CDM, what would Popper do?

This is a guest post by Prof. Jim Schombert

Greetings Stacy’s blog readers! My name is Jim Schombert, I’ve been working with Stacy for about a quarter of a Hubble time. I knew him when his hair was brown.

In any case, I asked him if I could make a contribution to the dark matter discussion. Partly this is motivated by the LUX-ZEPLIN ‘detection’, but also cause there are some clear red flags from a philosophy of science viewpoint. I was a philosophy major in college, switched to astronomy when I 1) realized there was no money in philosophy, and 2) people at parties would talk to an astronomy major, not so much a philosophy major. I never lost my interest in the philosophy of science, tried to keep up with the literature, so how should we view the current MOND vs dark matter debate from a philosophical view? (and for a more sophisticated analysis of the topics below, I direct the reader to Merritt’s superb books and articles on MOND).

There are basically two types of philosophers when it comes to science, one type is heavy in mathematical logic and deep discussions of foundations of science, the other type is more concerned about how science is done, i.e. the history and impact side of science. All of us know the key writers of this second type of philosophy of science; Popper, Kuhn, Lakatos (hereafter PKL). You will know them cause when you read their works you will immediately recognize yourself and colleagues in their descriptions. You basically have this strong sense that we operate on our data and computer simulations just as described by these famous thinkers. In short form, Popper = ‘falsification’, Kuhn = ‘paradigms’, Lakatos = ‘research programs’, we will expand upon this below.

We all know there is no ‘truth’ to our science, you can never prove a scientific idea, you can only test it, hard, in an attempt to falsify it (Popper). The more the idea stands up to testing, the ‘stronger’ it gets, where stronger means more used/accepted by the community. Paradigms (Kuhn) are now thought of more as frameworks (think Newtonian mechanics), and research programs (Lakatos) are like giant factories of ideas that define how we come up with new ideas to test. Key to all this was that PKL insisted that this only worked when discussing the history of science, and the philosophy community was deeply disappointed in the works of PKL because they wanted criteria that could be applied in the moment (i.e., some system to assist the science community in directing science to more productive methods and goals). It’s obvious to all of us (who have been doing science for awhile) that this description is more of an idealization and usually doesn’t work (pick any observing or funding review panel as an example of how this just fails in the moment, but works well in retrospect).

Let’s take a very simple example (that almost everyone is familiar with) to see how the methods proposed by PKL work best in hindsight and are fairly useless in the moment. Consider Newtonian gravity in the mid-19th century (by the way, did you know that Ptolemy’s system of epicycles was used by the makers of almanacs into the 20th century? The math was simpler and just as accurate as using Newton. But I digress). So, a wobble in the orbit of Uranus. Abandon Newton? Hell no, the framework/paradigm has solved so many problems. Consider an auxiliary hypothesis, i.e. another planet. Right there, Neptune! Boy, how strong is your paradigm that predicts new planets. Better telescopes, more accurate data, hmm, wobble in Mercury. Obviously, try the same hypothesis (the so-called planet of Vulcan). Nothing found, fortunately relativity came along pretty soon (although it’s goal was not to solve Mercury’s orbit, it had bigger issues to address). Newtonian physics survives, but now with limits on its applications (stick to low energies). In the moment, it was hard to see the depth of the problem or the need for radical change.

In any case, us working astronomers immediately recognize this kind of scientific effort; the nuts-and-bolts, thinking about new ideas, new technology, new data – in our own research. Now let’s turn to the dark matter paradigm. For the readers of this blog, I don’t need to discuss the details of the LCDM paradigm. In brief, Oort discrepancy (1960’s), flat rotation curves (1970’s), cluster M/L’s (1980’s), large scale structure (1990’s) and, of course, the capstone CMB+BBN+Omega_Lambda (2000’s). Historically, its important to remember that a great deal of work in the 1980’s eliminated anything made of baryons for CDM (low mass stars, black holes, planets, rocks, etc). In fact, most of us stopped even trying to explain CDM by the launch of HST, there were more interesting problems to address. We just took CDM as a black box solution; those interacting galaxy simulations looked amazing.

In the end, CDM was expected to be a gravitationally interacting, not short-lived, not hot, non-baryonic particle. With the closure of the SSC in 1992, a large number of particle physicists were suddenly out of jobs, dark matter looked like the clearest path to reach new physics (and a paycheck). And astronomers were just as guilty, Fig. 1 is a old plot from an Oemler workshop paper showing the state of the art in 1985. Notice the implication that dark matter could close the Universe (i.e., Omega = 1 all in mass; there was no Lambda back then). And a closed Universe was, for some reason*, a massive desire for the astro theoretical community. And we see the beginnings of how cosmological issues override galaxy issues for the community (there’s a nobility to doing cosmology, galaxies are dirty).

There is also no need to re-iterate the failure of the high energy community to find any dark matter particles. The amount of money and people in CDM makes the Manhattan project look like a garage shop effort (only CDM did not deliver a new weapon by the 2020’s). During this time, a few of us noticed that the baryons in galaxies were strongly tied to the ‘dark matter’. We were told that low mass dwarf galaxies were dark matter dominated, but a few of us mentioned that they are also strongly gas dominated, i.e. how did the dark matter know to dominate in galaxies with the highest gas fractions? These kinds of anomalies began to build up for observers, didn’t really appear in the theory papers.

Now let’s consider how PKL would interpret the current status of the dark matter paradigm. The emphasis by all three philosophers was to examine the historical record. This frustrates mainstream philosophers who would, ideally, like apply some criteria to the current state of affairs rather than only look at hindsight. But there may be enough historical record to, at least, determine the current trajectory for the dark matter paradigm.

Perhaps the first ‘crisis’ to the LCDM paradigm would be the discovery of the baryonic Tully-Fisher relation. The luminosity Tully-Fisher, which was only a measure of the stellar mass by way of stellar luminosity, was not very surprising to the astro/physics community for a larger galaxy would have more stars and dark matter, thus higher rotation velocities. The mass discrepancy was obvious, vindicating the need for some sort of invisible dark matter, but the scatter was large and the relationship was distinctly non-linear at the low mass end. However, the TF relation was well mapped at the high mass end (early-type cluster galaxies) and was extremely useful at locking down cluster distances and mapping large scale structure.

