub-community – those of us more concerned with cosmology rejected baryonic dark matter as nonviable while it remained reasonable to those of us more concerned with the dynamics of individual galaxies. I had a foot in both camps, and it metaphorically tore me apart.

In the mid-1990s, I was wrestling with our new data for low surface brightness galaxies. It did not make sense in terms of dark matter. Any kind of dark matter. I began to fear that the entire dark matter paradigm was no longer viable, a conclusion I fought tooth and nail to avoid. I worked much harder trying to save dark matter than I ever did subsequently working on MOND. More importantly, I think I was only receptive to MOND^{^} because I was already deeply concerned for the viability of dark matter. There lies the great schism: to me, dark matter was already practically falsified. No one else had that horrible, visceral experience – it was like losing a dear friend – so most of the community glibly ignored the surprising successes of MOND.

Maybe I was wrong to doubt dark matter? This is why I challenge other scientists to state their own criteria for its falsification. I don’t expect them to accept something so important just because I say so. But I do say so, and for good reasons – reasons most of them seem to be unaware of, so we’re starting from very different places. But if dark matter is a scientifically valid physical hypothesis, then it should be falsifiable. How can we tell if it is wrong?

I’m old enough to remember when cuspy dark matter halos were an absolute prediction of cold dark matter. That prediction failed, yet we find ways for CDM to persist. If we gave up on CDM as easily as we give up on MOND, we would have stopped talking about it thirty years ago.

Right. Where were we?

Yes, yes, the Bullet Cluster is moving awfully fast. Yes, yes, it’s a problem for LCDM. But this isn’t the same level of problem as it poses for modified physics. For LCDM, it is simply an unusual entry on the tail of a known velocity distribution, while the problems it poses for MOND-ish issues is more central.

I agree with this and I don’t. The issue is more central for MOND because a theory that seeks to supplant dark matter appears to need dark matter. So why bother?

It’s a good point, and as I’ve said over and over and over again, I find the situation in clusters profoundly dissatisfactory for MOND. Perhaps it even falsifies it. But a loss for MOND isn’t an automatic win for non-baryonic dark matter (which also requires new physics), so we bother because of everything else MOND does right that dark matter does not.

A common misconception here is that the unseen mass in MOND is necessarily non-baryonic. That’s a logical fallacy that stems from the sloppiness of the term “dark matter” which many of us equate with non-baryonic dark matter. Non-baryonic dark matter requires new physics. MOND requires new physics. So it sounds like we need double-new physics here when in fact we “just” need some undetected baryons^$ – something we need in both paradigms:

The agreement or mismatch between baryonic mass and observed velocity in LCDM (top) and MOND (bottom). As discussed before, the accounting of baryonic mass LCDM is better in galaxy clusters while MOND is better in all other gravitationally bound extragalactic systems.

What about the collision velocity? How bad a problem this is for LCDM depends on how unusual an entry it is in the tail of a known velocity distribution. There are many assessments of this; they range from “pretty unlikely” to “oh hell no.” For example, Lee & Komatsu found “the probability of finding 3000 km s^-1 in (2-3)R ₂₀₀ is between 3.3 × 10^-11 and 3.6 × 10^-9.” That’s worse than an unusual entry, that’s “oh hell no.” Various other workers hedge this way and that so it comes across as less improbable, but the most optimistic assessment I saw recently (and cannot now relocate) is that there is only a 10% chance of the Bullet Cluster existing in the volume of the universe that contains it. That’s over the whole sky, not just the 5% of the sky that has been surveyed in a way that could find it. So pretty unlikely, but not necessarily fatal. Ascencio et al. attempt to quantify this, finding that the “Bullet Cluster is in 2.78σ tension” with LCDM – that’s bad, but not fatal – but that including El Gordo results in a “combined tension” “estimated as 6.43σ”. That exceeds the usual 5σ threshold for fatal.

There is the obvious temptation to believe the assessment we prefer. There are analyses in which the Bullet Cluster collision speed doesn’t seem all that bad, though it never looks right. I’ve worked on this too, and we found it was pretty much impossible, and it looks to me like the more favorable analyses may be squeezing the toothpaste tube to make the collision lees improbable by shifting that to improbability in other parameters. One I recall requiring a ridiculously perfect, head-on bullseye collision.

In contrast, the high collision speed is entirely natural in MOND. That’s what happens as a consequence of the enhanced long-range force law. It just falls out, no muss, no fuss. So it isn’t just improbable in LCDM; it happens because of MOND, just like every other surprising result for the past forty years. This is why I say galaxy clusters ruin everything. We can’t say they favor one paradigm or the other without under-weighting some aspect of the data.

I guess that was enough science, because now we get to sociology:

to characterize those who disagree with you as somehow not being thoughtful is just…sloppy.

I may be wrong, but I am pretty much the antithesis of sloppy. Sloppy is describing me of characterizing those who disagree with me as not being thoughtful. I do not doubt the intellectual energy of people working on dark matter. I do doubt how deeply most of them have allowed themselves to think about MOND. I know this problem well, as I suffered it myself early on. MOND is too horrible to contemplate. Surely we can’t be that wrong! So we find some reason not to do so. The most common reason such people cite is the Bullet Cluster. It is my lived experience that lots of scientists (though certainly not all) fall into that category. If anybody finds that insulting, then they should do better to not be that person. I find it strange if not surprising that a scientist is offended at being challenged to think more about a topic about which he is, to use his word, dismissive.

FWIW, it will be difficult to convince me of >>ANY<< solution without DM being produced in particle accelerators. Indirect measurements are background-prone. And direct measurements, while super helpful, will tell us where to look in accelerators. (Think the DAMA debacle.)
My fear is that dark matter is real, but it only interacts gravitationally or far weaker than the weak force. If that’s the case, our grandkids will be having this argument.

OK, so this makes sense, but just let me note that it assumes that the solution to the mass discrepancy problem is a novel form of dark matter that can be created in an accelerator. That would be great if it could be done. All plausible dark matter particle candidates for which such a detection is plausible are pretty much excluded at this point, and building a bigger accelerator promises zero surety of success.

More generally, I don’t accept that the solution is a novel form of particle dark matter. That’s very much not in evidence! But I do share his fear of a non-interacting particle. I call that the Angel Particle because, indeed, we can argue about how many angels can dance in the core of a neutron star forever and ever. I have a greater fear, that by attending only to the failings of MOND while ignoring its successes – and a lot of scientists are guilty of that – we help to usher in a new age of dark epicycles.

I will not be available to respond to comments for a while, so I have deactivated them for this post.

^#Initially it came as a surprise how often claims to falsify MOND did not hold water. That has since become my experience with the vast majority of such claims. There is a powerful temptation to see in the data what one wants to see.

*For the most part, I’ve kept to this vow. It is hard not to check what MOND predicts when one hears claims that “it can’t do this!” when my experience, over and over again, is that usually it can. Not always, but usually.

^&I decline the opportunity to refine the prediction by chasing the continually changing distance estimate. Regardless of how close the current numbers are to the underlying truth, this is certainly not a five sigma exclusion of MOND.

^%The prediction cannot be made with dark matter. I’ve tried. Indeed, I’ve tried many things – there are multiple paths by which one might attempt to do this, and they do not yield a consistent answer. Worse, the one thing that is clear is that for low mass galaxies like these, there is a lot of scatter in DM halo mass at a given luminosity. The natural prediction from this is that galaxies of the same (low) luminosity have very different velocity dispersions. Yet the opposite is observed; luminosity is strongly predictive of kinematics. So we can fault MOND for predicting 3 km/s for Antlia 2 when 6 km/s is observed because it makes a prediction. We don’t fault LCDM because it does not make a comparably precise prediction to test.

^{^}Unlike Dr. Lincoln, I mean MOND specifically. I considered other modified gravity theories, but they fail. It is very obvious when the force law is wrong. Only MOND worked at the time and continues to work today. That it even comes close is telling us something profound, a lesson the community steadfastly refuses to learn.

^$I can hear from across the ocean Dr. Kroupa shouting that they’ve solved this problem.

Missing baryons: LCDM and MOND compared

In the last few posts we’ve discussed the local missing baryon problem in extragalactic objects spanning over ten orders of magnitude in mass from tiny dwarfs to rich clusters of galaxies. This discussion has so far been entirely in the context of LCDM. So – how does LCDM compare with MOND?

As a refresher, these are the data we’re trying to understand:

*The Extended Baryonic Tully-Fisher Relation (BTFR) for extragalactic objects. Rotating galaxies are shown as circles; objects dominated by pressure support as squares.* Adapted from Fig. 3 of McGaugh et al. (2026).

The flat rotation speed V_f is an indicator of the dynamical mass – that of the dark matter halo and all the baryons it contains in LCDM, and that of all the (presumptively baryonic) mass in MOND. In LCDM, it would be satisfactory for the baryon fraction of each object, m_b = M_b/M₂₀₀, to be equal to the cosmic baryon fraction (f_b = 0.157 according to Planck). For MOND, what you see is supposed to be what you get, so the baryon fraction should be one.

As we saw previously, m_b = f_b for rich clusters of galaxies. There is no local missing baryon problem for galaxy clusters: a satisfactory result. However, as we look at smaller systems, observations depart from this ideal. They do so systematically, with our accounting of baryons falling progressively shorter of our expectation as we examine progressively lower mass objects. This deficit is illustrated by the gray region here:

*The baryonic mass fraction as a function of baryonic mass. The horizontal line is the cosmic baryon fraction f_b = 0.157; the shaded region depicts the quantity of baryons that are missing.* Adapted from Fig. 4 of McGaugh et al. (2026).

Everything is fine for clusters at the high mass end (M_b > 10¹⁴ M_☉), and many people reasonably interpret that as corroboration of LCDM. For lower mass groups and bright galaxies, there is a deficit of a factor of two or three: an issue, but nothing too concerning by the standards of extragalactic astronomy, so this is widely ignored outside the community that works on it. The implicit assumption is that it’ll work out. But the magnitude of the problem continues to grow for smaller objects, becoming already an order of magnitude for intermediate mass galaxies. Not tiny dwarfs, just middle of the road spirals. The smallest mass dwarfs are worse off yet, missing over 90% of the baryons, approaching 98% or 99%. That is not satisfactory.

Making a straight-up comparison with MOND is a little tricky because the concept of a baryon fraction is a non-sequitor. There is no dark matter halo to compare against. Instead, we return to the concept of the velocity factor. In LCDM, we relate the observed flat rotation speed to that of the total dynamical mass through V_f = f_vV₂₀₀. Indeed, we can ask what velocity factor we need to explain away the missing baryon problem: maybe there are no missing baryons, just a systematic divergence of the observed V_f from the halo V₂₀₀. This can’t work, but it is useful to think about and provides a direct comparison with MOND.

In MOND, M_b = AV_f⁴ where A is the normalization^& of the BTFR. We can thus define an equivalent to the velocity factor, the residual velocity, taken here to be the ratio of the observed velocity to that expected for the observed mass, Δ_M = V_f,obs/V_f,pred. If the mass is a good predictor of the flat velocity, then Δ_M = 1. This leads to

**Figure 8** from McGaugh et al. (2026): The velocity factor in ΛCDM (top panel) and the residual velocity in MOND (bottom panel) as a function of baryonic mass. The gray region illustrates where each theory gets it wrong. The limits of this log-log plot are identical so that the areas of the shaded regions are directly comparable.

This is a straight-up comparison between the theories. Both theories suffer a missing baryon problem, but at different scales. The magnitude of each problem is indicated by the area of the shaded regions. (There is a dearth of data in our study^* from 10¹³ < M_b < 10¹⁴ M_☉, so we’ll just ignore that here.)

LCDM is spot on for clusters over the range 10¹⁴ < M_b < 10¹⁵ M_☉: f_v = 1 suffices to explain the data. Outside of that range, f_v must increase systematically to make up for what we previously attributed to missing baryons. In effect, we’re making the dark matter halos smaller so that the baryon fraction works out. As noted before, this can’t work, as rotation curve fits restrict the viable range of the velocity factor to 1 < f_v < 1.4, but we need it to grow to f_v = 5. That’s silly: at that point, the dark matter halo is contributing so little to the observed dynamics that we wouldn’t infer its existence at all.

MOND is spot on over the range 5 x 10⁵ < M_b < 5 x 10¹² M_☉: the data are consistent with Δ_M = 1. It falls short for rich clusters, where the observed mass of baryons in the intracluster medium (ICM) and the stars in galaxies predicts only ~80% of the observed velocity. This is the residual mass discrepancy in MOND.

For perspective, it helps to plot the linear baryon fraction. The astronomical scales of astronomical data oblige us to use logarithmic scales in many circumstances, but this may lead one to under-appreciate the scale of the issue. So here is the baryon fraction again, in both LCDM and MOND, this time with a linear scale:

The baryon fraction in LCDM (top) and MOND (bottom) as a function of mass. The scatter is an artifact of the propagation of errors when dividing one large, uncertain number (baryonic mass) by another large, uncertain number raised to a power (V_f³ in the top panel, V_f⁴ in the bottom). The data and their intrinsic scatter are the same but the scatter looks worse in the bottom panel because of the extra power of *V_f*. *(I ran out of patience translating every single datum; some of the least accurate data fall off the edge of this plot.)*

Individual galaxies and groups of galaxies are missing a lot of baryons in LCDM. This is not a subtle problem. It is not explained by simulations, nor am I aware of a satisfactory^% explanation. Worse, the apparent reason that we infer all these missing baryons is because the BTFR looks like the M_b ~ V_f⁴ of MOND rather than the M₂₀₀ ~ V₂₀₀³ of LCDM. With dark matter, we can accommodate pretty much any power law, or none at all – a lot of scatter would be more natural. So why did it have to be MOND? Even in ignorance of MOND the data pose a fine-tuning problem for LCDM. But it isn’t just a fine-tuning problem; it is a fine-tuning that arises because of MOND. To be successful, a LCDM model must be tuned to look like MOND. If it doesn’t, it’s wrong. If it does, why should we prefer a fine-tuned model to the theory that predicted the correct behavior in the first place?

MOND is not perfect here: it suffers a missing baryon problem in rich clusters. Since M_b ~ V_f⁴, predicting only ~80% of the observed velocity translates to missing ~60% of the mass. That’s a lot! But it could be worse: if, like Zwicky, we had done this experiment before the advent of X-ray observatories, we would be unaware of the mass of gas in the ICM, and infer that MOND was missing practically all (~96%!) the mass. That would seem utterly ridiculous, and we would conclude that MOND is wrong when much of the problem would have been that we were missing an important reservoir of baryons. Perhaps we still are. I do not like this possibility – there is still a lot of ground to make up, and I am not aware of a satisfactory solution. I guess I’m just a skeptic that way.

If we think the residual mass discrepancy problem MOND suffers in rich clusters is serious and perhaps fatal, should we not also conclude the same from the local missing baryon problem in LCDM?

But the bullet cluster double-secret falsifies MOND!

Let’s examine that assertion in the context of what we learned above.