It was Stacy’s superb observer’s insight to think about adding the gas mass to the stellar mass producing a total baryon mass to compare to the rotational velocity (I helped a little, the so-called BTFR). I say ‘observer insight’ for its one of those things that if you work at optical and radio telescopes you are comfortable thinking in terms of different kinds of data, and at ease turning luminosities into stellar masses (i.e., M/L) and HI fluxes into gas masses (lots of details, but that’s what data reduction and analysis is all about). The scatter was greatly reduced and the relationship was increasingly close to a linear fit in log-log space. Regardless
of your MOND vs dark matter bias, the BTFR quite simply says the total baryon mass is related to the total dark matter mass, which would make sense if you scoped a piece of the early Universe (equal proportions of baryons and dark matter) and made a galaxy.

But that’s not how galaxy formation was suppose to work (you don’t scope a piece of the Universe, those CDM halos form first, baryons fall in later), and this baryon-dark matter relationship varied in a surprising coherent fashion across Hubble types. Stacy being a deep reader of obtuse literature in the 80’s taught us that this type of behavior was exactly predicted by basic MOND, especially for the galaxies in my new LSB catalog (to which I remember responding “You’re using my galaxies to disprove Newton? Only death can save you from my wrath”). While MOND was designed to explain flat rotation curves, its key underlying premise is that there is no dark matter, baryons decide how things move, which is what the BTFR said.

Now, in a strictly Popperian methodology, is the BTFR sufficient to falsify the dark matter paradigm? No, of course not, as Kuhn points out, a paradigm that has been so successful in making predictions, and offering research paths, is not simply dropped, that would be intellectually short-sighted. There a lots of historical examples in science to support the continued investigation into dark matter after the discovery of the BTFR. In the field of electromagnetism, one does not actually observe the magnetic fields, we observe its effects and build tools to explore its behavior and derive out rules, theories and models. There is nothing unbalanced about studying invisible dark matter.

In a Popperian fashion, we continued to ‘ruthlessly test’ the BTFR with better samples, improved stellar population models, improved distances. The BTFR survived all this examination, although it risked refutation at every point (for example, many claims that it deviated from a linear power-law fit at the low and high mass ends). The current version by Duey et al. (2026) is by far one of the tightest corrections in astronomy, especially considering all the possible underlying astrophysical processes (e.g., feedback) that could easily distort any kinematic relation. And, as often stated, the slope is 4 and MOND predicted 4, before we looked.

By the time of the BTFR discovery, the CDM community was deep into normal science mode. As Kuhn would state, lots of problem-solving, training students, textbooks and workshops to give. However, MOND was having its breakthroughs. The second crisis will occur with the SPARC sample where Federico Lelli showed that not only does the total baryon mass determine the total dynamical mass (BTFR), but also on the local scale within a galaxy, the baryon density is also correlated with the kinematics point by point within a galaxy’s gravitational potential. This will become the awkwardly phrased radial acceleration relation (RAR).

The smooth change, in acceleration space, displayed by the RAR is well predicted by MOND, but is difficult to explain/reproduce in LCDM without feedback from star formation processes. And the process of adding ‘feedback’ to CDM simulations begins to feel a lot like adding epicycles to ‘save the phenomenon’. Kuhn would say that we need a shift to a new disciplinary matrix, which asks new questions, and will recognize the competence of these new baryon-to-dark matter connection. But to threaten the paradigm, anomalies like RAR and BTFR must persist across different methods, resist easy explanation, and be critical to central themes. Which they seem to do, for many observers, not so much in the theory community.

Theorists, understandably, resist the MOND framework for 1) must explain more, 2) make more precise predictions, 3) coordinate research and 4) open new avenues. These criteria were not met. Also, switching to Milgromian physics will require a complete re-training of techniques, tools and students. A huge investment in time and effort that does not immediately appeal to the theory community given the success of decades of galaxy simulations.

To most of the community, the discovery of the BTFR and RAR do not ‘feel’ like a Kuhnian science revolution seen in the historical record (e.g. Galileo discovering the phases of Venus). A better explanation of the state of current affairs is to adopt the ideas of Lakatos. The heart of Lakato’s philosophy is the role of research programs. A research program is a more expansive view of Kuhn’s paradigms that includes much of the day-to-day operation of science; teaching, grants, textbooks, workshops, presentations. For Lakatos, it was all about whether a research area was progressive or degenerate (possibly even in a zombie stage). To be progressive, for Lakatos, the dark matter framework must be predictive, successful, expanding, yet CDM appears to be in ‘zombie’ stage, ceasing to generate novel predictions, responding to failure with ad hoc adjustments, and no clear path to increase its value as a line of thinking.

MOND work seems more progressive, clean explanations for new discoveries. But it also is struggling, the path forward is not obvious. The accelerated galaxy formation model is the most interesting discovery, and maps into our newest technology, JWST. However, it has been extremely difficult to acquire funding or observing time to address MOND ideas. It’s surprising how conservative funding and telescope panels have become (let’s do one more H_o project). The risk/reward dynamic is clearly in MOND’s favor, but not until more MOND worker are appointed to review panels. I know what I’m talking about, I ran NASA ADP review for two years. So many smart people in one room, overwhelmed by the work, reduced to the easiest path of proposal ranking.

I predict the near future will be one of continued stagnation of CDM research, no progress with many excuses. Some of the excuses looking like new physics, some perhaps even getting the fever that string theory had. The ‘successes’ for CDM have all been indirect (my observations are completely explained by X type of CDM, rather than here are my observations of X type of CDM). And there is still a role for CDM to play in astronomy. Since MOND has a non-linear Poisson equation, it does not lend itself to N-body simulations with ease. And N-body work using CDM has been extremely successful in our understanding of tidal features in interacting galaxies and shell galaxies. It just has to be remembered that when one uses CDM in your simulation, you are basically allowing a proxy for different gravity, and you call it CDM. Kinda of like the softening parameters using early N-body to avoid the wacky math singularities as ‘particles’ in your galaxy simulation got too close.

*Inflation. It was Inflationary cosmology that insisted on Ω_m = 1, and theorists chugged that Kool-Aid hard – Stacy

The External Field Effect and Tests of MOND on Cosmological Scales

The effective force law in galaxies is MOND. At high accelerations, this is the same as Newton’s inverse square law: $g = GM/r^2$ . At low accelerations, the deep MOND limit is $g = \sqrt{a_0GM}/r$ . The transition between the regimes occurs at $g = a_0 = 1.2 \times 10^{-10}\;\mathrm{m}\,\mathrm{s}^{-2}$ .

*The effective force law in galaxies (data) looks like MOND (line).*

This is well established in galaxies. The question naturally arises whether this holds on larger scales. A recent paper by Gallardo et al. says no:

On scales from 30 – 230 megaparsecs, we constrain the gravitational acceleration between pairs of halos^$ at separation r to be g ∝ 1/rⁿ with n=2.1±0.3

so Newton’s n = 2, not the n = 1 of the deep MOND limit. Here is their result for the kinematic SZ effect*.

**Fig. 1** from Gallardo et al.: *Pairwise kSZ measurements [μK] as a function of the physical separation of galaxy clusters [Mpc].*

Here the blue line represents Newton’s n = 2 with the boost of dark matter as appropriate for LCDM. This matches the data better than the yellow line with n = 1 representing MOND. But is that the right representation?

Although the bulk of our analysis is model independent, we have also shown the theoretical curves appropriate for MOND. This test is the largest-scale direct test of MOND to date. Our formalism is an excellent approximation to MOND in the low-acceleration regime

This much is good. It’s a nice, general approach to represent 1/r force laws like the deep MOND limit. But is that the right thing to do here?

that said, we have not included the external field effect (EFE) in this analysis. This can modify the acceleration law in the case where the baryonic gravity of an object is less than the gravity of its larger environment, and has been used to explain^# the velocity dispersion of satellite galaxies. However, this effect likely does not affect our analysis.

because the EFE absolutely affects this analysis.

They go on to say words about how the EFE is a thing that applies on small scales (hence the allusion to dwarf galaxies) but they’re looking at large scales so it shouldn’t matter. If only that were true.

The length scale does not matter in MOND. The acceleration scale matters. Is the chunk of the universe over which they’re integrating in the low acceleration regime? Yes. Is the EFE relevant on these scales? Also yes.

Some EFE from large scale structure is unavoidable. Everything feels the tug of everything else. In the deepest depths of the intergalactic medium, the EFE is tiny (maybe 1% of a₀) but also ubiquitous. In the absence of a prominent mass, it dominates. That’s exactly the regime this experiment probes.

The force in the MOND EFE regime looks like a boosted version of Newton: $g \approx (a_0/a_{\mathrm{EFE}})\,GM/r^2$ . The boost factor $a_0/a_{\mathrm{EFE}}$ is what we interpret as dark matter: the total mass that we need in excess of what we observe.

So, what Gallardo et al. falsify is a straw man version of MOND in which the force law has n = 1 forever and always. That’s only true in the absence of the EFE, for which the prediction is n = 2 – as observed.

There may be a test in the amplitude of the boost factor. We already know that is in the ballpark that’s required for the cosmic dark matter, $\Omega_m/\Omega_b \approx a_0/a_{\mathrm{EFE}}$ , as this is one of the first things I checked when I was surprised to encounter MOND in the previous century. So I don’t see much hope in distinguishing between the two this way.

Reality is more complicated. The amplitude of the EFE throughout intergalactic space depends on the cosmic distribution of mass. This was calculated by Chae et al. (2021), who found it to vary with both distance (from us) and position on the sky:

**Fig. 5** from Chae et al. (2021): Variation of e_N,env (the Newtonian^% amplitude of the EFE in units of a₀) with distance for the galaxies in the NSA and Karachentsev catalogs. Individual galaxies are color-coded by right ascension (R.A.). The black lines show the mean trend (solid) and standard deviation (dashed) in bins of distance.

It’s even worse than this, because most of the baryons are in the IGM. It makes a difference to the amplitude of the EFE how clumpy these are. They’re probably somewhat clustered into filaments and walls, but we don’t really have a great empirical map of that. So: the EFE is definitely there at a level that matters, but precisely what that level might be is rather hard to say. My best guess today is ~2% of a₀, but it could be more, and probably is in places. Could be less in the midst of the deepest voids.

Gallardo et al. do not cite Chae et al. (2021), nor evince awareness^& that there are relevant constraints on the EFE on the scales they probe. Note that the distance range of the figure from Chae et al. (2021) goes out to 150 Mpc, which is where the relevant data of Gallardo et al. are (their last two points are largely irrelevant). So we are talking about very much the same length scales, which does matter to the integration they do. That integration per force averages over any real variations in the EFE.

There is nowhere you can go to completely escape the EFE. A rather profound fact that appears not to be widely appreciated is that there is a minimum cosmic acceleration of order $\sim 10^{-12}\;\mathrm{m}\,\mathrm{s}^{-2}$ .

^$Saying “halos” here is quite the linguistic bias since there are no dark matter halos in the theory they’re testing.

*There are less obscure ways to do this, but this is what they chose to do and I’m not going to attempt to unravel it here. Ask them.

^#Not just explain, but predict. Repeatedly. Until the same ability to predict kinematics in advance of observation is demonstrated by LCDM, I’m going to remain underwhelmed by post-facto tests that claim to favor LCDM over MOND.

^%If I recall the notation correctly, the actual EFE is $\sqrt{a_0\,e_{\mathrm{N,env}}}$ and in these units $a_0 = 1$ so $\log(e_{\mathrm{N,env}}) = -3$ corresponds to an EFE that is 3% of $a_0$ . This plot was made assuming maximal clumping of the IGM baryons which gives an upper limit, so the reality is probably less; see Fig. 6 of Chae et al.

^&I wonder what the refereeing process looked like. I can imagine there being no mention of the EFE initially, with a referee (or perhaps one of the many coauthors) asking if they ought to maybe worry about it and the majority choosing to wave their hands through it. That’s what it looks like.

This is not the first time I’ve encountered the misconception that the EFE can be ignored on cosmic scales. It happened in the submitted version of Aguirre et al. (2001). In that case, I was the referee, and pointed out that the EFE had a profound effect on what they were saying about Lyman-alpha absorbers. To their credit, they listened and corrected it. I think they still kinda low-balled the amplitude of the EFE, but it went from a factor of tens problem to a small problem or maybe not a problem at all.