The Bullet Cluster, which is made up of two galaxy clusters that collided a few billion years ago. The pink is the ICM observed by the Chandra X-ray Observatory. JWST provides the image of the many galaxies and also provides the data to map the mass through gravitational lensing (blue). Note that most of the mass indicated by lensing is centered on the galaxies, not the ICM. Image: NASA, ESA, CSA, STScI, CXC; Science: James Jee (Yonsei University/UC Davis), Sangjun Cha (Yonsei University), Kyle Finner (IPAC at Caltech)

The bullet cluster is composed of two clusters that collided and passed through one another. The collision segregated the gas of the ICM (pink above) from the galaxies. This happens because gas is diffuse and collisional. The gas of the two clusters can’t help smacking into each other, slowing down and forming the shock front visible in the shape of the gas of the smaller cluster on the right. Galaxies, on the other hand, have lots of empty space between them. They are collisionless and pass right by each other. In doing so, they are slowed less than the gas, getting ahead of it, leading to the separation that we observe.

OK, cool. The argument one usually hears against MOND based on this is that the baryonic mass in gas outweighs that in galaxies, so the lensing signal should be centered on the gas: the blue should align with the pink, not with the galaxies. Instead, we see the opposite, so the mass has to be dark matter.

This would be a good argument if the gas were all of the baryonic mass. This is a common assumption that makes sense in LCDM, where the baryon fraction checks out, so most people seem to stop thinking at that point. But each theory needs to be considered in its own context, and it cannot be the case in pure^# MOND that we see all the baryons^## in the picture above. That’s what we learned above. It may be unsatisfactory, but we knew this already before the bullet cluster was discovered (e.g., Sanders & McGaugh 2002). So the only new thing we learn from this aspect of the bullet cluster is that if there is an additional reservoir of baryonic mass, it is collisionless. It didn’t collide like the gas, it passed through like the galaxies. There are lots of candidate baryonic objects that fit that requirement: brown dwarfs, neutron stars, black holes, very small rocks^{^}. There is no requirement that the unseen mass be non-baryonic; we do not need the new physics of a new dark matter particle from beyond the Standard Model of particle physics on top of the new physics of MOND.

Now, as I think I’ve made clear, I am very uncomfortable with the apparent requirement that there is lots of undetected baryonic mass in clusters. If I were the MOND partisan that lots of people seem to assume I am, then I guess I’d portray this as a bold prediction. The dark baryons have to be there, and we should be turning all possible resources to detecting them, rather like we have for WIMPs. But I’m not that person. I am also not a person who sees this missing baryon problem for MOND as automatically worse than the missing baryon problem for LCDM. There is a much bigger deficit to be made up in LCDM, in many more systems^### of very different types over a larger dynamic range in mass. The missing baryon problem in LCDM looks worse to me than that in MOND. Yet the community attitude seems to be largely unaware of it. Those who are seem mostly to presume that it’ll work out. Maybe, but this should not be accepted by assumption, it needs to be demonstrated. It has yet to be.

If you think the missing baryon problem in clusters is a terrible problem for MOND, then you should be similarly worried that LCDM evinces the same kind of problem – one that is objectively larger in amplitude. It seems that, having accepted that there is dark matter, people don’t much care what it is. I do. The dark matter paradigm has obliged us to abandon parsimony. Not only does LCDM need two novel substances, dark matter and dark energy, it requires two kinds of dark matter: baryonic dark matter and non-baryonic dark matter.

There is a communal failure of objectivity about this. The thought process is both transparent and simple: MOND doesn’t explain clusters; it requires dark matter. Therefore dark matter^#### exists and it is silly to think about MOND. That would make sense if it weren’t a logical fallacy. Instead, it provides a permission structure to remain ignorant of what MOND gets right. I get that; there’s a lot to know. But I would also suggest that ignorance does not provide a strong basis for drawing scientific conclusions, especially for a subject so rife with confirmation bias and cognitive dissonance.

^&The normalization is related to Newton’s constant and Milgrom’s constant through A = ζ/(a₀G) where ζ is a factor of order unity that depends on the geometry of the system. It is one for spheres, and always approaches the limit ζ → 1 at sufficiently large radii, but observations are usually obtained at radii where the flattened geometry of disk galaxies is relevant, so in practice ζ ≈ 0.8. This can be derived from the geometry (all purely conventional; nothing to do with MOND) or one can obtain it empirically by comparing A = 50 M_☉ km^-4 s⁴ from fitting the BTFR to data for galaxies with known a₀; for a₀ = 1.2 x 10^-10 m s^-2, (a₀G)^-1 = 63 M_☉ km^-4 s⁴, so ζ =A(a₀G) = 50/63 = 0.8.

^*There remains room for improvement for poor clusters (here I call 10¹³ < M_b < 10¹⁴ M_☉ objects “poor clusters” because astronomical terminology can always be made worse). A particular issue is the quantity of intracluster gas, which dominates rich clusters (and is readily detected in X-rays), but seems to be absent in the smallest groups. There has to be a transition in between, but is it smooth so that all poor clusters have the same amount, or is there a huge variation in ICM mass among poor clusters? I have seen anecdotal indications that poor clusters that are detected in X-rays extend the trend of rich clusters while those that aren’t don’t, as if the residual mass discrepancy MOND evinces in clusters is somehow related to the presence of X-ray gas.

^%There are lots of unsatisfactory explanations. Some sound more plausible than others, but all fail to engage with the underlying prompt: why do the data look like MOND if we live in a universe made of dark matter?

^#It is possible that the problem MOND faces in clusters might not be one of missing mass, but rather it could be an indication of a deeper theory that is not exactly like pure MOND.

^##If there is additional mass in clusters, it doesn’t necessarily have to be baryonic. It could, in part, be neutrinos or sterile neutrinos or other more exotic beasts of the unknown meagerie of our enormous universe. However, there is no requirement that the unseen mass be anything other than mundane, ordinary matter.

^{^}Though an amusing thought, very small rocks do not make a viable candidate dark matter object any more than witches float because they weigh the same as a duck.

^###I have heard otherwise brilliant scientists dismiss the successes of MOND as a fluke. MOND has made too many successful predictions for that to be a reasonable assertion; it is a good example of what Putnam meant by “no miracles.” Yet the same scientists will cite the consistency of the baryon fraction in clusters to the cosmic baryon fraction as something that cannot be a fluke, ergo LCDM must be right. So which fluke is worse? I do not have patience to list all of MOND’s successful predictions here, though there are many reviews that do so and there will be a long paper soon that does more. What I will note here, having just done the exercise, is that the cluster baryon fraction is more likely to be a fluke. In order to estimate a baryonic mass for each cluster, we extrapolate the so-called beta profile that describes the distribution of X-ray gas. That’s a reasonable thing to do, and when we do it, we get an answer that is satisfactory in LCDM. However, it is not a small extrapolation. We are inferring a lot of baryonic mass at large radii from the fit of the beta profile at smaller radii. That’s the obvious thing to do, and I think it is probably correct, but it is also something that could go badly wrong. We experimented with other plausible gas mass profiles, and the answer can vary a lot, often leading to considerably fewer baryons than the cosmic fraction. That would be bad for LCDM, and also make the problem MOND suffers (too few baryons) worse, so it doesn’t help anything. But if there is a fluke here, it is more likely to be the coincidence of the cluster baryon fraction with the cosmic baryon fraction than is the consistency of the observed BTFR with the prediction of MOND for most of the rest of the universe.

^####This is where sloppy terminology leads to a logical fallacy: people equate “dark matter” with non-baryonic cold dark matter. The latter is a subset of the former; the unseen mass in MOND need not be the same as the non-baryonic stuff that we commonly assume the dark matter is.

Local baryons in simulations and reality

Our cosmology du jour, LCDM, suffers a local missing baryon problem: we don’t detect all of the baryons we expected to find associated with the dark matter halos of individual galaxies. Should we?

Empirically, yes. The stars and cold (atomic and molecular) gas appear to be all that there is to see in late type galaxies. Having additional large reservoirs of baryons results in a fine-tuning problem: the amount of extra stuff must vary precisely with galaxy mass so as not to impact the remarkably tight mass-rotation speed relation.

In theory, no. There are lots of places to stash extra baryons in phases where they are hard to detect. Warm/hot ionized gas in the circum-galactic medium (CGM) is one obvious place to harbor hard-to-detect phase of baryonic material that might add up to a lot of mass. Indeed, in many galaxy formation simulations, lots of mass winds up in the CGM. So what should we expect in LCDM?

That depends on who you ask. I went down a deep rabbit hole about this, both covering many different types of modern hydrodynamical simulations and how what we’ve expected has varied historically. It is a mess. But there are consistent threads: we do expect that galaxies might harbor extensive CGM (for reasons that vary) or that the missing baryons might not be in galaxies at all, having been ejected to the intergalactic medium (IGM) or prevented from accreting in the first place.

Rather than attempt a systematic survey of simulations, I’ll focus here on one example, EAGLE. The reason for this choice is that Mitchell & Schaye (2022) address exactly this subject. Here is their Fig. 1, which shows the baryon content of various components as a function of dark halo mass in the top panel. The bottom panel tracks where the heavy elements are, which is interesting, but I won’t address that here.

**Figure 1** from Mitchell & Schaye (2022): The median total baryonic mass (top panel) and metal mass (bottom panel) associated with the haloes of central galaxies at z = 0, normalized by the available baryon mass and plotted as a function of halo mass (M₂₀₀). Line colours indicate the mass in different components, including the CGM (cyan), ISM (green), stars (black), gas that has been ejected beyond R₂₀₀ (red), and gas that we estimate has been prevented from being accreted due to feedback effects (blue). Grey lines show the total mass, adding together each of these components. Solid lines show masses associated with the central subhalo, whereas dashed lines also include the masses associated with satellite subhaloes. For 10¹¹ < *M₂₀₀* < 10¹³ M_☉, most of the baryons that have ever been accreted on to haloes have since been ejected and reside outside R₂₀₀ by z = 0. Preventative feedback is important for *M₂₀₀* < *10¹² M_☉*. About half of the metals produced by stellar evolution are then ejected beyond R₂₀₀, apart from in very massive haloes.

There’s a lot going on here! For reference, our Milky Way resides in a ~10¹² M_☉ halo. Above M₂₀₀ > 10¹³ M_☉, most objects are groups and clusters where the distinction between centrals and satellites starts to matter. It’s complicated enough without that, so I’ll stick to individual galaxies. On the lower mass end, there’s a huge compression of the large range in stellar mass exhibited by dwarf galaxies into a relatively narrow range of halo mass, so the lower limit of M₂₀₀ = 10¹⁰ M_☉ captures many but not all dwarf galaxies.

That’s just the x-axis. The y-axis of the top panel shows the fraction of baryons in each component relative to the amount available in each halo (the product of the cosmic baryon fraction and the halo mass). The colored lines denote five different baryonic mass components. We readily observe two: the stars (black line) and cold gas (green line, denoted ISM in the figure legend). Additional components include gas in the CGM (cyan line) and gas that has been ejected to the IGM (red line) or was never accreted in the first place (blue line).

From this perspective, it seems hopeless to account for all the baryons on a halo-by-halo basis. Large galaxies (10¹¹ < M₂₀₀ < 10¹³ M_☉) eject most of their baryons (the red line exceeds all others). Lower mass galaxies also eject a lot of baryons, but most of them never get accreted in the first place (the blue line). There are a couple of reasons why accretion might be precluded. One is cosmic reionization: heating of the gas in the early universe by the first UV sources makes the gas too hot to stick to low mass halos: its thermal velocity exceeds their escape speed*. Another is feedback, in which the stars that do form in a galaxy return enough energy to the surrounding gas heat it up enough to prevent it from accreting. The latter process apparently dominates in the EAGLE simulations but which effect really dominates is a topic that simulators love to debate.

The largest reservoir of baryons that sticks to its dark matter halos is the CGM. This exceeds both stars and cold (ISM) gas for all halo masses. The CGM mass fraction increases with mass, which appears to be the opposite of what we need empirically, but really we need to sum up all three of the non-observed components to compare with the unseen baryonic mass that we infer:

**Figure 7** from McGaugh et al. (2026): *The ratio of missing-to-observed baryonic mass as a function of baryonic mass.*

That seems unlikely to add up, but it isn’t really possible to check. We don’t measure the missing component (by definition); we only infer its existence from the cosmic baryon fraction. One could laboriously check each simulation to see if the various missing components that should not be detected add up in the right way to explain the data, but one could always wave away any inconsistency by tweaking how many baryons get lost to the IGM. Between ejection to the IGM, prevention of accretion in the first place, and a quasi-undetectable CGM, the prospects for rigorously testing simulations are limited. However, each of these are distinct effects that occur in combination. This exacerbates the fine-tuning problem: not only does the unaccounted-for mass have to vary just so, these different mechanisms must somehow conspire to makes it so. It does not inspire confidence that this will work out when one realizes that these different mechanisms behave differently in different simulations.

We are not able to directly test the fraction of baryons that are prevented from accreting or that are ejected to the IGM. We have only the vaguest of constraints on the CGM restricted to massive galaxies. But we do measure the stellar and ISM gas mass, so we can compare the EAGLE simulation above to the data:

**Figure 6** from McGaugh et al. (2026): The stellar mass fraction (left panel) and gas mass fraction (right panel) as a function of mass M₂₀₀ with the equivalent V₂₀₀ on the top axis. Blue points are star-dominated spirals, green are gas rich dwarf irregulars, and yellow points are Local Group rotators. The lines show the expectation for central subhalos in the EAGLE simulations (Mitchell & Schaye 2022) with the width of the gray bands representing the range of the velocity fudge factor f_v = V_f/V₂₀₀ from *f_v* = 1 (bottom edge) to 1.4 (top edge). The dotted line in the right panel denotes the limit where gas is precluded from accreting onto halos in the EAGLE simulations (the blue line in the EAGLE figure).

To the eye habituated to astronomical accuracy, the stellar mass fraction in the left panel works out pretty well. The gray band representing the simulations does more or less the same thing as the data. However, this is one of the occasions on which we can fool ourselves with log-log plots. The bands are offset from the data by a factor that is not modest. The width of the bands already accounts for the plausible variation in the velocity fudge factor. One can of course consider implausible values of f_v, but the shape is also a problem. If we make an adjustment to match intermediate mass galaxies, the difference from high mass galaxies gets worse. One could make further tweaks, but this is a hopeless game as the shape problem stems from the curvature that is inevitable in abundance matching relations and the lack thereof in Tully-Fisher.

The gas content of EAGLE simulated galaxy-like objects does not compare well to the observed ISM in real galaxies (right panel). Gas is historically the hardest part to do in large magnetohydrodynamical cosmological simulations, so I’ve cut simulators a lot of slack, only occasionally pointing out that this doesn’t work out. But it really doesn’t work out, so if they want me to cut them slack then they should refrain^% from asserting that everything works out. It has become a tiresome, decades-long refrain that has never panned out.

The problem for cold ISM gas in massive galaxies in EAGLE is that there isn’t enough of it. The problem in intermediate mass galaxies is that there really isn’t enough of it. The typical value is off by an order of magnitude at M₂₀₀ ~ 10¹¹ M_☉. The problem in low mass galaxies is that it isn’t there at all. They typical EAGLE object with M₂₀₀ < 10^10.5 M_☉ has no cold gas at all. Such objects should not exist, apparently. But gas rich, low masses galaxies are boilerplate examples of observational reality, so it is a substantial problem for a simulation if such things are predicted to be rare^$.