While on the topic of refereeing, I note that papers that find MOND wanting generally get less scrutiny than those which find that it works. Hardly surprising, once framed that way. I only mention this because there are certain toxic science communicators who rush to social media to denounce the incompetence of the referee any time a MOND-positive paper appears. Funny how they can be sure of the competence of the referee in a process they are not in any way privy to. If you take the time to think about it, you can more likely presume that the MOND paper has been held to a higher, not lower, standard, and weigh its credence appropriately.

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.

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!

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.

Very thin galaxies

The stability of spiral galaxies was a foundational motivation to invoke dark matter: a thin disk of self-gravitating stars is unstable unless embedded in a dark matter halo. Modified dynamics can also stabilize galactic disks. A related test is provided by how thin such galaxies can be.

Thin galaxies exist

Spiral galaxies seen edge-on are thin. They have a typical thickness – their short-to-long axis ratio – of q ≈ 0.2. Sometimes they’re thicker, sometimes they’re thinner, but this is often what we assume when building mass models of the stellar disk of galaxies that are not seen exactly* edge-on. One can employ more elaborate estimators, but the results are not particularly sensitive to the exact thickness so long as it isn’t the limit of either razor thin (q = 0) or a spherical cow (q = 1).

Sometimes galaxies are very thin. Behold the “superthin” galaxy UGC 7321:

*UGC 7321 as seen in optical colors by the Sloan Digital Sky Survey.*

It also looks very thin in the infrared, which is the better tracer of stellar mass:

**Fig. 1** from Matthews et al (1999): *H-band (1.6 micron) image of UGC 7321. Matthews (2000) finds a near-IR axis ratio of 14:1. That’s super thin (q = 0.07)!*

UGC 7321 is very thin, would be low surface brightness if seen face-on (Matthews estimates a central B-band surface brightness of 23.4 mag arcsec^-2), has no bulge component thickening the central region, and contains roughly as much mass in gas as stars. All of these properties dispose a disk to be fragile (to perturbations like mergers and subhalo crossings) and unstable, yet there it is. There are enough similar examples to build a flat galaxy catalog, so somehow the universe has figured out a way for galaxy disks to remain thin and dynamically cold^# for the better part of a Hubble time.

We see spiral galaxies at various inclinations to our line of sight. Some will appear face on, others edge-on, and everything in between. If we observe enough of them, we can work out what the intrinsic distribution is based on the projected version we see.

First, some definitions. A 3D object has three principle axes of lengths a, b, and c. By convention, a is the longest and c the shortest. An oblate model imagines a galaxy like a frisbee: it is perfectly round seen face-on (a = b); seen edge-on q = c/a. More generally, an object can be triaxial, with a ≠ b ≠ c. In this case, a galaxy would not appear perfectly round even when seen perfectly face-on^{^} because it is intrinsically oval (with similar axis lengths a ≈ b but not exactly equal). I expect this is fairly common among dwarf Irregular galaxies.

The observed and intrinsic distribution of disk thicknesses

Benevides et al. (2025) find that the distribution of observed axis ratios q is pretty flat. This is a consequence of most galaxies being seen at some intermediate viewing angle. One can posit an intrinsic distribution, model what one would see at a bunch of random viewing angles, and iterate to extract the true distribution in nature, which they do:

**Figure 6** from Benevides et al. (2025): Comparison between the observed (projected) $q$ distribution and the inferred intrinsic 3D axis ratios for a subsample of dwarfs in the GAMA survey with $M_{⋆} = 10^{9}$ – $10^{9.5} M_{⊙}$ . The observed shapes are shown with the solid black line and are used to derive an intrinsic $c / a$ (long-dashed) and $b / a$ (dotted) distribution when projected. Solid color lines in each panel corresponds to the $q$ values obtained from the 3D model after random projections. Note that a wide distribution of $q$ values is generated by a much narrower intrinsic $c / a$ distribution. For example, the blue shaded region in the left panel shows that an observed $5 %$ of galaxies with $q < 0.2$ requires $41 %$ of galaxies to have an intrinsic $c / a < 0.2$ for an oblate model. Similarly, for a triaxal model (right panel, red curve) $43 %$ of galaxies are required to be thinner than $c / a = 0.2$ . The additional freedom of $b \neq a$ in the triaxial model helps to obtain a better fit to the projected $q$ distribution, but the changes mostly affect large $q$ values and changes little the $c / a$ frequency derived from highly elongated objects.

That we see some thin galaxies implies that they they have to be common, as most of them are not seen edge-on. For dwarf^$ galaxies of a specific mass range, which happens to include UGC 7321, Benevides et al. (2025) infer a lot^% of thin galaxies, at least 40% with q < 0.2. They also infer a little bit of triaxiality, a ≈ b.

The existence and numbers of thin dwarfs seems to come as a surprise to many astronomers. This is perhaps driven in part by theoretical expectations for dwarf galaxies to be thick: a low surface brightness disk has little self-gravity to hold stars in a narrow plane. This expectation is so strong that Benevides et al. (2025) feel compelled to provide some observed examples, as if to say look, really:

**Figure 8** – images of real galaxies from Benevides et al. (2025): Examples of $10$ highly elongated dwarf galaxies with $q \leq 0.2$ and $M_{⋆} = 10^{7}$ – $10^{8.5} M_{⊙}$ . They resemble thin edge-on disks and can be found even among the faintest dwarfs in our sample. Legends in each panel quote the stellar mass, the shape parameter $q$ , as well as the GAMA identifier. Objects are sorted by increasing $M_{⋆}$ , left to right.

As an empiricist who has spent a career looking at low mass and low surface brightness galaxies, this does not come as a surprise to me. These galaxies look normal. That’s what the universe of late type dwarf^$ galaxies looks like.

Edge-on galaxies in LCDM simulations

Thin galaxies do not occur naturally in the hierarchical mergers of LCDM (e.g., Haslbauer et al. 2022), where one would expect a steady bombardment by merging masses to mess things up. The picture above is not what galaxy-like objects in LCDM simulations look like. Scraping through a few simulations to find the flattest galaxies, Benevides et al. (2025) find only a handful of examples:

**Figure 11** – images of simulated galaxies from Benevides et al. (2025): *Edge-on projection of examples of the flattest galaxies in the TNG50 simulation, in different bins of stellar mass.*

Note that only the four images on the left here occupy the same stellar mass range as the images of reality above. These are as close as it gets. Not terrible, but also not representative^&. The fraction of galaxies this thin is a tiny fraction of the simulated population whereas they are quite common in reality. Here the two are compared: three different surveys (solid lines) vs. three different simulations (dashed lines).