There are many other LCDM simulations on the market. At most one of them can be correct. EAGLE is a reasonable example for illustrating what galaxy formation should plausibly do. Though not perfect, it is a reasonable representative of the LCDM brand. In this context, it makes sense to me that there would be all these various baryonic components and reservoirs. But reality doesn’t look like that. We add up the stars and cold gas and we’re done; anything extra involves fine-tuning. Maybe there should be more stuff associated with galaxies, but the fine-tuning problem this entails augers otherwise.

*I started to say a lot more about this here, but decided it was too deep a rabbit hole, so instead refer to a note about a conversation I had with Colin Norman on what the reionization scale should be.

^%The sociology in the simulation community seems to be to assert complete success in explaining everything at all times until the next batch of simulations completes running, then point out all the improvements. Everything is explained all the time, only more so as time goes on.

^$There is one caveat of comparing apples and oranges. The galaxies for which we have gas data are generally blue, late type (mostly dwarf irregular) galaxies. So we should make this comparison to similar objects in the simulation, but this distinction was not made by Mitchell & Schaye (2022). Persisting in my habit of giving the LCDM paradigm every benefit of the doubt, one can imagine that there is an as-yet undiscovered population of very low surface brightness galaxies that are red and gas poor pervading the universe, and that EAGLE is predicting these things are out there waiting to be discovered. The gas fractions are low because there are a lot of gas poor galaxies that we haven’t discovered yet. Having spent much of my career seeking low surface brightness galaxies, I’ve never been disappointed that there are more of them out there. I have, however, routinely been disappointed that there are enough of them to solve huge numerical discrepancies like this.

The local missing baryon problem

Last time, we started talking about the data in the recent paper The Baryonic Mass-Halo Mass Relation of Extragalactic Systems. Here, we’ll put on our dark matter hat, and use the data to make an accounting of the mass – both the dark matter and the baryons in all their various forms. From this conventional perspective we will obtain a method for relating what we see to what we don’t. In the context of LCDM cosmology, this provides an alternative approach to abundance matching. It also provides a test: are the two consistent?

The conventional picture we have in mind is a baryonic galaxy residing in a dark matter halo bathed in a background of intergalactic matter.

**Fig. 1** of McGaugh et al. (2026): Conceptual elements of a galaxy: the stars (yellow/blue) and atomic gas (green) of NGC 6946 (Spitzer 3.6µ and 21 cm data: F. Walter et al. 2008) are shown embedded in an extended dark matter halo (black). The dark matter density decreases continuously with radius so the halo has no hard edge, but for convenience we adopt the common convention that the radius r₂₀₀ marks the boundary of the dark matter halo and the dividing line between the circumgalactic medium (CGM) and the intergalactic medium (IGM; orange). The stars and atomic gas illustrated here appear within r < 20 kpc while r₂₀₀ ≈ 220 kpc (not shown to scale).

I’ve talked here about the stars and gas a lot because that’s what we see. These are the essential components that define a galaxy and comprise the mass that correlates with rotation velocity to make the baryonic Tully-Fisher relation (BTFR). I’ve talked a bit about the stuff between the galaxies, the intergalactic medium (IGM), but I don’t think I’ve previously had cause to talk much about the circumgalactic medium (CGM). As the name implies, this is gas in the vicinity of a galaxy, but not in the galaxy itself – at least not the part we can readily see. In the notional picture above, the distinction between the CGM and the IGM is the boundary of the dark matter halo that nominally demarcates gravitationally bound from unbound material.

Notional is doing a lot of work here. There’s a lot of gas in the IGM, and some of it is certainly in the vicinity of galaxies, so in that regard counts as circum-galactic. But there’s no hard and fast distinction between these components just as there’s no hard edge to a dark matter halo. Our brains don’t like that, so we impose notional boundaries and proceed as if these are meaningful.

Proceeding thus, we expect our dark matter halo* to contain its fair share of the cosmic baryon fraction, f_b = M_b/M₂₀₀ = 0.157 according to the Planck flavor of LCDM cosmology. We can test this by adding up all the baryons and comparing that to the total mass enclosed by r₂₀₀. This is straightforward for the stars and gas we see, but not for the stuff we don’t see – both dark matter and the gas in the CGM.

There are some measurements of the CGM, but these tend to be statistical in nature (if we stack data for a bunch of galaxies, we sorta see something), not the precise, individual, galaxy-by-galaxy measurements that we have for the stars and atomic gas. The stars and atomic gas are the mass in the extended Tully-Fisher relations we discussed previously, and are the bulk of the normal material in the galaxies we see. The bulk of the CGM lies at much larger radii, beyond the stars and atomic gas, but within the notional edge of the dark matter halo, as depicted above. Since we don’t measure it directly in individual galaxies, we’re gonna leave the mass of the CGM as an open question rather than something to be included in the sum of known baryonic mass.

The situation is even murkier for the dark matter, which we don’t see at all, so we don’t have a good way to measure the “total” mass of dark matter halos. This isn’t even a well-defined quantity in principle since halos are not expected to have a hard edge. Conventionally, we adopt the mass within a radius that contains a density two hundred times the cosmic critical density, r₂₀₀, as the notional edge. There are obscure historical reasons for this choice that I do not have the patience to describe. One could make other choices, arguably better choices, but r₂₀₀ is the most common choice used in the literature so we’ll stick with it here. The halo mass is the mass enclosed by this radius, M₂₀₀. If one goes through the math, it turns out that the circular speed of a test particle, V₂₀₀, orbiting at r₂₀₀ scales with the Hubble parameter [h = H₀/(100 km/s/Mpc)] such that V₂₀₀ = h r₂₀₀ when V₂₀₀ is in km/s and r₂₀₀ is in kpc. The dynamical mass (rV²/G) can then be written

M_{200} = (3.3 \times 10^5\;\mathrm{M}_{\odot}\,\mathrm{km}^{-3}\,\mathrm{s}^3)\,V_{200}^3.

That is a lot of huffing and puffing to get a way to relate the halo mass to something we can (kinda sorta) measure. The flat rotation velocity V_f has always been taken as the signature of the dark matter halo. One therefore expects V₂₀₀ ~ V_f. Indeed, these quantities cannot differ by much if dark matter is what explains flat rotation curves. However, the notional radius of the dark matter halo where V₂₀₀ occurs is much larger, by roughly an order of magnitude or more, than the radius where V_f is measured. So they need not be identical, depending on the halo model. So to relate what we measure to what we’d like to know we define a little ol’ fudge factor, f_v, such that:

V_f = f_v V_{200}

If a rotation curve stays flat indefinitely (as our empirical experience suggests), f_v = 1. If instead dark matter halos behave as they should in LCDM, then the rotation speed should gradually decline as we approach the halo’s edge so that f_v > 1. How much greater?

One way to estimate the fudge factor f_v is to fit dark matter halo models to data. This process does not directly measure V₂₀₀, but it does provide an estimate of that quantity based on the data available a smaller radii. One can do this for as many halo models as one has the patience to consider. For example, here are the results for two common halo models, the traditional pseudo-isothermal halo first adopted to explain flat rotation curves and the CDM-expected NFW halo:

**Figure 2** from McGaugh et al. (2026): The observed flat velocity V_f as it relates to the fitted V₂₀₀ for pseudo-isothermal (left panel) and NFW (right panel) halos (Li et al. 2020). Filled points have formal uncertainties <20% in V₂₀₀; open points are less accurate fits. The solid line shows V_f = V₂₀₀. The gray line in the right panel shows Equation (2a) of Katz et al. (2019), which corresponds roughly to f_v ≈ 1.4.

The result for pseudo-isothermal halos is consistent with f_v = 1, as expected – this model was adopted to make flat rotation curves. There is nevertheless some scatter. This typically happens because the observed rotation is not observed to be flat over a large enough range of radii to enforce flatness further out (as often happens in dwarf galaxies) or because the stars account for so much of the mass over the observed range that the inferred dark matter component is still rising (as often happens in bright, high surface brightness galaxies). This sort of haziness is inevitable when one only measures the inner few percent of the notional virial radius.

The result for NFW halos is approximately f_v = 1.4, albeit with a lot more scatter. This happens for the same reasons as above, with the additional problem that the dark matter profile in real galaxies rarely looks like NFW. Of all the many halo models considered by Li et al. (2020), NFW consistently performs the worst. One is forcing a fit of a function that would rather not. One signature of this misfit is the occurrence of very large V₂₀₀ for dwarf galaxies with small V_f. Taken literally, this would mean that some of the smallest dwarf galaxies reside in dark matter halos that outweigh those of giants like the Milky Way. This seems absurd, and it is. For example, by this approach, the dwarf galaxy NGC 3109 residing just outside the Local Group outweighs the Local Group and both its giants, Andromeda and the Milky Way, put together. But it is pretty clear from the local velocity field that the entire Local Group is not orbiting this little dwarf.

The estimation of huge V₂₀₀ for galaxies with small V_f happens because of the cusp-core problem. The density cusp predicted by NFW expects a curved shape for the inner rotation curve while the data show a more gradual, quasi-linear rise. Any decent fitting program will realize that it can make a curve look like a straight line if it stretches it out enough, so it does exactly this by making the halo very large. That sorta fits the data, but it makes no physical sense. Between this systematic effect and the large scatter induced by the other effects discussed above, one is better off inferring V₂₀₀ from V_f with a fixed fudge factor. So we’ll do that, leaving the exact value of f_v as an open question, but noting that for most objects it almost certainly resides in the narrow range

1 \le f_v \le 1.4.

That’s a lot of words to say the observed flat rotation speed gives us our best kinematic estimator or the dark matter halo mass. In this context, bear in mind the small scatter in the extended Tully-Fisher relations. This contrasts with the large scatter seen in the fits above. This strongly implies that V_f is more closely tied to the underlying mass^{^} than are the model-specific halo fits to the entire rotation curve. That might seem counterintuitive given that V_f is only a portion of the rotation curve (albeit a well-defined portion). However, it makes more sense when one considers that rotation curve fits must consider the contribution of stars as well as dark matter. Since the stellar mass-to-light ratio is never perfectly known, there is a degeneracy between the two that contributes to the scatter seen above. That variation is not real, it’s just an artifact of the fitting procedure. But when we get to large radii, beyond the confounding effects of the stellar population, the signature of the dominant mass becomes apparent in the flat rotation speed.

We saw above that we expect the halo mass M₂₀₀ to correlate with V₂₀₀. We observe that baryonic mass M_b correlates with the flat rotation velocity V_f. The natural assumption is that the stuff we see is proportional to the total (mostly dark) mass while the observed flat velocity is a property of the halo. Hence M_b ~ M₂₀₀ and V_f ~ V₂₀₀. This simple argument has been the basis for many papers claiming to explain the Tully-Fisher relation over the course of many years. This would be entirely satisfactory if it weren’t so completely wrong.

Here we need to introduce another fudge factor, m_b, that relates the mass we see to the halo that spawned each galaxy:

M_b = m_b\,M_{200}

The obvious assumption is that m_b is a constant for all galaxies, in which case Tully-Fisher follows because M_b ~ M₂₀₀ ~ V₂₀₀³ and V₂₀₀ ~ V_f. The wee problem is that this predicts a Tully-Fisher relation with slope 3: M_b ~ V_f³ when we observe one with slope 4: M_b ~ V_f⁴. In order to reconcile these two, our new fudge factor cannot be a constant. Worse, we need to fine tune it to transform the predicted power law into the observed one: m_b ~ V_f. That… doesn’t make any sense.

We can refrain from thinking and plunge ahead to simply plot the baryon fraction. While we’re at it, let’s also plot the stellar mass fraction m_* = M_*/M₂₀₀ because that is more commonly discussed in the literature. (Often stellar masses are available for galaxies without the corresponding gas mass measurements.) These fractions have to be increasing functions of circular velocity, or equivalently, mass (m_b ~ V_f ~ M_b^1/4):

**Figure 4** from McGaugh et al. (2026): The stellar mass fraction as a function of stellar mass (top) and the baryonic mass fraction as a function of baryonic mass (bottom). Data and symbols as in Figure 3 with the additional distinction that large squares in the top panel represent the sum of the stellar mass of all galaxies in a group or cluster while small squares are the stellar mass of the brightest galaxy only. The horizontal line is the cosmic baryon fraction f_b = 0.157 (Planck Collaboration et al. 2020). The colored lines in the top panels show the stellar mass–halo mass relations from abundance matching given by B. P. Moster et al. (2013; dashed–dotted green line), P. S. Behroozi et al. (2013; dashed–triple dotted pink line), and A. V. Kravtsov et al. (2018; red dashed line). The black line in the lower panel is m_b = f_b tanh(M_b/M₀)^1/4 where f_b is the cosmic baryon fraction (0.157) and M₀ = 5 x 10¹³ M_☉.

To be specific, I’ve computed the halo mass assuming f_v = 1. Different assumptions just slide the data up and down; the trend persists. This is discussed more in the paper if you’re interested in such details.

This gives a nifty way to relate what we can see to what we can’t. There’s a simple formula:

m_b = f_b \tanh\left(\frac{M_b}{M_0}\right)^{1/4}

where f_b = 0.157 is the cosmic baryon fraction and and M₀ = 5 x 10¹³ M_☉ is the scale where the function bends, transitioning from the M_b ~ V_f⁴ of the BTFR that holds over most of the mass range to the m_b = f_b of rich galaxy clusters. The precise value of the turnover mass is not well constrained, as it happens in the one place that is not well sampled by the available data. Indeed, there is nothing special about the functional form; it is simply a choice that transitions nicely from one regime to the other. There’s no physics in it^&. Still, this is a useful way to estimate the halo mass of pretty much any extragalactic object just by summing up its observed baryonic mass.

Indeed, this kinematic mass-matching relation is better than the widely used abundance matching relations in that it has less scatter. Abundance matching generally relies on stellar mass; that results in more scatter for the same reasons discussed for Tully-Fisher. This is particularly apparent at the low mass end of the top panel above, where galaxies of the same circular velocity (halo mass) have very different stellar masses. This goes away when baryonic mass is used instead.

There is reasonable agreement between abundance matching and kinematics at intermediate masses. The lines representing various abundance matching relations parallel the kinematic data. The offsets that are apparent can be cured by an appropriate choice of f_v. Always a free parameter to the rescue there is.

At the high mass end, things go amiss again. Partly this is because abundance matching relations reference the stellar mass of the “central” galaxy. The picture is that each halo contains one central galaxy with many satellite galaxies in subhalos, so what matters is the stellar mass of the central. This is overly simplistic: galaxy clusters are messy, the brightest galaxy isn’t necessarily at the center, and most have substructure with multiple groups rather than a single hierarchy. Besides that, the stellar mass tells you little about the halo mass without further environmental context: a galaxy with M_* ~ 4 x 10¹¹ M_☉ could reside in halo masses spanning a couple of orders of magnitude.