**Figure 9** from Benevides et al. (2025): Fraction of galaxies that are derived to be intrinsically thinner than $c / a \leq 0.2$ as a function of stellar mass. Thick solid lines correspond to our observational samples while dashed lines are used to display the results of cosmological simulations. Different colors highlight the specific survey or simulation name, as quoted in the legend. In all observational surveys, the frequency of thin galaxies peaks for dwarfs with $M_{⋆} \sim 10^{9} M_{⊙}$ , almost doubling the frequency observed on the scale of MW-mass galaxies. Thin galaxies do not disappear at lower masses: we infer a significant fraction of dwarf galaxies with $M_{⋆} < 10^{9} M_{⊙}$ to have $c / a < 0.2$ . This is in stark contrast with the negligible production of thin dwarf galaxies in all numerical simulations analyzed here.

Note that the thinnest galaxies in nature are dwarfs of mass comparable to UGC 7321. Thin disks aren’t just for bright spirals like the Milky Way with log(M_*) > 10.5. They are also common^*$ for dwarfs with log(M_*) = 9 and even log(M_*) = 8, which are often gas dominated. In contrast, the simulations produce almost no galaxies that are thin at these lower masses.

The simulations simply do not look like reality. Again. And again, etc., etc., ad nauseam. It’s almost as if the old adage applies: garbage in, garbage out. Maybe it’s not the resolution or the implementation of the simulations that’s the problem. One could get all that right, but it wouldn’t matter if the starting assumption of a universe dominated by cold dark matter was the input garbage.

Galaxy thickness in Newton and MOND

Thick disks are not merely a product of simulations, they are endemic to Newtonian dynamics. As stars orbit around and around a galaxy’s center, they also oscillate up and down, bobbing in and out of the plane. How far up they get depends on how fast they’re going (the dynamical temperature of the stellar population) and how strong the restoring force to the plane of the disk is.

In the traditional picture of a thin spiral galaxy embedded in a quasi-spherical dark matter halo, the restoring force is provided by the stars in the disk. The dark matter halo is there to boost the radial force to make the rotation curve flat, and to stabilize the disk, for which it needs to be approximately spherical. The dark matter halo does not contribute much to the vertical restoring force because it adds little mass near the disk plane. In order to do that, the halo would have to be very squashed (small q) like the disk, in which case we revive the stability problem the halo was put there to solve.

This is why we expect low surface brightness disks to be thick. Their stars are spread thin, the surface mass density is low, so the restoring force to the disk should be small. Disks as thin as UGC 7321 shouldn’t be possible unless they are extremely cold^*# dynamically – a situation that is unlikely to persist in a cosmogony built by hierarchical merging. The simulations discussed above corroborate this expectation.

In MOND, there is no dark matter halo, but the modified force should boost the vertical restoring force as well as the radial force. One thus expects thinner disks in MOND than in Newton.

I pointed this out in McGaugh & de Blok (1998) along with pretty much everything else in the universe that people tell me I should consider without bothering to check if I’ve already considered. Here is the plot I published at the time:

**Figure 9** of McGaugh & de Blok (1998): Thickness q = z₀/h expected for disks of various central surface densities ₀. Shown along the top axis is the equivalent B-band central surface brightness ₀ for _* = 2. Parameters chosen for illustration are noted in the figure (a typical scale length h and two choices of central vertical velocity dispersion _z). Other plausible values give similar results. The solid lines are the Newtonian expectation and the dashed lines that of MOND. The Newtonian and MOND cases are similar at high surface densities but differ enormously at low surface densities. Newtonian disks become very thick at low surface brightness. In contrast, MOND disks can remain reasonably thin to low surface density.

There are many approximations that have to be made in constructing the figure above. I assumed disks were plane-parallel slabs of constant velocity dispersion, which they are not. But this suffices to illustrate the basic point, that disks should remain thinner^&% in MOND than in Newton as surface density decreases: as one sinks further into the MOND regime, there is relatively more restoring force keep disks thin. To duplicate this effect in Newton, one must invent two kinds of dark matter: a dissipational kind of dark matter that forms a dark matter disk in addition to the usual dissipationless cold dark matter that makes a quasi-spherical dark matter halo.

The idea of the plot above was to illustrate the trend of expected thickness for galaxies of different central surface brightness. One can also build a model to illustrate the expected thickness as a function of radius for a pair of galaxies, one high surface brightness (so it starts in the Newtonian regime at small radii) and one of low surface brightness (in the MOND regime everywhere). I have chosen numbers^** resembling the Milky Way for the high surface brightness galaxy model, and scaled the velocity dispersion of the low surface brightness model so it has very nearly the same thickness in the Newtonian regime. In MOND, both disks remain thin as a function of radius (they flare a lot in Newton) and the lower surface brightness disk model is thinner thanks to the relatively stronger restoring force that follows from being deeper in the MOND regime.

The thickness of two model disks, one high surface brightness (solid lines) and the other low surface brightness (dashed lines), as a function of radius. The two are similar in Newton (black), but differ in MOND *(blue)*. The restoring force to the disk is stronger in MOND, so there is less flaring with increasing radius. The low surface brightness galaxy is further in the MOND regime, leading naturally to a thinner disk.

These are not realistic disk models, but they again suffice to illustrate the point: thin disks occur naturally in MOND. Low surface brightness disks should be thick in LCDM (and in Newtonian dynamics in general), but can be as thin as UGC 7321 in MOND. I didn’t aim to make q ≈ 0.1 in the model low surface brightness disk; it just came out that way for numbers chosen to be reasonable representations of the genre.

What the distribution of thicknesses is depends on the accretion and heating history of each individual disk. I don’t claim to understand that. But the mere existence of dwarf galaxies with thin disks is a natural outcome in MOND that we once again struggle to comprehend in terms of dark matter.

*Seeing a galaxy highly inclined minimizes the inclination correction to the kinematic observations [V_rot = V_obs/sin(i)] but to build a mass model we also need to know the face-on surface density profile of the stars, the correction for which depends on 1/cos(i). So as a practical matter, the competition between sin(i) and cos(i) makes it difficult to analyze galaxies at either extreme.

^#Dynamically cold means the random motions (quantified by the velocity dispersion of stars σ) are small compared to ordered rotation (V) in the disk, something like V/σ ≈ 10. As a disk heats (higher σ) it thickens, as some of that random motion goes in the vertical direction perpendicular to the disk. Mergers heat disks because they bring kinetic energy in from random directions. Even after an object is absorbed, the splash it made is preserved in the vertical distribution of the stars which, once displaced, never settle back into a thin disk. (Gas can settle through dissipation, but point masses like stars cannot.)

^Oval distortions are a major source of systematic error in galaxy inclination estimates, especially for dwarf Irregulars. It is an asymmetric error: a galaxy with a mild oval distortion can be inferred to have an inclination (i > 0) even when seen face-on (i = 0), but it can never have an inclination more face-on (i < 0) than exactly face-on. This is one of the common drivers of claims that low mass galaxies fall off the Tully-Fisher relation. (Other common problems include a failure to account for gas mass, bad distance estimates, or not measuring V_flat.)

^$In a field with abominable terminology, what is meant by a “dwarf” galaxy is one of the worst offenders. One of my first conference contributions thirty years ago griped about the [mis]use of this term, and matters have not improved. For this particular figure, Benevides et al. (2025) define it to mean galaxies with stellar masses in the range 9 < log(M_*) < 9.5, which seems big to me, but at least it is below the mass of a typical L* spiral, which has log(M_*) ~ 10.5. For comparison, see Fig. 6 of the review of Bullock & Boylan-Kolchin (2017), who define “bright dwarfs” to have 7 < log(M_*) < 9, and go lower from there, but not higher into the regime that we’re calling dwarf right now. So what a dwarf galaxy is depends on context.

^%Note that the intrinsic distribution peaks below q = 0.2, so arguably one should perhaps adopt as typical the mode of the distribution (q ≈ 0.17).

^&Another way in which even the thin simulated objects are not representative of reality is that they are dynamically hot, as indicated by the κ_rot parameter printed with the image. This is the fraction of kinetic energy in rotation. One of the more favorable cases with κ_rot = 0.67 corresponds to V/σ = 2.5. That happens in reality, but higher values are common. Of course, thin disks and dynamical coldness go hand in hand. Since the simulations involve a lot of mergers, the fraction of kinetic energy in rotation is naturally small. So I’m not saying the simulations are wrong in what they predict given the input physics that they assume, but I am saying that this prediction does not match reality.

^*$The fraction of thin galaxies observed by DESI is slightly higher than found in the other surveys. Having looked at all these data, I am inclined to suspect the culprit is image quality: that of DESI is better. Regardless of the culprit for this small discrepancy between surveys, thin disks are much more common in reality than in the current generation of simulations.

^*#There seems to be a limit to how cold disks get, with a minimum velocity dispersion around ~7 km/s observed in face-on dwarfs when the appropriate number, according to Newton, would be more like 2 km/s, tops. I remember this number from observations in the ’80s and ’90s, along with lots of discussion then to the effect of how can it be so? but it is the new year and I’m feeling too lazy to hunt down all the citations so you get a meme instead.

^&%In an absolute sense, all other things being equal, which they’re not, disks do become thicker to lower surface brightness in both Newton and MOND. There is less restoring force for less surface mass density. It is the relative decline in restoring force and consequent thickening of the disk that is much more precipitous in Newton.

^**For the numerically curious, these models are exponential disks with surface density profiles Σ(R) = Σ₀ e^-R/R_d. Both models have a scale length R_d = 3 kpc. The HSB has Σ₀ = 866 M_☉ pc^-2; this is a good match to the Eilers et al. (2019) Milky Way disk; see McGaugh (2019). The LSB has Σ₀ = 100 M_☉ pc^-2, which corresponds roughly to what I consider the boundary of low surface brightness, a central B-band surface brightness of ~23 mag. arcsec^-2. For the velocity dispersion profile I also assume an exponential with scale length 2R_d (that’s what supposed to happen). The central velocity dispersion of the HSB is 100 km/s (an educated guess that gets us in the right ballpark) and that of the LSB is 33 km/s – the mass is down by a factor of ~9 so the velocity dispersion should be lower by a factor of $\sqrt{9}$ . (I let it be inexact so the solid and dashed Newtonian lines wouldn’t exactly overlap.)

These models are crude, being single-population (there can be multiple stellar populations each with their own velocity dispersion and vertical scale height) and lacking both a bulge and gas. The velocity dispersion profile sometimes falls with a scale length twice the disk scale length as expected, sometimes not. In the Milky Way, R_d ≈ 2.5 or 3 kpc, but the velocity dispersion falls off with a scale length that is not 5 or 6 kpc but rather 21 or 25 kpc. I have also seen the velocity dispersion profile flatten out rather than continue to fall with radius. That might itself be a hint of MOND, but there are lots of different aspects of the problem to consider.

The odd primordial halo of the Milky Way

The mass distribution of dark matter halos that we infer from observations tells us where the dark matter needs to be now. This differs form the mass distribution it had to start, as it gets altered by the process of galaxy formation. It is the primordial distribution that dark matter-only simulations predict most robustly. We* reverse-engineer the collapse of the baryons that make up the visible Galaxy to infer the primordial distribution, which turns out to be… odd.

The Gaia rotation curve and the mass of the Milky Way

As we discussed a couple of years ago, Gaia DR3 data indicate a declining rotation curve for the Milky Way. This decline becomes more steep, nearly Keplerian, in the outskirts of the Milky Way (17 < R < 30 kpc). This is may or may not be consistent with data further out, which gets hard to interpret as the LMC (at 50 kpc) perturbs orbits and the observed motions may not correspond to orbits in dynamical equilibrium. So how much do the data inform us about the gravitational potential?

Milky Way rotation curve (various data) including Gaia DR3 (multiple analyses). Also shown is the RAR model (blue line) that was fit to the terminal velocities from 3 < R < 8.2 kpc (gray points) and predates other data illustrated here.

I am skeptical of the Keplerian portion of this result (as discussed at length at the time) because other galaxies don’t do that. However, I am a big fan of listening to the data, and the people actually doing the work. Taken at face value, the Gaia data show a Keplerian decline with a total mass around 2 x 10¹¹ M_☉. If correct, this falsifies MOND.