Setting aside the issue of centrals, there is a serious tension for individual high mass galaxies. The stellar mass fraction suggested by kinematics keeps going up where that of abundance matching turns over. This is due to the linearity of the Tully-Fisher relation compared to the knee in the Schechter function shape of the stellar mass function. The two don’t match up, as discussed previously. This same tension has long been with us; in the ’90s we were concerned with the difference between “the luminosity function normalization” and “the Tully-Fisher normalization.” This tension never went away. Still, the tension between abundance matching and kinematics doesn’t seem tragic, and might be remedied with some appropriate finagling of both the baryon fraction and the velocity fudge factor.

But where are all the baryons? They’re all accounted for in clusters, which reach the cosmic baryon fraction. But in no other system is the checksum complete. There is a missing baryon problem locally in each and every dark matter halo below the cluster scale. To confound matters further, there is a fine-tuning problem: the amount of missing baryons scales precisely with the amount of observed baryons.

The logarithmic plot above may understate the magnitude of the problem. To clarify this, we can plot the ratio of missing-to-observed baryons on a linear scale, at least in part:

**Figure 7** from McGaugh et al. (2026): The ratio of missing-to-observed baryonic mass as a function of baryonic mass. Data and symbols are the same as above. The ratio is linear in the bottom half of the diagram, then switches to logarithmic in the top half. Spiral galaxies are shown twice: once with f_v = 1.0 (solid blue circles) and again with f_v = 1.4 (small open circles). The Milky Way is the yellow point at the top of the gray band, which shows the range from zero CGM to that required to explain all of the locally missing baryons when f_v = 1. Stars represent the CGM measurements of Milky Way–mass galaxies by Miller & Bregman (2015), Bregman et al. (2022), and Zhang et al. (2026) from bottom to top. These suffice to explain the missing baryons provided that f_v ≈ 1.4. This explanation becomes progressively less plausible for lower mass galaxies.

The scatter blows up when we plot linear ratios; this is an artifact of error propagation. Nevertheless, it is helpful to see that the local missing baryon problem is not subtle. It is already a factor of ~2 for groups and ~3 for bright galaxies. It’s not as if we’ve misplaced a few percent of the baryons. Most of the baryons that should be associated with galaxy dark matter halos are not in evidence.

This problem has been known for a while, but doesn’t seem to be acknowledged to be a problem. Not all baryons need condense down into the central galaxy; some might be left behind, still mixed in with the dark matter halo. The widespread assumption seems to be that the missing baryons are probably in the CGM.

Accounting for the missing baryons with gas in the CGM almost works in bright galaxies like the Milky Way where we need “only” a factor of a few. Recent estimates suggest that the CGM is comparable in mass to the stars, or even somewhat more. These are very uncertain, as this mass is dispersed in diffuse gas over an enormous volume, and the total mass estimates often involve large extrapolations: the CGM is detected most readily nearby the central galaxy, but most of its implied mass is way far out near r₂₀₀. Accepting these estimates at face value leads to the star symbols in the plot above. This makes the checksum complete provided the halo is not too massive, as happens if f_v ≈ 1.4. This is what we expect for NFW halos, so it might work out if those were viable. However, there is a bigger issue.

The local missing baryon problem gets progressively worse for lower mass galaxies. For 10¹⁰ M_☉ galaxies – not all that much smaller than the Milky Way (M_b = 7 x 10¹⁰ M_☉), the problem isn’t a factor of two or three: there are ~6 baryons missing for every one that is observed. For 10⁹ M_☉ galaxies, the deficit is an order of magnitude. For even lower mass galaxies, the difference is so large we have to abandon the linear plot lest the interesting parts for bright galaxies get scrunched into invisibility. By the time we get to small dwarf galaxies of 10⁶ M_☉, the ratio of missing-to-observed baryons approaches 100:1. It is not plausible to imagine that the CGM of dwarf galaxies explains this deficit. (And yes, we’ve looked.)

A common explanation for this variation is that low mass dark matter halos have shallower potential wells, so have a harder time holding onto their baryons. Supernova can drive material out of galaxies; these go off with the same energy regardless of the galaxy they’re in so they may be more effective at blowing baryons out of lower mass systems. There is sufficient energy (IF properly^% distributed) to completely unbind the baryons, so they might wind up in the IGM, defeating any hope of completing the checksum. This is the sort of argument that sounds clever but fails to address the real problem. The difficulty isn’t just ridding ourselves of these meddlesome baryons, it is getting rid of exactly the right amount each and every time.

As awkward as it is to realize that most of the baryons that should be in low mass halos are not in evidence, it is not difficult to imagine ways in which this might happen, like the aforementioned supernova-driven galactic winds. The more dire aspect of the problem is the fine-tuning. Galaxies of the same observed baryonic mass are always missing the same amount of baryons, whether that’s a factor of 2 or 10 or 100. If the visible parts of a dwarf galaxy are only 1% of the available baryons, you’d expect a lot of scatter. Sometimes a halo of that mass might have 2% or even 3% of its baryons condense to the parts we see. That would show up in the scatter in a way it does not: galaxies of the same circular velocity (halo mass) have the same baryonic mass every time. They don’t vary by factors of two (or more). So while we can build models that makes the baryon fraction just so, the fact that we can write a simple equation for it with practically zero scatter is profoundly uncomfortable.

An extra bit of weirdness is that in LCDM, galaxies are built hierarchically by merging small objects into large ones. This poses a teleological problem. Consider a small halo at high redshift. If it remains alone, then it it will contain a dwarf galaxy at low redshift that has a low baryon fraction. But if it mergers into a larger system, then by the current time that larger system has to have a larger baryon fraction. In effect, a low mass halo has to know where it will end up some billions of years in the future. Will it remain alone and unmerged? Better blow out all those baryons! Will it merge into a larger system? Better hang on to the right amount of baryons. Does that system merge into a still larger object? Hope it held onto even more baryons, in exactly the right amount at every step along dozens of mergers.

I can imagine all this happening in a stochastic fashion with the net result being that more massive systems wind up with a higher baryon fraction, at least on average. I cannot give credence to this process resulting in the small observed scatter. As people are always telling me, “galaxies are complicated.” Indeed, they should be – in LCDM. But in reality they’re not! They obey simple scaling laws, laws that do not follow naturally from LCDM.

The local missing baryon problem encapsulates one of the fine-tuning problems that has never been satisfactorily explained. This alone would be considered fatal for most theories. For LCDM, it is just another problem to be addressed through the eternal tweaking of models and simulations.

*Strictly speaking, M₂₀₀ refers to all mass within r₂₀₀, baryons as well as dark matter. I’m going to call it halo mass anyway, because that’s what we mean, the baryons are a small fraction of the total, and because that’s what everybody does in the literature. If we make some other choice for the definition of the mass of the halo, M_Δ, then the inferred baryon fraction of an objects scales by M₂₀₀/M_Δ. The cosmic baryon fraction does not care what choice we make, so the implicit assumption is that one asymptotes to the cosmic fraction if one gets far enough out, irrespective of what r_Δ we adopt. While this is a sensible assumption – individual objects must merge into the larger cosmos at some point – there is no guarantee that the universe cooperates. For example, the baryon fraction in galaxies declines with increasing radius, but that in galaxy clusters increases with radius. I’ve seen hints that it doesn’t really settle down to the cosmic (or any particular) value. These are only hints – considerable extrapolation is involved – so we’ll ignore this inconvenience and assume that the baryon fractions of individual objects do in fact converge to the cosmic value far enough out.

^{^}It makes the most sense if the underlying total mass is the observed baryonic mass.

^&I made a very similar fit in McGaugh et al. (2010) but didn’t publish it because there was no physics in it. Since then the field has been awash in abundance matching relations that were similarly fit sans physics. There has been much ink spilled justifying it post-facto with feedback, but I have refrained from this exercise in intellectual onanism.

^%It is common to assume in simulations that a large fraction (50 – 100%) of the energy from supernovae is returned to the surrounding gas. This process is not resolved in cosmological simulations, all the energy return happens as part of the “subgrid” physics, so the feedback efficiency is set, in practice, to make things work out as well as possible.

Observationally, most of the SN energy finds its way out along the path of least resistance where the density of the surrounding gas is smallest (“chimneys”). This process couples to the surrounding gas with only a few percent efficiency.

Extended Tully-Fisher relations

Previously I had alluded to some of the major projects I’ve been working on. One has come to fruition and can be found on the arXiv and in the Astrophysical Journal^&. It has taken many years to assemble the data in this paper, during which time the models purporting to explain some of it have evolved considerably while consistently failing to address the real problems they raise. There is a lot to explore, so it will take more than one post.

Here I start with the empirical basis: the stellar mass and baryonic Tully-Fisher relations. The Tully-Fisher relation was originally discovered as a relation between luminosity and linewidth in rotationally supported galaxies – spirals and irregulars. It immediately proved useful as an extragalactic distance indicator. As such, it was instrumental in breaking the impasse in the Hubble constant^* debate (back when it was 50 vs. 100, not 67 vs. 73), and it remains useful in this role.

Physically, the obvious interpretation was that luminosity is a proxy for stellar mass and linewidth^{*^} is a proxy for rotation speed. This is correct. Of the various rotation speeds one can define and measure, the one that works best, in terms of minimizing the scatter in the relation, is the flat rotation speed measured in the outer parts of extended rotation curves. See Stark et al. (2009) and Trachternach et al. (2009) for further examples. The scatter is basically a function of data quality.

On the mass axis, converting measured flux to luminosity to mass is a bit dicier, as we need to know the distance for the first step and the stellar mass-to-light ratio for the second. There is inevitably some intrinsic scatter in the mass-to-light ratio of a stellar population. While I don’t doubt that luminosity is a proxy for stellar mass, improving on it is hard to do: there are many instances in which simply assuming a straight mapping of light to mass can be as effective as applying fancier population models. We might^{^} finally be getting past that, so it is worth discussing a bit.

The procedure to convert starlight into stellar mass involves the construction of stellar population models that use the color(s) or spectral energy distribution of a galaxy to infer the types of stars that make the light. This is a long-argued subject; suffice it to say there are a number of points where it can go wrong. The most obvious is the IMF; the initial spectrum of masses with which stars are born. Most of the light we see from galaxies is produced by its higher mass stars, which are disproportionately bright (there is a steep scaling of stellar luminosity with mass). But most of the mass is locked up in low mass stars that contribute little to the total luminosity. So we are, in effect, using the light of the few to represent the mass of the many. That would go badly wrong if we don’t know the relative mix, i.e., the shape of the IMF. This has been the subject of much research, and over many decades has been narrowed down pretty well. While I hope that this is almost settled, the specter of the IMF lurks as a menace to all stellar mass determinations.

There is a lot else we need to know to build a stellar population model. This includes such essentials as the spectra of individual stars of each and every type and stellar evolution as a function of mass and composition including exotic phases like the asymptotic giant branch. There are a lot of places where this can go badly wrong, and sometimes^{^%} does. So I wouldn’t say we know how to do this perfectly, but we have become pretty good at it.

Converting light to mass suffices to plot the stellar mass Tully-Fisher relation. That accounts for most of the baryonic mass of high mass spirals, but it ignores the mass of the interstellar gas. This can be appreciable in lower mass systems. Indeed, the standard issue dwarf galaxy in the field is more gas than stars:

**Figure 1** from McGaugh et al. (2019): The gas and stellar masses of rotating galaxies. Blue points are galaxies in the SPARC database (Lelli et al. 2016b) and the gas rich galaxies discussed by McGaugh (2012). The location of the Milky Way is noted in red (McGaugh 2016): it is a typical bright spiral. Grey points are the sample of Bradford et al. (2015). The line is the line of equality where M_* = M_g.

With measurements of mass and rotation speed, we can construct the Tully-Fisher relation:

**Figure 4** from McGaugh et al. (2019): The stellar mass (left) and baryonic Tully-Fisher relation (right). Data from Lelli et al. (2016b) and McGaugh (2012) are shown as blue points if both axes are measured with at least 20% accuracy; less accurate data are shown in grey. The latter include cases for which the rotation curve does not extend far enough to measure V_f, in which case the last measure point is used. These cases are systematically offset to lower velocity. Inclination uncertainties and distance errors also contribute to the scatter. The better the data, the tighter the relation. *The location of the Milky Way is noted in red (you are here).*

The stellar mass Tully-Fisher relation is a good correlation by the standards of extragalactic astronomy. The majority of studies in the literature are restricted to massive^% galaxies, mostly those with M_* > 10¹⁰ M_☉ where stars dominate the baryonic mass budget so the omission of gas is not obvious. As we look to lower masses, the relation bends and the scatter increases. That this happens right where gas starts to become important to the mass budget suggests that we’re missing an important component, and voila – a nice, continuous relation that is linear in log space is restored when we plot the baryonic mass Mb = M_*+M_g. Indeed, the data are consistent with a simple power law

M_b = A \, V_f^4

with A = 50 M_☉ km^-4 s⁴. The intercept A has consistently been measured within 10% of this value over the past couple of decades. That this is an integer power law so that the intercept has real physical units is intriguing. That doesn’t happen in most astronomical scaling laws, which are usually more happenstance, like the mass-luminosity relation for main sequence stars.

Why limit ourselves to rotationally supported galaxies? Let’s plots every known type of gravitationally bound extragalactic object, from the smallest ultrafaint dwarfs to the largest clusters of galaxies. Note that I’ve flipped the axes to accommodate the huge dynamic range in baryonic mass, roughly twelve (12) orders of magnitude. This is like having gnats at one end of the scale and blue whales at the other. On that scale, a person is a regular galaxy like the Milky Way.

**Figure 3** from McGaugh et al. (2026): Extended Tully-Fisher relations plotting the flat-equivalent circular velocity of extragalactic systems as a function of stellar mass (top panel) and baryonic mass (bottom panel). Data for rotationally supported galaxies are depicted by circles; squares represent pressure supported systems. The blue circles are galaxies with directly measured distances, V_f from rotation curves, and stellar masses from WISE photometry from Duey et al. (2026, in preparation). Green circles are gas-rich galaxies (M_g > M_*; Stark et al. 2009; Trachternach et al. 2009; Bernstein-Cooper et al. 2014; McNichols et al. 2016; Iorio et al. 2017; Namumba et al. 2025; Xu et al. 2025) not already in Duey et al. (2026). Yellow points are Local Group galaxies, both spirals and dwarfs (McGaugh et al. 2021); gray squares are ultrafaint dwarfs (Lelli et al. 2017). Lensing results for early- and late-type galaxies (Mistele et al. 2024a) are shown as pink squares and magenta circles, respectively. Red squares are clusters of galaxies (Mistele et al. 2025), and purple squares are groups of galaxies (McGaugh et al. 2026). The orange line is the BTFR fit only to rotating galaxies over a more limited range (about three orders of magnitude in baryonic mass, from M_b ~ 4 x 10⁸ to 4 x 10¹¹ M_☉) by McGaugh (2005).

One improvement from twenty years ago, aside from the greater number of objects and the increase in dynamic range, is the accuracy of the mass measurements. I tried a number of prescriptions for the stellar mass-to-light ratio in McGaugh (2005), which resulted in a range of possible slopes. Now we just use the stellar mass from precise population models (Duey et al. 2025) and recover my best estimate from back then. The room to dodge the obvious conclusion about the slope of the relation by complaining about the choice of stellar mass estimator – a popular course of action back then – is gone. Another technical issue we’ve spent a lot of effort working on is how to put all these very different systems on the same scale of V_f. I won’t elaborate on this here: if you’re interested in that level of detail, you can go read the paper and references there in. If we got this wrong, it would add to the scatter in the relation, and/or create offsets between different types of data.

Both of the extended Tully-Fisher relations, that in stellar mass (top panel) and that in baryonic mass (bottom panel, the extended BTFR) are good correlations. That in baryonic mass is clearly better in the sense that it is tighter over a larger dynamic range. From small dwarf galaxies (M_b ~ 5 x 10⁵) to groups of galaxies (5 x 10¹² M_☉), the data are consistent with a single power law (M_b ~ V_f⁴) for all systems with remarkably little scatter. Outside this range, the data for both the lowest and the highest mass systems deviate from a straight line towards higher mass at a given flat velocity. I don’t put much credence in the smallest systems as I think there is little chance that their measured velocity dispersions are representative of their equilibrium gravitational potential. For all practical purposes, our knowledge runs out as we hit the regime of ultrafaint^# dwarfs. The deviations of the most massive systems, clusters of galaxies, are more difficult to dismiss.

Restricting our attention for the moment to the range where a single power law suffices to describe the data, we note that there is not much scatter in the BTFR. Some of it is from random uncertainties; these dominate most studies and lead to a lot more scatter than seen here: these data are very good. We can account for the known observational errors and subtract off their contribution to estimate the intrinsic scatter in the relation. This is the variance of the data from a perfect line. The intrinsic scatter for the best data (the WISE-SPARC sample of Duey et al. 2026) is about 0.11 dex in mass – about what we expect^$ for stellar populations. That doesn’t leave much room for other sources of scatter, so the underlying physical relation has to be very tight indeed: essentially perfect over the range 5 x 10⁵ < M_b < 5 x 10¹² M_☉.

Scatter will also occur if our mass budget is incomplete. We can see this in the transition from the stars-only relation to the BTFR. There is a lot of scatter in the stellar mass Tully-Fisher relation around 10⁷ < M_b < 10⁹ M_☉. Galaxies in this mass range are sometimes star-dominated and sometimes gas-dominated. The gas fraction is all over the place. This shows up as scatter in the stellar mass Tully-Fisher relation. That’s not real; it is a sign that we’ve missed an important mass reservoir. This is cured when we add in the gas mass, which is dominated by atomic gas (HI to spectroscopists and astronomers). That this addition removes the scatter and restores a single power law relation strongly suggests that there are no further substantial reservoirs^** of baryonic material that we’re missing.

This logic applies to other systems as well. Bright spirals do not need much correction because their baryonic mass is dominated by stars. Their stellar mass Tully-Fisher relation is pretty much already their BTFR.

Perhaps this applies to clusters of galaxies as well? There was a huge correction from stars-only to stars plus gas. The gas in this case is the hot, ionized plasma of the intracluster medium (ICM) that belongs to the cluster itself and not any individual galaxy within it. That goes most of the way to close the gap between the stars-only cluster data and the extrapolation of the BTFR fit to individual galaxies, but not all the way. So perhaps we are still missing an important baryonic mass component? It happened before – we didn’t know about the ICM for decades after Zwicky first identified the missing mass problem in clusters – so perhaps there are still more baryons to discover there.

It could also be that the apparent offset occurs because we’ve failed to put clusters on the same V_f scale as galaxies. This is not easy to do, and we’ve spent a lot of time worrying about it. I don’t think this is what’s going on, though it would make my life a lot simpler if it were. Different indicators – dynamics vs. ICM hydrostatics vs. gravitational lensing – can give somewhat different answers, but not in a way that “fixes” the problem: I see no viable path in which the offset turns out to be a simple difference in the way the depth of the gravitational potential is measured. I would love to be wrong here, but I’m not dismissing the offset for clusters as I am for ultrafaint dwarfs (which don’t do lightly).

Perhaps the extrapolation of the BTFR from individual galaxies to clusters is simply not appropriate. They’re very different kinds of systems, after all. To dig into that, we need some theoretical perspective – why does the observed power law happen? Should we expect different systems to share the same BTFR?

Theory is something I’ve studiously avoided in this post: the possibility that there are baryons that remain to be discovered in clusters can be inferred empirically. All the other data line up, so why not clusters? But unless and until these hypothetical additional baryons are discovered, that’s just one possibility. How likely this possibility seems to be diverges rapidly once we overlay a theoretical preference, which I will leave to future posts. (I did warn it would take more than one.)

^&This paper appears in ApJ volume 1001. The literature has grown quite a bit since I started contributing to it in volume 342. The Astrophysical Journal was founded in 1895. So I’ve been contributing to it for a little over a quarter of its temporal existence, but nearly twice the number of volumes have been published in that shorter time. It’s no wonder none of us can keep up.

^*Indeed, Tully & Fisher’s “preliminary estimate of the Hubble constant is H₀ = 80 km/s/Mpc” remains correct to this day, within the uncertainties (hard to estimate at the time, but roughly ±10 km/s/Mpc).

^{*^}There appears to be an irreducible intrinsic scatter in the linewidth: it is not a perfect proxy for rotation speed. Linewidths are observationally easier to obtain than resolved, extended rotation curves, so the numbers of galaxies in samples using linewidths can be very large without ever approaching the quality provided by resolved interferometric observations. Bigger samples are not necessarily better.

^{^}I emphasize might here because the community seems to have moved towards reporting stellar masses as if we observe these rather than the luminosities and colors/SEDs that the mass estimates are based upon. The latter are data – observed quantities – while stellar masses are a derived quantity that is inevitably model dependent. This doesn’t stop being true just because we decide to invest a lot of faith in our models.

^{*^}The Sloan Digital Sky Survey provides stellar masses based on models that are known to be wrong in the near infrared. Since SDSS itself is entirely optical, one might not notice. If one mixes SDSS data with near-IR data, one will get the wrong answer.

^%This is a classic selection effect. Brighter objects can be seen at a much greater distance than dim ones, so probe a much larger volume. Consequently, their raw numbers always dominate surveys even if their number density is low. Stars are a great example: most of the stars you can see at night are intrinsically luminous: bright stars that are rather far away. Mundane, low mass stars do not stand out even when nearby.

^#This isn’t for lack of observations of ultrafaint dwarfs, it’s the underlying assumptions.

^$No amount of information suffices to perfectly specify the stellar mass that produces an observed luminosity and SED (spectral energy distribution/set of colors), so one always expects at least some intrinsic scatter in the stellar mass-to-light ratio. I’ve seen estimates that range from 0.1 – 0.2 dex for near-IR colors. That’s as good as it can get as there is always some transient population (e.g., AGB stars) that produce an amount of light that depends on the star formation rate some time ago, not what we measure now. Optical colors are worse in the sense of having more intrinsic scatter, as they are more susceptible to the comings and goings of bright but short-lived stars whose numbers fluctuate with the stochastic star formation rate. Finding 0.11 dex intrinstic scatter is pretty much as good as it can get. (By dex we mean the scatter in log space.)

^**We noted this effect in the original BTFR paper to argue that it was unlikely that we were missing substantial amounts of molecular gas (H₂), which was a concern at the time. Flash forward, and we were right: the molecular gas mass is almost always a distant third behind stars and atomic gas in the baryonic mass budgets of individual galaxies. Nowadays, the concern is about the mass of baryons in the circumgalactic medium (CGM). That’s getting ahead of the story, which I’ll save for a future post. For now, it suffices to note that any baryonic mass in the CGM is far beyond the radius where the flat velocity is measured, so is not relevant to the sums here.

Yep, it’s a religion

I have been concerned for years that dark matter was morphing from legitimate science into a cold, dark religion. I have been reluctant to put it that way, because there are lots of scientists who work on dark matter that have not fallen entirely down that rabbit hole and who continue to make valuable contributions working in that context. But a recent experience reminded me that my concerns were not misplaced, and there are plenty of scientists who have fallen irredeemably down this rabbit hole. No matter what answer the future holds to be correct, many current scientists will have gone to their graves in denial of it.

Where is the boundary between science and religion? It is hard to assess where the borderline is. But it is easy to see when people are far over the line – so far over that it doesn’t really matter where exactly the line is. One can attend any conference on the subject to find people who unabashedly assert that dark matter exists without question. Not just that acceleration discrepancies have been amply demonstrated empirically, but that the only possible interpretation is dark matter. If asked whether this invisible mass is in the room with us now, they will enthusiastically^# answer yes! Since dark matter has not been detected in the laboratory, this assertion is an expression of faith – the hallmark of religion – not of an established scientific fact. What we have established is that there are discrepancies between what we see and what we get when we assume Newtonian gravity (or GR, if needed). What we don’t know is whether the cause of these discrepancies is some form of invisible mass (dark matter) or if the equations we employ are inadequate (modified gravity [or more generally, dynamics]).

Indeed, these days many people will assert that dark matter has already been detected, usually citing astronomical evidence that used to be considered too feeble to merit a Nobel prize. Funny how repeating a mantra long enough morphs an aspiration into accepted reality. Modern physics is not providing a strong falsification of the supposition that science is a social construct.

A prominent example of an observation of the sky that is frequently cited as absolutely requiring cold dark matter is the acoustic power spectrum of the cosmic microwave background. Quoting clayton from a few years ago:

the primary reason to believe in the phenomenon of cold dark matter is the very high precision with which we measure the CMB power spectrum, especially modes beyond the second acoustic peak. There is a stone-cold, qualitative, crystal clear prediction of CDM about the relative sizes of the second and third peaks that modified gravity profoundly and irredeemably gets wrong: it thinks the third peak should be relatively larger* than the second… whereas CDM thinks they should be about the same

I would accept that this were conclusive proof of dark matter if this were the unique prediction of dark matter: that there was no other way to do it, so all other approaches were indeed irredeemable. (Quite the strong language, eh?) The problem is that CDM is not the one unique was to fit these data. Skordis & Zlosnik showed that it is possible to write a modified gravity theory that also fits the CMB data:

*CMB power spectrum observed by Planck fit by AeST (Skordis & Zlosnik 2021).*

This does not prove the AeST theory of Skordis & Zlosnik is correct, but it does demonstrate that it is possible to write a modified gravity theory that does indeed do what it is frequently asserted to be impossible for a modified gravity theory to do. I’ve heard of a couple of other theories that can also do this (the relativistic Khronon theory of Blanchet and nonlocal MOND as discussed by Deffayet & Woodard), so clearly this success is not uniquely limited to cold dark matter, or even a particular modified gravity theory. The work of Skordis & Zlosnik (2021) was known and in the literature before clayton made the assertion above in late 2022, so either he wasn’t paying attention (likely) or is convinced that it is impossible so doesn’t even consider the possibility (also likely). The former just says we’re all too busy, but the latter is a mark of religious thinking: my god is the only god, thou shalt have no other hypotheses before^& me.

Many people are very impressed with the quality of the LCDM fit to the CMB. That is indeed very good, but there are enough free parameters that we were going to get a fit to any physically plausible power spectrum. If not, we’ve never been shy about making up new parameters. (Evolving dark energy, anyone? How about a running power spectrum? There’s a whole bag of possibilities!) What I’ve been more impressed with is the consistency of the fit to the CMB data with the many independent constraints on conventional cosmology. Or at least it was, until it wasn’t.

The Hubble tension has gotten steadily worse (in terms of statistical significance), and it really does not look like local measurements are to blame, nor is it the only tension. People seem to miss that it is the CMB-fitted value of the Hubble constant that has evolved over time to spoil the concordance that got us to believe in LCDM in the first place. But if the CMB is the cornerstone of your religion, all other data must inevitably be at fault and can be ignored: there is an entire community of cosmologists who choose to believe the best-fit Planck cosmology to the exclusion of all other data. It’s like the bad old days of the Hubble tension all over again, with the physics community choosing to believe the lower value of H₀ because it makes more sense for the aspects of cosmology that they care about while those in the astronomical community who actually measure H₀ find a persistently higher value.

A real tension in LCDM implies the need for new physics of the unknown variety. One doesn’t want to go there if it can be helped. I didn’t consider MOND until I was already concerned for the viability of dark matter. There are real problems for the paradigm that its more intense advocates simply deny, brush aside without real thought, or choose to remain ignorant of. When they are confronted with a problem, they are pretty creative about making stuff up on the spot. Anything to avoid having to confront the unspeakable – another hallmark of religion.

For example, cold dark matter is scale free. That’s foundational to the hypothesis. So the existence of an acceleration scale in the kinematic data is anathema to CDM. When I first pointed this contradiction out, there were a variety of assertions to the effect of “does too!” One example is provided by Kaplinghat & Turner, who claim to show “how Milgrom’s law comes about in the cold dark matter theory of structure formation.” That would, indeed, be ideal, and is a requirement for any theory to be successful.

Wee problem: they demonstrat no such thing. CDM is scale free, yet K&T claim that it explains Milgrom’s Law, which is predicated on the existence of an acceleration scale. Well, which is it? Is CDM scale free? Or does it explains the acceleration scale? We can’t have it both ways: their very premise is self-contradictory. It is absurd on its face.

The acceleration scale is defined by baryons, for which K&T have no model. To connect baryons with dark matter, they make a hand-waving argument about galaxies reaching a₀ at the edge of their disks. This is not even a concept of a model and does not begin to suffice as an explanation for many reasons, a prominent one being that low surface brightness galaxies have accelerations less than a₀ everywhere:

Centripetal acceleration curves color coded by galaxy surface brightness. Low surface brightness galaxies (blue colors) have low (sub-a₀) accelerations everywhere: there is no edge at which they reach a₀. (Adapted from McGaugh 2020.)

Milgrom pointed out this and many other shortcomings of their scenario, so I feel no need to elaborate further. Milgrom eviscerated their paper so thoroughly that the proper course of action would have been to retract it. Instead, they simply never acknowledge the criticism, and persist to this day in pushing it as some sort of valid scientific explanation. It is not; it does not withstand even mild critical scrutiny. But it doesn’t need to: it reassures the faithful that all is well. They hear what they want to hear without questioning its veracity. That’s another hallmark of religion.

I have refrained from saying these things in the past because I’m too nice. For example, a few years ago I started then abandoned the draft text below, which I simply cut & paste:

One of the things that attracted me to a career in science is the notion of objectivity. I grew up for a time in the bible belt, where people earnestly believed things that were obviously untrue, even to the eyes of a small child. On the occasions that I had the temerity to point out the obvious, the contradictions posed by facts never had an impact on their belief system. Rather, it inevitably earned me a warning that I was going to hell. No few of these people seemed to think it was their religious duty to send me there prematurely, or at least to make life on Earth a living hell.

Scientists eschew such behavior, but are also human, so often engage in it anyway. I’ve encountered it a lot. I get it; I went through the same denial, grief, and anger over the prospect of losing my good friend cold dark matter. The stages of grief never brought something back from the dead, but it has engendered a lot of blame-the-messenger.