How does dark matter fare? There is an implicit assumption made by many in the community that any failing of MOND is an automatic win for dark matter. However, it has been my experience that observations that are problematic for MOND are also problematic for dark matter. So let’s check.

Short answer: this is really weird in terms of dark matter. How weird? For starters, most recent non-Gaia dynamical analyses suggest a total mass closer to 10¹² M_☉, a factor of five higher than the Gaia value. I’m old enough to remember when the accepted mass was 2 x 10¹² M_☉, an order of magnitude higher. Yet even this larger mass is smaller than suggested by abundance matching recipes, which give more like 4 x 10¹² M_☉. So somewhere in the range 2 – 40 x 10¹¹ M_☉.

The Milky Mass has been adjusted so often, have we finally hit it?

The guy was all over the road. I had to swerve a number of times before I hit him.
Boston Driver’s Handbook (1982 edition)^&

If it sounds like we’re all over the map, that’s because we are. It is very hard to constrain the total mass of a dark matter halo. We can’t see it, nor tell where it ends. We infer, indirectly, that the edge is way out beyond the tracers we can see. Heck, even speaking of an “edge” is ill-defined. Theoretically, we expect it to taper off with the density of dark matter falling as ρ ~ r^-3, so there is no definitive edge. Somewhat arbitrarily,** we adopt the radius that encloses a density 200 times the average density of the universe as the “virial” radius. This is all completely notional, and it gets worse, as the process of forming a galaxy changes the initial mass distribution. What we observe today is the changed form, not the primordial initial condition for which the notional mass is defined.

Adiabatic compression during galaxy formation

To form a visible galaxy, baryons must dissipate and sink to the center of their parent dark matter halo. This process changes the mass distribution and alters the halo from its primordial state. In effect, the gravity of the sinking baryons drags some dark matter along^# with them.

The change to the dark matter halo is often called adiabatic compression. The actual process need not be adiabatic, but that’s how we approximate it. We’ve tested this approximation with detailed numerical simulations, and it works pretty well, at least if you do it right (there are boring debates about technique). What happens makes sense intuitively: the response of the primordial halo to the infall of baryons is to become more dense at the center. While this makes sense physically, it is problematic for LCDM as it takes an NFW halo that is already too dense at the center to be consistent with data and makes it more dense. This has been known forever, so opposing this is one thing feedback is invoked to do, which it may or may not do, depending on how it really works. Even if feedback can really turn a compressed cusp into a core, it is widely to expected to be important only in low mass galaxies where the gravitational potential well isn’t too deep. It isn’t supposed to be all that important in galaxies as massive as the Milky Way, though I’m sure that can change as needed.

There are a variety of challenges to implementing an accurate compression computation, so we usually don’t bother: the standard practice is to assume a halo model and fit it to the data. That will, at best, given a description of the current dark matter halo, not what it started as, which is our closest point of comparison with theory. To give an example of the effect, here is a Milky Way model I built a decade ago:

**Figure 13** from McGaugh (2016): Milky Way rotation curve from the data of Luna et al. (2006, red points) and McClure-Griffiths & Dickey (2007, gray points) together with a bulgeless baryonic mass model (black line). The total rotation is approximately fit (blue line) with an adiabatically compressed NFW halo (solid green line) using the procedure implemented by Sellwood & McGaugh (2005). The primordial halo before compression is shown as the dashed line. The parameters of the primordial halo are a concentration c = 7 and a mass *M₂₀₀ = 6 x 10¹¹ M_☉*. *Fitting NFW to the present halo instead gives c = 14, M₂₀₀ = 4 x 10¹¹ M_☉, so the difference is appreciable and depend on the quality and radial extent of the available data.*

The change from the green dashed line to the solid green line is the difference compression makes. That’s what happens if a baryon distribution like that of the Milky Way settles in an NFW halo. The inferred mass M₂₀₀ is lower and the concentration c higher than it originally was – and it is the original version that we should compare to the expectations of LCDM.

When I built this model, I considered several choices for the bulge/bar fraction: something reasonable, something probably too large, and something definitely too small (zero). The model above is the last case of zero bulge/bar. I show it because it is the only case for which the compression procedure worked. If there is a larger central concentration of baryons – i.e., a bulge and/or a bar – then the compression is greater. Too great, in fact: I could not obtain a fit (see also Binney & Piffl and this related discussion).

The calculation of the compression requires knowledge of the primordial halo parameters, which is what one is trying to obtain. So one has to guess an initial state, run the code, check how close it came, then iterate the initial guess. This is computationally expensive, so I was just eyeballing the fit above. Pengfei has done a lot of work to implement a method that iteratively computes the compression and rigorously fits it to data. So we decided to apply it to the newer Gaia DR3 data.

Fitting the Gaia rotation curve with adiabatically compressed halos

We need two inputs here: one, the rotation curve to fit, and two, the baryonic distribution of the Milky Way. The latter is hard to specify given our location within the Milky Way, so there are many different estimates. We tried a dozen.

Another challenge of doing this is deciding which data rotation curve data to fit. We chose to focus on the rotation curve of Jiao et al. (2023) because they made estimates of the systematic as well as random errors. The statistics of Gaia are so good it is practically impossible to fit any equilibrium model to them. There are aspects of the data for which we have to consider non-equilibrium effects (spiral arms, the bar, “snails” from external perturbations) so the usual assumptions are at best an approximation, plus there can always be systematic errors. So the approach is to believe the data, but with the uncertainty estimate of Jiao et al. (2023) that includes systematics.

For a halo model, we started with the boilerplate LCDM NFW halo^$. This doesn’t fit the data. Indeed, all attempts to fit NFW halos fail in similar ways for all of the different baryonic mass models we tried. The quasi-Keplerian part of the Gaia rotation curve simply cannot be fit: the NFW halo inevitably requires more mass further out.

Here are a few examples of the NFW fits:

**Fig. A.3** from Li et al. (2025). Fits of Galactic circular velocities using the NFW model implementing adiabatic halo contraction using 3 baryonic models. [Another 9 appear in the paper.] Data points with errors are the rotation velocities from Jiao et al. (2023), while open triangles show the data from Eilers et al. (2019), which are not fitted. [The radius ranges from 5 to 30 kpc.] Blue, purple, green and black solid lines correspond to the contributions by the stellar disk, central bar, gas (and dust if any), and compressed dark matter halo, respectively. The total contributions are shown using red solid lines. Black dashed lines are the inferred primordial halos.