Here’s an example, from a review by Mike Turner:

There is a lot of misinformation packed into this short paragraph.

The first clue is right there at the beginning, in red: the heading “False starts.” This is false framing, a classic tool of propagandists. It starts from the outset by asserting that the topic to be discussed is wrong at a level of knowledge so common it requires no justification. This is not the way one starts an objective discussion, much less a scientific one.

Turner then misconstrues what Milgrom did. He didn’t notice the scale a₀ in the data, for which there was scant evidence at the time. Rather, Milgrom made the obvious statement that the inference of dark matter relied on the assumption that dynamics, as encapsulated by the laws of inertia and gravity, is the same on the very different scales of galaxies as in the solar system where they were established, so we ought to consider if dynamics might change in some way. He quickly excluded a size dependence as a possibility. How he settled on acceleration is beyond the scope of this post, and not for me to say. Neither is it for Turner to say.

After a brief and incomplete description of what MOND is, Turner allows that “this one-parameter model fits all the rotation-curve data”. Even in making this admission, he chooses to call it a model rather than a theory. A model is something specific you build in the context of a theory, like a halo model in CDM. MOND is more than that.

Turner quickly moves on without contemplating any meaning that rotation curves might hold. Let’s pause to consider that.

First, I would not say that MOND fits all the rotation curve data. It fits most galaxies, but there are a minority of weird cases that are not well fit. The weird cases inevitably don’t make sense in terms of dark matter either, so on the whole I interpret this to be the usual price of dealing with astronomical data – some of it is just goofy. Setting such cases aside, I can and have fit the same data with all sorts of dark matter halo models. MOND requires fewer parameters, which is important, but the difference isn’t in the fitting. The difference is in predictive ability. I can use MOND to predict the dynamics of galaxies a priori, and have done so many times. I cannot use any flavor of dark matter theory to do the same, and it’s not for lack of trying.

The predictive power of MOND must be telling us something, even if it is something about the nature of dark matter or the process of galaxy formation. There are many papers written on this, some deep and profound, others absurd and banal. Turner cites none of them, nor displays any awareness that such work exists. I would venture to guess that is because acknowledging such work would imply that there is something to debate here, something he would apparently rather not admit.

That’s where I left off. It’s exhausting deciphering other people’s false assertions. Moreover, I just don’t like criticizing other people, no matter how richly they deserve it. (Turner has never refrained from criticizing me in ad hominem terms: on one occasion^$ he showed my picture to an audience and called me “the enemy.”) A large segment of the particle physics and cosmology community appears to think this way, and has succumbed to a scientific version of bible thumping in which you can assert any absurd thing so long as it falls within the framework of the holy LCDM. They really need to find something better to do.

I had hoped we were past this, but I heard a talk last week that was exactly in this mode. To paraphrase, the talk went

We’re sure dark matter exists. We have been sure about it for decades. In that time, we have been repeatedly proven wrong about what it is. Rather than re-think our paradigm in the face of these repeated failures, we double down yet again on the existence of this invisible, undetected mass, asserting aggressively^% that it must be true while eliding or misrepresenting the evidence that it is not. This enables us to make up a whole lot of exciting new possibilities for what the dark matter might be and conceive of ever more grandiose experiments to continue not to detect it. You must believe in dark matter!

This was not a science talk so much as an indoctrination session. It was as if I had stumbled into a revivalist tent where some hothead was preaching to the choir. This is the kind of talk that misled an entire generation into wasting their careers at the bottom of a mine shaft searching for WIMPs. At least WIMPs were a well-motivated hypothesis; this kind of talk could lead a new generation down an even greater variety of garden paths.

I am well aware that I might fall prey to this attitude myself. That’s why I set criteria by which I would change my mind: detect dark matter already, or at least provide a satisfactory explanation as to how MOND comes about. Neither of those criteria have been met. There are claims to do the latter, but so far these are just variations on models I tried and found to fail long ago. If I thought these could work, I would have said so. At the same time, I don’t see any dark matter advocates taking up the challenge to specify what would change their minds. When I ask them what could falsify dark matter, I get dumbfounded looks – the deer-in-the-headlight face one gets when the immediate response why would you even ask that? is checked by a distant memory that scientific theories are supposed to be falsifiable.

Personally, I found it humbling to encounter MOND in my own data. I too thought we understood the universe with dark matter. But who ordered this? Certainly not me: my own conventional, dark-matter based predictions were falsified. No one else working in the context of dark matter had got it right at the time either. Only Milgrom ordered this.

And what is this? There is a direct connection between what we see and what we get. Even in ignorance of MOND, the radial acceleration relation encodes a one-to-one relation between the distribution of baryons and the effective force. This is so direct that one can right down a single equation connecting the two:

g_{obs} = F(g_N/a_0)\,g_N.

The observed acceleration is a simple function of that predicted by Newton for the stars and gas that we see. There is no mention of unseen mass; everything is specified by what we can see is there.

I’ve sometimes heard astronomers complain about the reductionist ethos of physics, trying to cram all the complexity of the entire universe into a theory of everything. But here it is appropriate: there is a single, apparently universal force-law at work in galaxies. That’s telling us something profound. And yet if questioned about this, the physicists are the ones who will complain that galaxies are complicated, so they should be exempted from having to explain them. Galaxies should be complicated – in LCDM. But they’re observed not to be, in the sense that a single equation suffices to describe their kinematics. The problem isn’t that galaxies are inexplicably complicated, it’s that they should be but aren’t.

I am deeply disappointed that many scientists apparently lack the physical intuition to immediately recognize the import of the simple relation between what we see and what we get. It is the same sort of thing Newton noticed in the solar system: everything happens as if the gravitational force is proportional to the product of the masses and the inverse square of their separation. He didn’t understand why at the time, and was criticized for indulging in magical thinking: how can there be action at a distance? But that’s what the data were saying, and the same applies now. We might not yet understand the why, but that the data look as if MOND is what’s happening in this universe.

^#The framing has morphed over the years. A recent advent is that some people have started proactively asserting that invisible mass is in the room with us now in order to avoid having to answer it as a question that makes them sound like loonies.

*He means the third peak should be smaller than the second, not larger, if by “it” he means modified gravity with the baryon density expected from big bang nucleosynthesis, which was the hypothesis that correctly predicted the first-to-second peak ratio but does indeed get the second-to-third peak ratio wrong. Funny how the CMB community was able to completely ignore the successful prediction for several years, but were then suddenly all over the latter failure. The third peak falsifies the ansatz on which that particular prediction was built, not the entire concept of modified gravity. This would be like asserting that all possible forms of dark matter are excluded because we haven’t yet detected WIMPs. It is a classic failure of objectivity, which is another hallmark of faith-based argumentation: we know His name is [insert favorite deity], not [insert any other deity].

^&Or after me. Dark matter was my first hypothesis, and I’m here to tell you that True Believers do not suffer second hypotheses or those who stray from the fold. I guess that’s why so many scientists who are MOND-curious keep it on the down low. Wise, perhaps (that’s why tenure needs to be a thing), but hardly the ideal of the open and free exchange of scientific ideas.

^$I wasn’t there, but one audience member (not someone I knew) thought it was so over the top that he told me about it, sharing a link with a video. (I did not retain that link, and doubt the hosting conference website is still active.)

^%Argument weak here. RAISE VOICE!

The well of inspiration

AP News recently ran a story on the interface of religious inspiration and the search for dark matter. I thought about commenting on it, then thought better of it, but now here I am again. I know some but not all of the people who are quoted, all good scientists. The article ends with a nice quote from Jennifer Wiseman:

“Studying the deep universe may make us feel insignificant,” Wiseman said. “But it also gives us a sense of unity that we’re all on the same planet. … The hope is we get a sense of joy, humility and love from these contemplations.”

I’ve known Dr. Wiseman since undergrad days, before either of us were Ph.Ds. We don’t agree on much that is specific to religion, but somehow manage to be friends anyway, and I completely agree with the sentiments she expresses here.

There is a tendency to portray religion as being at odds with science, and that can certainly happen. But they share more in common than this trope implies. Both arise from a deep wellspring in the human spirit, the same wellspring: the desire to know. How does the universe work? How did it come to be so? Why? and on and on. Where they differ is in approach: when encountering an unknown, especially things that are fundamentally unknowable (e.g., does God exist?), religion asserts an answer and asks that we accept it on faith. Faith is anathema to the scientific process; one is, in principle, to search for the truth through observation and experimentation.

Religion and science come into conflict when scientific knowledge encroaches on religion’s known unknowns. There was a time when the structure of the heavens must have seemed unknowable; the realm of the sacred, safe from access by mundane human knowledge. So it is easy to see how Galileo came into conflict with the Church despite being a faithful member:

Church: Scripture teaches us that the Earth is the unmoving center of mortal corruption; the heavens above us are perfect and unchanging.

Galileo: I can see spots on the sun and mountains on the moon.

Church: We are the center about which the heavens rotate every day, clearly the center of all rotation.

Galileo: I see satellites circling Jupiter.

Church: God is great; He could do one thing one day and an entirely different thing another day*.

Galileo: It appears that the Earth revolves around the Sun, never the other way around.

Church: Do you see these instruments of torture?

This is a tongue-in-cheek portrayal of a serious historical incident, but my point is that conflict arises when scientific knowledge encroaches onto turf that had formerly been the exclusive province of religion. Addressing our spiritual need to know can be inspirational, but it can also be a let down. Giving specific answers to formerly unknowable questions can be a bit of a mood killer. Is that all there is?

Another downside of scientific knowledge is that it can never be complete. There is always something left unknown, and we are not supposed to fill that void with something we take on faith. Yet we are profoundly uncomfortable with not knowing. A little extrapolation at and beyond the fringes of current knowledge is natural, and can sometimes provides a useful way forward in driving new discoveries. But it can get carried away, e.g., string theory. So extraordinary caution is also warranted: the temptation to fill in the blank is where empiricism transitions to theology.

Understanding that we know a lot – as we do at this juncture in the history of science – and yet still don’t know something important, like what most of the universe is made of, is hard to accept. It becomes even harder when admitting something we thought we understood is wrong, or is less complete than we thought. We know most of the mass in the universe is made of non-baryonic cold dark matter^&. Don’t we?

The dynamical evidence for acceleration discrepancies is abundant. That these must be caused by non-baryonic cold dark matter is less clear. This is where cosmology intrudes. Cosmology has always been the nexus where science and religion meet, with philosophical imperatives often obscuring essential observational facts. Before Kepler, orbits had to be circular. After Inflation, the density parameter had to be one. (We meant Ω_m = 1, not the modern weak-sauce version Ω_m + Ω_Λ = 1.) The mass density is larger than the baryon density from BBN (Ω_m > Ω_b), so there has to be non-baryonic dark matter. That such a substance is also required to fit the acoustic power spectrum of the CMB amplifies our faith in the existence of such stuff.

We think we’ve solved cosmology, and that solution requires non-baryonic cold dark matter. So to admit that maybe we were wrong about dark matter just because it persistently remains undetected and provides unsatisfactory explanations of many astronomical observations and is consistently outperformed in predictive capacity by an alternative theory is to admit that we don’t understand as much about the universe as we thought. That’s really, really hard. Our unwillingness to admit that maybe we have been wrong about such an important issue is where human nature kicks in to blur the line between scientific knowledge and religious faith. It’s much easier to ignore those nagging doubts^% and have faith that we were right all along.

Cosmology works so well that dark matter has to exist. I’ve head this sentiment expressed over and over by many different scientists, yet this is an assertion of faith. A more conservative statement is that cosmology as we currently conceive it works if, and only if, an appropriate form of non-baryonic dark matter exists with the required cosmic density. If not, then we need a new model – something that presumably^{^} stems from a more general underlying theory.

Since both religion and science arise from the same desire to know, it is easy to have faith that we know more than we actually do.

This is the sticking point we’ve hit in the dark matter debate. We’ve been calling the acceleration discrepancy the dark matter problem for so long that this linguistic mistake has morphed into an absolute certainty that invisible mass exists. It has become a matter of faith.

*Of course an omnipotent deity could do that. Apparently He chooses not to do so, sparking the schism between theists, for whom God actively intervenes in miraculous ways in real world affairs, and deists, who view God more as the great watchmaker, setting creation in motion but not interfering with its operation. Deism is conducive to science, as the search to identify the rules by which the universe works is to gain insight – however remote – into the mind of God. I suspect this attitude informed Einstein’s complaint against quantum mechanics: “God does not play dice with the universe.”

^&I certainly thought so before I didn’t.

^%The social pressure to conform to the preferred cosmology is enormous. I know I’m only making it worse for myself. But an explanation that omits MOND is a lie of omission. Here science and religion certainly overlap in the sentiments expressed by the seventeenth century cleric Paul Gerhardt:

“When a man lies, he murders some part of the world.”

Assertions that somehow “feedback” explains MOND are simply a numerical form of magical thinking; an excuse to not have to explain the inexplicable. By eliding MOND, we murder a part of the world.

^{^}Here I am extrapolating at the fringes of knowledge.

Paradigm Shifts in Modern Astrophysics

I see that I’ve been posting once a month so far in 2026. I’ve lots to say but no time to say it. Some of it good, some of it bad, maybe sometime I’ll get around to it. No guarantees. On the good side, I’ve been working on a big project or two; may have something to say about those soon. I’ve also been meaning to write about the Planet 9 anomaly for months stretching into years now. Fascinating stuff related to MOND but not something I’ve worked on myself. On the bad side, I’ve been obliged to waste yet more time on my university administration’s insistence on merging our department into physics based on a snap decision made by a disinterested leader who employed all the forethought typically reserved for bombing a random country in the Middle East.

So I have had no time for novel posts lately, and today is no different. However, I thought readers of this blog would appreciate the post Paradigm Shifts in Modern Astrophysics: Applying Thomas Kuhn’s The Structure of Scientific Revolutions to Dark Matter at Heritage Diner that was pointed out to me by Moti Milgrom. Since I wouldn’t have seen it had he not mentioned it, perhaps that’s the case for you as well. I’m not gonna re-post it verbatim – you can read it there yourself – but I am going to offer a running commentary with a few observations, both personal and historical. So bring it up in a separate browser window and let’s read along…

This post riffs off of Kuhn’s The Structure of Scientific Revolutions as it pertains to dark matter and MOND. If you’re not familiar with it, Kuhn’s work on the philosophy of science is foundational to the way in which a lot of physical scientists approach their field (whether they realize it or not). Philosophers of science have done a lot more since then, but I’m not going to attempt to go there. I will look back to Popper* to note that I’ve heard Kuhn depicted as being some sort of antithesis to Popper. I don’t see it that way. To be pithy, Popper tells us how science should be done while Kuhn tells us how it is done. Who could have imagined that a human endeavor would be messy in practice and not always live up to its ideal?

I’m not sure how to do this; I guess I’ll excerpt relevant quotes and riff off those. The basic thesis is that dark matter is on the brink of a Kuhnian paradigm shift.

We are living through exactly that moment in modern astrophysics.

I certainly hope so! This moment in the history of science is taking a long damn time. A century ago, we went from “classical physics explains everything” to “quantum mechanics, WTF?’ in the space of about a decade. I’ve been working on matters related to MOND for over thirty years now, dark matter longer than that, and of course Milgrom started more than a decade before I did.

The essay discusses the “cartography of collapse,” which includes crisis and revolution:

The third stage is crisis — triggered when anomalies accumulate beyond the paradigm’s absorptive capacity. And the fourth is revolution, in which a new framework displaces the old not through incremental persuasion but through a gestalt shift, what Kuhn famously described as seeing the same duck-rabbit drawing and suddenly recognizing a rabbit where you had always seen a duck.

This resonated with me because I had exactly this experience. I started my career as much a believer in dark matter as anyone. I was barely aware that MOND existed (this seems to remain a common condition). But it reared its ugly head in my own data for low surface brightness galaxies. Try as I might – and I tried mighty hard, for a long time – I could not reconcile how the shapes of rotation curves depended on surface brightness as they should according to Newton while simultaneously lying exactly on the Tully-Fisher relation without any hint of dependence on surface brightness⁺. I could explain one or another, but not both simultaneously – at least, not without engaging in some form of tautology that made it so. I came up with a lot of those, and that has been a full-time occupation for many theorists ever since.

For me, this gradually became a genuine crisis. I pounded my head against the wall for months. Then, as I was wrestling with this problem, I happened to attend a talk by Milgrom. I almost didn’t go. I remember thinking “modified gravity? Who wants to hear about that?” But I did, and in a few short lines on the board, Milgrom derived from MOND exactly the result I found so confusing in terms of dark matter. This chance meeting in Middle Earth (Cambridge, UK) changed how I saw the universe. The change wasn’t immediate – it had to ferment a while – but ultimately I found myself asking myself over and over how this stupid theory could have its predictions come true when there was so much evidence for dark matter. Finally I realized that the evidence for dark matter assumes that gravity is normal; really it was just evidence of a discrepancy, and it could be that the assumption was at fault. That realization was sudden: where I’d always seen a duck, suddenly I could also see a rabbit.

Most scientists have not had this experience. What constitutes a crisis serious enough to contemplate a paradigm change is a highly personal matter of judgement. It happened in my data, so I took it seriously, but others didn’t care. So I made predictions for their data. Some of those came true, but they rejected the evidence of their own data. It just could not be so! At what point does a mere problem amount to a true anomaly?

Part of the sociological issue is that the dark matter paradigm has been in a constant state of crisis since its inception. The reasons vary over time. Sometimes valid solutions have been found to the crisis du jour, other times we’ve chosen to just live with it. It is much easier to live with a bad solution than to rethink one’s entire world view.

The problem with being in a constant state of crisis makes is that it seems like nothing can ever be a genuine crisis. Every foundational change is just another new normal. We complain, say it can’t be so, argue, offer bad ideas, reject them, get used to them, then eventually accept that one of them maybe isn’t so bad, so that must be what is going on. After a few years It is Known and people convince themselves that we expected just that all along.

It takes a lot of evidentiary weight for a paradigm to change, and it takes a lot of time for that to accumulate. But, as Kuhn recognized, mere facts are not enough. Humans and their attitudes matter. As Feyerabend noted,

The normal component [i.e. the accepted paradigm and its adherents] is large and well entrenched. Hence, a change of the normal component is very noticeable. So is the resistance of the normal component to change. This resistance becomes especially strong and noticeable in periods where a change seems to be imminent.

P. Feyerabend in Criticism and Growth of Knowledge

The post correctly points out that dark matter itself was an anomaly going back to Zwicky in 1933. This is often depicted^ as the first detection of dark matter, but it was also noted by Oort in 1932. Zwicky was aware of Oort’s work and cited him, but they’re very different results. Oort was worried about a factor of ~2 discrepancy in stellar dynamics in our local chunk of the Milky Way; Zwicky discovered a discrepancy of a factor of ~1000 in the Coma cluster of galaxies. These both imply the need for unseen mass, but the results are not at all the same. In retrospect, Oort’s discrepancy is a subtle detection of a flat rotation curve while Zwicky’s discrepancy was (at least) two distinct discrepancies: what we now consider the usual cosmic dark matter, but also missing baryons: most of the normal matter in clusters is in the hot, diffuse intracluster medium, not in the stars in the galaxies that Zwicky could see and account for. The modern discrepancy is only a factor of ~6, which is rather less than 1,000. (The distance scale also played a role in exaggerating Zwicky’s result.)

This all seemed crazy in the 1930s, even in the immediate aftermath of the quantum revolution. Consequently, Zwicky’s work was mostly ignored^$. The subject of dark matter didn’t really take off until the 1970s. Considerable credit goes (rightly) to Vera Rubin, though many others made essential contributions – just on the subject of rotation curves, Albert Bosma, Mort Roberts, and Seth Shostak all made important contributions, the relative importance of which depends on who you ask.

An important aspect of scientific revolutions is persistence. Vera was persistent. She was fond of relating the story of showing her first (1970) flat rotation curve of Andromeda to Alan Sandage, only to have him dismiss it as “the effect of looking at a bright galaxy.” What the heck did that mean? Nothing, of course – it is the sort of stupid thing that smart people say when confronted with the inconceivable. So Vera persisted, and by the end of the decade had shown that flat rotation curves were the rule, not some strange exception. They became accepted as a de facto law of nature, and the dark matter interpretation was solidly in place by 1980.

The scientific community absorbed this anomaly not by questioning Newtonian gravity or Einstein’s general relativity, but by proposing an invisible scaffolding — a halo of non-luminous, non-interacting matter surrounding every galaxy. Dark matter became not a crisis but a patch.

Indeed, this seemed the most appropriate (scientifically conservative) course of action at the time, as summarized in this exchange (also from the early 1980s):

To emphasize the essence of what is said here:

Tohline: I might be so bold as to suggest that the validity of Newton’s law should now be seriously questioned.

Rubin: The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.

This was a very reasonable attitude, at the time. But I’ve heard the phrase “only as a last resort” many times now over the course of many years from many different scientists. At what point have we reached the last resort? In the case of dark matter, once we’ve convinced ourselves that invisible mass has to exist, how can we possibly disabuse ourselves of that notion, should it happen to be wrong?

In Kuhnian terms the last resort is reached when the weight of anomalies in the standard paradigm become too great to sustain. But that point is never reached for many die-hard adherents. Whatever the right answer about dark matter turns out to be, I’m sure many brilliant people will go to their graves in denial. Hence the more cynical phrase

Science progresses one funeral at a time.^%

But does it? What if the adherents of an ingrained but incorrect paradigm breed faster than they go away? I’ve seen True Believers train graduate students who’ve gone on to train students of their own. Each generation seems to accept without serious examination the inadequate explanations for the anomalies made by their antecedents, so the weight of the anomalies doesn’t accumulate; instead, each one gets swept separately under the proverbial rug and forgotten. Forgetting is important: when new anomalies come to light, hands are waved and new explanations are promulgated; no one chekcs if the new explanations contradict the previous generation of explanations. What passed before is a solved problem, and we need never speak of it again.

This is not a recipe for a scientific revolution, but for a thousand years of dark epicycles.

Returning to the post,

By the late 1980s and early 1990s, dark matter had been formally incorporated into the reigning cosmological framework. Lambda-CDM — where Lambda refers to the cosmological constant (a proxy for dark energy) and CDM stands for Cold Dark Matter — became the standard model of cosmology.

The essence of this statement is correct but some of the details are not. Dark matter was widely accepted by 1980. That’s still a little before my time, but my impression is that the magnitude of the discrepancy was at first a factor of two, so it could simply have been normal baryons that were hard to see. However, the discrepancy rapidly snowballed to an order of magnitude, so we needed something non-baryonic. This was happening simultaneously with talk of supersymmetry and grand unified theories in particle physics that could readily provide new particles to be candidates for the dark matter, leading to the shotgun marriage of particle physics and cosmology, two communities that had had little to do with each other before then, and which still make an odd couple. Cosmology as traditionally practiced by astronomers needed dark matter but didn’t much care what it was; particle physics was all about the possibility of new particles but didn’t care about the details of the astronomical evidence.

To rephrase the above quote, I think it is fair to say that “by the late 1980s and early 1990s, cold dark matter had been formally incorporated into the reigning cosmological framework.” But that framework was not yet LCDM, it was Ω_m = 1 SCDM. The Lambda only came to prominence by the end of the 1990s, as I’ve related elsewhere. This process is depicted by many scientists as a revolution in itself, and in many regards it was. The cosmological constant had been very far out of favor; rehabilitating it was a grueling experience and no trivial matter. But it wasn’t really a scientific revolution in the sense that Kuhn meant: our picture didn’t fundamentally change, we just learned to accept a parameter^& that was already there but that we didn’t like.

The post goes on to note the absence of dark matter detections:

This silence is itself an anomaly… as the silence deepens, the null result itself becomes harder to dismiss.

This is correct, and yet… Physicists have built many experiments that have achieved extraordinary sensitivities. If cold dark matter was composed of WIMPs as originally hypothesized, we would have detected them long ago. Initially, the reaction was to modify WIMPs. Did we say the cross-section would be 10^-39 cm²? We meant 10^-44 cm². When that was excluded, we slid the cross section still lower, but people also started giving themselves permission to think the unthinkable. By unthinkable I mean a particle that can’t be detected, not modified gravity. That’s more unthinkable. So the anomaly isn’t dismissed, but it is treated with less gravity than it should be, and certainly with less import than a positive detection would have been granted. Did we say WIMPs? We didn’t mean just WIMPs. It could be anything. (They damn well meant WIMPs and only WIMPs^#. Anyone who tells you otherwise is gaslighting^*% you, and probably themselves.)

The post goes on to talk about MOND. It gives me too much credit for the gravitational lensing work. This was done by Tobias Mistele, and our work is based on that of Brouwer et al. But it is correct to note that these data are a problem for the dark matter paradigm. Rotation curves remain flat beyond where dark matter halos should end. If correct, this is a genuine anomaly. Perhaps in some distant future it will be recognized** as such in retrospect; at present it seems mostly to be ignored.

It goes on to talk about the JWST observations. Yeah, that part is correct. The community seems to be in the usual process of gaslighting itself into denial of the anomaly. For the first two years after JWST started returning images of the deep universe, people were aghast. How can this be so? It was all anyone could talk about. But then the unexpected became the new normal. Hands were waved, star formation was accepted to be absurdly efficient, and people accepted the impossible. I no longer hear the talk of how problematic the JWST observations are; this chatter simply stopped.

Anomalies don’t weigh a paradigm down if we don’t accept that they’re anomalies. But I’ve lived through the revolution, it’s hard to see a positive outcome while it is still ongoing. For it is certainly true that

What waits on the other side of the dark matter revolution — if that is what is coming — we cannot yet know.

The future is the unknown territory. We don’t know, and can never know, if dark matter doesn’t exist – it is impossible to prove the negative. But we do know MOND works much better than it should in a universe made of dark matter. That demands a scientific explanation that is still wanting. But MOND by itself is not a complete answer, so we are like the parable of the blind men and the elephant, each sensing a different part of reality but as yet unable to see the whole.

Still, there is reason for optimism. The article closes by noting that

Kuhn’s deepest insight was not that science changes. It is that the change, when it comes, is never merely technical. It is a reorganization of the world itself — the universe seen suddenly whole in a configuration it has always had, but that we had simply lacked the paradigm to perceive.

Not knowing how things ultimately work out is good, actually. One way or the other, there is still fundamental science to be done. We have not reached the stage of looking for our discoveries in the sixth place of decimals.

*Trivia I just learned looking at Popper’s wikipedia page: he was spending his last days in London around the same time I was a postdoc in Cambridge just starting to struggle with the scientific and philosophical implications of the dark matter-MOND miasma.

Unrelated trivia: I was at a workshop in Jerusalem early in the century but missed the opportunity to meet Jacob Bekenstein because I was too shy to bother the great man.

⁺If you do not find this confusing, you are not thinking clearly.

^A nice, brief summary of this early history is related by Einasto. This is the first place I’ve seen the citation to Opik (1915) written out. I’ve only heard mentioned verbally before, so I’ll have to try looking that up later.

The full story is way more complicated than this sounds, and still gets debated off and on. The amplitude of the Oort discrepancy is much smaller today. Locally, the 3D density of mass seems to be accounted for by known stars, gas, and stellar remnants (which were still a new thing in the 1930s). So this Oort limit shows no discrepancy. There remains a modest discrepancy in the 2D dynamical surface density. It appears to me to boil down to the vertical restoring force having a (sometimes ignored) term that depends on the gradient of the rotation curve. Were that falling in a normal Newtonian way, there would be no discrepancy. But it isn’t; this deviation from Newton in the radial direction leads to the Oort discrepancy in the vertical direction. Instead of being as negative as Newton predicts, dV/dR is close to zero, hence my description of this as an indirect detection of a[n almost] flat rotation curve. (dV/dR = -1.7 km/s/kpc, so not exactly zero, but a lot closer to zero than Newton without dark matter would have it be.) The vertical discrepancy is nevertheless much reduced, now being well below a factor of two.

^$To his apparently great embitterment. He had some choice things to say about astronomers of his time. I am inclined to suspect that those who praise Zwicky the loudest today would have been among those he had reason to complain about had they been contemporaries.

^%This is attributed to Planck, but he had a lot more nuanced things to say about it in his Nobel Prize lecture.

^&Einstein disavowed the cosmological constant as his “greatest blunder,” so one argument against it was (for a long time) that it should never have been a part of the theory of General Relativity in the first place. I wonder how things might have gone had that been the case – that he had never introduced Lambda. Perhaps then the data that led to us accepting Lambda would have required a genuine revolution, but it isn’t obvious that we would have accepted it (we might still be debating it), nor is it apparent that LCDM is what comes out of such a revolution. But we don’t get to do that experiment: the Great Man had suggested Lambda, so it was OK to bring it back: we weren’t wrecking his theory by introducing a crazy new entity, we were just admitting an unlikely (antigravity-like) component thereof.

^#Or axions! Or warm or self-interacting dark matter. Or macros nee strange nuggets! Or or or… Sure, there have been lots of ideas for what the dark matter could be. But when we say that “by the late 1980s and early 1990s, cold dark matter had been formally incorporated into the reigning cosmological framework” what the vast majority of scientists working on the topic (including myself) meant was that CDM == WIMPs. We were aggressively derisive of other ideas, and these are only dredged up again now because of the experimental non-detection of WIMPs. WIMPs are still a better dark matter candidate than the others for the same reasons that we were derisive of the others back in the day. We haven’t been looking as hard for the others, so comparable experimental limits do not yet exist. To quote myself,

The concept of dark matter is not falsifiable. If we exclude one candidate, we are free to make up another one. After WIMPs, the next obvious candidate is axions. Should those be falsified, we invent something else. (Particle physicists love to do this. The literature is littered with half-baked dark matter candidates invented for dubious reasons, often to explain phenomena with obvious astrophysical causes. The ludicrous uproar over the ATIC and PAMELA cosmic ray experiments is a good example.)