LCDM as represented by NFW suffers the same failure mode as seen in MOND (plot at top): both theories overshoot the Gaia rotation curve at R > 17 kpc. This is an example of how data that are problematic for MOND are also problematic for dark matter.

We do have more freedom in the case of dark matter. So we tried a different halo model, Einasto. (For this and many other halo models, see Pengfei’s epic compendium of dark matter halo fits.) Where NFW has two parameters, a concentration c and mass M₂₀₀, Einasto has a third parameter that modulates the shape of the density profile^%. For a very specific choice of this third parameter (α = 0.17), it looks basically the same as NFW. But if we let α be free, then we can obtain a fit. Of all the baryonic models, the RAR model+compressed Einasto fits best:

**Fig. 1** from Li et al. (2025). Example of a circular velocity fit using the McGaugh19^$$ model for baryonic mass distributions. The purple, blue, and green lines represent the contributions of the bar, disk, and gas components, respectively. The solid and dashed black lines show the current and primordial dark matter halos, respectively. The solid red line indicates the total velocity profile. The black points show the latest Gaia measurements (Jiao et al. 2023), and the gray upward triangles and squares show the terminal velocities from (McClure-Griffiths & Dickey 2007, 2016), and Portail et al. (2017), respectively. The data marked with open symbols were not fit because they do not consider the systematic uncertainties.

So it is possible to obtain a fit considering adiabatic compression. But at what price? The parameters of the best-fit primordial Einasto halo shown above are c = 5.1, M₂₀₀ = 1.2 x 10¹¹ M_☉, and α = 2.75. That’s pretty far from the α = 0.17 expected in LCDM. The mass is lower than low. The concentration is also low. There are expectation values for all these quantities in LCDM, and all of them miss the mark.

**Fig. 2** from Li et al. (2025). Halo masses and concentrations of the primordial Galactic halos derived from the Gaia circular velocity fits using 12 baryonic models. The red and blue stars with errors represent the halos with and without adiabatic contraction, respectively. The predicted halo mass-concentration relation within 1 σ from simulations (Dutton & Macciò 2014) is shown as the declining band. The vertical band shows the expected range of the MW halo mass according to the abundance-
matching relation (Moster et al. 2013). The upper and lower limits are set by the highest stellar mass model plus 1 σ and the lowest stellar mass model minus 1 σ, respectively.

The expectation for mass and concentration is shown as the bands above. If the primordial halo were anything like what it should be in LCDM, the halo parameters represented by the red stars should be where the bands intersect. They’re nowhere close. The same goes for the shape parameter. The halo should have a density profile like the blue band in the plot below; instead it is more like the red band.

**Fig. 3** from Li et al. (2025). Structure of the inferred primordial and current Galactic halos, along with predictions for the cold and warm dark matter. The density profiles are scaled so that there is no need to assume or consider the masses or concentrations for these halos. The gray band indicates the range of the current halos derived from the Gaia velocity fits using the 12 baryonic models, and the red band shows their corresponding primordial halos within 1σ. The blue band presents the simulated halos with cold dark matter only (Dutton & Macciò 2014). The purple band shows the warm dark matter halos (normalized to match the primordial Galactic halo) with a core size spanning from 4.56 kpc (WDM5 in Macciò et al. 2012) to 7.0 kpc, corresponding to a particle mass of 0.05 keV and lower.

So the primordial halo of the Milky Way is pretty odd. From the perspective of LCDM, the mass is too low and the concentration is too low. The inner profile is too flat (a core rather than a cusp) and the outer profile is too steep. This outer steepness is a large part of why the mass comes out so low; there just isn’t a lot of halo out there. The characteristic density ρ_s is at least in the right ballpark, so aside from the inner slope, the outer slope, the mass, and the concentration, LCDM is doing great.

What if we ignore the naughty bits?

It is really hard for any halo model to fit the steep decline of the Gaia rotation curve at R > 17 kpc. Doing so is what makes the halo mass so small. I’m skeptical about this part of the data, so do things improve if we don’t sweat that part?

Ignoring the data at R > 17 kpc allows the mass to be larger, consistent with other dynamical determinations if not quite with abundance matching. However, the inner parts of the rotation curve still prefer a low density core. That is, something like the warm dark matter halo depicted as the purple band above rather than NFW with its dense central cusp. Or self-interacting dark matter. Or cold dark matter with just-so feedback. Or really anything that obfuscates the need to confront the dangerous question: why does MOND perform better?

*This post is based on the recently published paper by my former student Pengfei Li, who is now faculty at Nanjing University. They have a press release about it.

^&A few months after reading this in the Boston Driver’s Handbook, this exact thing happened to me.

**This goes back to BBKS in 1986 when the bedrock assumption was that the universe had Ω_m = 1, for which the virial radius was 188 times the critical density. 200 was close enough, and stuck, even though for LCDM the virial radius is more like an overdensity close to 100, which is even further out.

^#This is one of many processes that occur in simulations, which are great for examining the statistics of simulated galaxy-like objects but completely useless for modeling individual galaxies in the real universe. There may be similar objects, but one can never say “this galaxy is represented by that simulated thing.” To model a real galaxy requires a customized approach.

^$NFW halos consistently perform worse in fitting data than any other halo model, of which there are many. It has been falsified as a viable representation of reality so many times that I can’t recall them all, and yet they remain the go-to model. I think that’s partly thanks to their simplicity – it is mathematically straightforward to implement – and to the fact that is what simulations predict: LCDM halos should look like NFW. People, including scientists, often struggle to differentiate simulation from reality, so we keep flogging the dead horse.

^%The density profile of the NFW halo model asymptotes to power laws at both small and large radii: ρ → r^-1 as r → 0 and ρ → r^-3 as r → ∞. The third parameter of Einasto allows a much wider ranges of shapes.

^$$The McGaugh19 model user here is the one with a reasonable bulge/bar. This dense component can be fit in this case because we start with a halo model with a core rather than a cusp (closer to α = 1 than to the α = 0.17 of NFW/LCDM).

Triton Station

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

Category: LCDM