McGaugh (2008)

^*%An easy way to deflate such gaslighting is to ask why so many experiments have been built to search for WIMPs but not all these other allegedly great dark matter candidates. After a pause and dismayed stare, you’ll probably get an answer about “looking under the lamp post” because that’s where it is possible to make detections. That’s sorta true, but it isn’t the real reason. The real reason is that we all drank the Kool-Aid of the WIMP miracle, so genuinely believed that the dark matter had to be WIMPS, not merely that they were a convenient experimental target. (I did not chug the kool-aid as hard as the people who based entire careers on building WIMP detection experiments, but I did buy into the idea to the exclusion of other possibilities for dark matter – as did most everyone else.)

**In retrospect, Galileo’s observations of the angular size and phases of Venus were utterly fatal to the geocentric paradigm. That’s easy to say now; at the time it was just another piece of evidence.

Asteroids that spin really fast

There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.

– Hamlet

One of the great privileges of astronomy is that there’s always something new and surprising to discover. It’s a big universe. There’s a lot in it.

In appreciation of this simple joy, I thought I’d share a nifty discovery made by the Rubin Observatory*. This is a big ground-based telescope with an absolutely mammoth camera that is conducting a decade-long survey to unprecedented depth. It basically images the sky over and over again every night. Since it is such a large telescope with a huge camera, each snapshot is a deep image of a large chunk of sky. That lets us see lots in co-added images that we couldn’t otherwise see. But it also adds a new dimension: time. By imaging the sky deeply every clear night, night after night, we see things that vary. That includes variable stars and things that move, like asteroids.

**Figure 2** from Greenstreet et al. (2026): A small region of the sky surveyed by the Rubin Observatory. The apparent lines are due to bright asteroids that moved between the individual exposures that were used to create the co-added image. A closer look reveals that they are composed of a set of closely spaced point sources, one from each input exposure. The color segments correspond to data taken in different bands: red representing the object’s flux in the i band, green corresponding to the r band, and the blue channel capturing a linear combination of g and u bands.

Most asteroids are in the main belt between Mars and Jupiter. The size distribution is a power law with lots of little asteroids for every big one. The small ones are faint, so hard to see, but exactly the sort of thing that is now made easy by the Rubin Observatory. Practically as soon as serious survey work began, well over a thousand new asteroids were discovered. Right away, in the first week of observation. So many new objects, so many orbits to trace. That’s pretty cool, especially if you’re a solar system geek. I am not; I am a jaded old man unamused to be witnessing a repeat of the decline and fall of both science and society. It takes more than a bunch of rocks in space to grab my attention.

Nevertheless, the thing about finding lots of new objects is that this often reveals some unexpected extremes. As Luca Rizzi at NSF put it, the “Rubin Observatory will find things that no one even knew to look for.” Amongst the new asteroids are some truly weird objects. In particular, Greenstreet et al. (2026) report the discovery of a number of rapidly rotating asteroids. As they state in their abstract,

We model lightcurves and derive rotation periods and colors for ∼2000 objects. We find 75 main-belt asteroids and one near-Earth object with reliable rotation periods spanning 0.031–21.3 hr … We find 19 superfast rotators with periods shorter than the 2.2 hr spin barrier. Rubin-discovered MBA 2025 MN₄₅ is the fastest-rotating d > 0.5 km known asteroid with a rotation period of 1.9 minutes; along with NEO 2025 MJ₇₁ (1.9 minutes) and Rubin-discovered MBAs 2025 MK₄₁ (3.8 minutes), 2025 MV₇₁ (13 minutes), and 2025 MG₅₆ (16 minutes), these five super- to ultrafast rotators join a couple of NEOs as the fastest-spinning subkilometer asteroids known.

Here is an example: the light curve of their fastest rotator, MN₄₅:

*Portion of Fig. 13 of Greenstreet et al. (2026) showing the light curve of newly discovered main belt asteroid 2025 MN₄₅ with its phenomenal 1.88-minute rotation period*.

Every once in a while, a result comes along that make stop and remember that heavenly objects can be pretty darn cool. This is an asteroid about half a kilometer in diameter that is spinning once every 1.88 minutes. That’s 112 seconds. That’s crazy talk.

Imagine standing on the surface of this object. The horizon is visibly curved. You could walk around its diameter (if you could walk at all) in less than an hour. But it would be a hike that traverses ~15 soccer pitches, so not nothing. So let’s just stand still, eh? Still relative to the surface, of course – you could not run fast enough to keep up with the rotation at the equator (about 14 m/s or 30 mph).

You see a sky full of stars. The sun is the brightest by far, but not quite as bright as we’re used to here on Earth, being much closer in at 1 AU than the asteroid belt (mostly between 2 -3 AU). More importantly for our eyes, there is no atmosphere (hope you remembered your spacesuit), so there is no scattered light obscuring the other stars when the sun is above the horizon. You can see all the stars all the time, better than the darkest night on Earth.

That sky full of stars is rising and setting at a furious pace. You can watch it happen, no need for the horizon to make clear that the sun is setting. It takes just under a minute for a star to traverse 180^o from rising on one horizon to setting on the opposite side of the sky. I imagine that’d be really cool to watch: the entire Galaxy of stars spinning over your head, rising, passing through the meridian, setting, then coming ’round again in less time than it takes to read this post. I wonder how fast the awe gets replaced by nausea (hope your space suit is equipped with a barf bag).

While it is fun to imagine circumnavigating an asteroid this small, it would be hard to walk as it has hardly any gravity. It’s mass is about a hundred million metric tonnes, depending on the density (more on that below). That sounds like a lot, but it ain’t – not for gravity, at any rate. The surface gravity would be less than 5 cm/s/s, a factor of 200 times less than we feel here on Earth. You could play superman and leap tall buildings with a single bound, but you’d never come down: it would be easy to achieve escape velocity with your Terran muscles. Simply walking would be hard: your first natural-strength step would be your last one.

Maybe it would be safer to stand at the pole than at the equator. Indeed, it is the only place you’d have a hope of standing. This thing is a giant merry-go-round in space. At the equator you’d get flung into space by the centrifugal force if you didn’t have a death grip on the surface. So how does this thing hold together?

That is something Greenstreet et al. (2026) wondered as well. They note that most main belt asteroids have periods longer than 2.2 hours (not minutes!). This is a “spin barrier” that most objects don’t cross. If they did, they would fly apart.

Binary asteroids are surprisingly common given their mutual lack of gravity. A possible origin of such systems is angular momentum-induced fissionning after a gradual spin-up by the YORP effect. The cohesion limit depends on the composition of the object:

**Figure 14** from Greenstreet et al. (2026): Asteroid size vs. rotation period for numbered asteroids (gray circles) and the 19 Rubin superfast rotators. Blue circles represent known objects with reliable spin periods (uncertainties below 30% based on data from NASA JPL (2025)) that rotate above the so-called “spin barrier” indicated by the level curves for various values of cohesive strengths. Rubin superfast rotator 2025 MG₅₆ lies beyond the Strong Lunar Cohesion model, and 2025 MV₇₁ along with ultrafast rotators 2025 MJ₇₁, 2025 MK₄₁, and 2025 MN₄₅ even farther beyond to required cohesive strengths of clay or rock.

Most asteroids and comets are rubble piles. That’s the technical term. Rather than being monolithic boulders in space, the typical asteroid is an agglomeration of many smaller chunks. We know this from a variety of lines of evidence, but a key one is that in the cases where their sizes and masses have been well measured, their densities are lower than that of solid rock. A typical rock has a density of ~3 g/cc while asteroids are half that, typically ~1.7 g/cc. There’s a fair amount of nothing mixed in with the solid material. That’s why they can sustain the impacts that make relatively large craters: they don’t shatter because seismic waves can’t propagate through the empty interstitial spaces. They’re basically big sandbags^$.

MN₄₅ is not a rubble pile. A rubble pile has no cohesive strength; it would fly apart if spinning so fast. We’ve observed rubble piles be shredded by tidal forces. None of the obvious space substances fit the bill to hold MN₄₅ together; even terrestrial clay would not hold together. MN₄₅ needs to be at least as cohesive as solid rock. The same holds for MK₄₁. So maybe there are big boulders?

I wonder what the surfaces of these super-fast rotators look like. The standard-issue asteroid surface is heavily cratered and covered with dust, regolith, small rocks, and the occasional boulder. Such loose material should get flung off into space when the spin rate is to high. Leaving what, pray tell?

There is a good reason asteroids are not, in general, solid rocks. They do run into each other on occasion. Not often; the asteroid belt is pretty empty, it is nothing like the dense hazard to navigation that is beloved of science fiction movies. But they spend billions of years orbiting around the sun, so their paths do sometimes cross. When they do, their relative speeds are typically several km/s, so they do not bump gently in the night – hence the heavily cratered surfaces they typically display. A big, monolithic rock would likely shatter as a result of the larger collisions. So I’m skeptical that MN₄₅ is a literal boulder. What is it then? Since it isn’t my specialty, I feel free to engage in some wild speculation.

Most of the meteorites that reach the Earth’s surface are stony, but a few percent are irons, composed of metal (mostly iron and nickel). It is quite an experience to pick one up, as they are quite dense (iron has a density of 7.9 g/cc). They feel much heavier than the eye expects for a rock that size. I love to pass examples from our meteorite collection around in class – the irons never fail to raise eyebrows.

Things in space have a size distribution. Small things are common and hit the earth all the time. Look up at a clear, dark sky for ten minutes and there’s a good chance you’ll see^ a shooting star. That’s typically the size of a grain of sand – it hits the atmosphere at high speed and gets incinerated as it zips across. Larger objects also burn up in the atmosphere; something has to be pretty hardy and/or large for a chunk of it to reach the ground as a meteorite. Big things are rare, but they’re out there.

So maybe MN₄₅ is a big iron? Just a giant nickel-iron nugget in the sky.

I wondered what might be known about such a hypothetical object, but there does not seem to be much in the literature. It is always hard to be sure one isn’t missing something important in a literature search, but perhaps this is a case of discovering a thing we didn’t know to look for. Though some things may just be forgotten: trying a search again, I find this gem from Chapman (1974) discussing the size distribution of asteroids that might be the sources of stony-iron and iron meteorites:

A class of apparently stony-iron asteroids shows a relative excess frequency between diameters of 100 to 200 km. It is proposed that the latter objects are predominantly metallic cores of a group of chemically differentiated planetesimals all of which (except Vesta) have suffered catastrophic collisions.

That sounds a bit like the Phaeton hypothesis in which a protoplanet formed where the asteroid belt is now but subsequently shattered into what we now see as asteroids. This idea fell out of favor because Jupiter’s gravity precludes things in the asteroid belt from clumping up; it acts for the sun like shepherd moons act for Saturn to maintain structure in its rings. But the rings had to come from somewhere, and the iron meteorites appear for have come from an object that had chemically differentiated into an iron core and a rocky mantle. It is now believed that Jupiter might have migrated in the very early days of the solar system when there was still a lot of the solar nebula to exchange angular momentum with, so maybe some protoplanets formed, collided, and shattered before Jupiter arrived to nail them down?

The curious thing I found in my first search is a report that main-belt asteroid (16) Psyche has a large metal content on its surface. That’s not the same as being one giant iron, and one always has to take these things with a grain of salt. There is a cool image at the top of that link that looks like a metallic (16) Psyche, but it is obviously fake: we have no picture of that asteroid at that resolution. Scroll down to the bottom to see the real data, which I would have found more compelling without the teaser slop image.

So maybe there are large nuggets of metal in the sky? I’m sure there are other possibilities. We should also bear in mind that this is astronomy. Things go wrong; maybe these results will change after months and years of repeated observation. These data were obtained over the course of a mere nine nights of observing; Rubin’s survey will continue for a decade. That means lots more data, so it should be come clear if something more mundane explains these extreme spin rates. Or something even more outrageous. Maybe to mess with us aliens painted black & white polka dots on MN₄₅ and its compatriots to make it look like they are spinning faster than they are: what we measure is the light curve, we don’t resolve the structure of the tiny asteroid. Of course, invoking aliens would be even more insane.

*I hear people refer to the telescope as “the Rubin.” Having known Vera Rubin personally, I can’t write that. Rubin was a vital, dynamic scientist and marvelous human being. It’s great to name⁺ it after her, but Rubin is a person, not an object.

⁺As an scientist, Vera was all about how things on the sky moved. As an observer, she made extensive use of spectroscopy to trace motion via the Doppler effect. The Rubin Observatory is strictly an imaging survey, which I find it a little ironic, as it operates in an observing mode that’s about as far removed from what she did as you can get with a ground-based telescope.

^$We learned a great deal from the Deep Impact mission, which collided with the nucleus of Comet 9P/Tempel 1 on July 4, 2005. Mike A’Hearn was the PI of that project; we were both at the University of Maryland when it happened. The mission had a normal space probe component and also an impactor – a copper bullet (easy to distinguish from cometary material) to collide with the comet. People involved with the project had bets all over the place as to what would happen, ranging from a big explosion that excavated a new crater to a clean penetration like a bullet through talcum powder. We got the former, consistent with a rubble pile. A surprising amount of tarry, carbon-rich material was ejected along with the expected ice and rocks. So this “dirty snowball” was a tarry mess.

The orbit of the mission put the spacecraft with its bullet in front of the comet, so Mike would say we didn’t crash into the comet; the comet ran over our spacecraft. Motion is relative, yes, but intent matters. As part of the outreach for the mission, people could record a message that was carried on the bullet to the comet on a CD. Knowing full well it was our intent to violently crash into the comet, I couldn’t resist contributing “We come in peace.”

^These days you’ll also see lots of satellites. Lots and lots of satellites. So many satellites messing up the sky. They look like points of light that move slowly but steadily across the sky. Slower than airplanes without the bright, blinking lights. Note the direction of travel: most commercial satellites travel east-west, following the economics of launching with the angular momentum granted by the planet’s spin. Spy satellites often pay extra to launch into polar orbits, moving north-south to get a peek at the whole globe as it rotates underneath.

For those who don’t recognize it, the image under the title is from a recreation of the classic early-days video game Asteroids, which I recall playing as a teenager. One flew a little ship around and shot asteroids before they ran into you. They break apart like monolithic boulders, a common misconception for what asteroids are like that I shared at the time. Being more like giant sandbags, you could shoot these things as much as you could, even with nukes, and the thing would still run you over.

It was gratifying to find the recreation of the classic game on the web, so had to play it some. I notice that this version encourages another misconception. After firing the rocket to start into motion, that motion does not persist: one grinds to a halt like there is friction in space. I don’t recall that mistake being made in the original. It has been more than a few decades since I played it, but my recollection is that you would just keep going once started, in accordance with Newton’s first law. Indeed, I liked to see how high I could the ramp up speed and still dodge asteroids. Not very fast. I’m more forgiving of the periodic boundary conditions. These rather undersell the immensity of space, but one can think of it as a mere re-centering of the coordinate system. The basic fact of Newton’s first law would be easier to get right in the code than the wrong that this realization implements.

Triton Station

A Blog About the Science and Sociology of Cosmology and Dark Matter