A Blog About the Science and Sociology of Cosmology and Dark Matter
Author: tritonstation
Stacy McGaugh is an astrophysicist and cosmologist who studies galaxies, dark matter, and theories of modified gravity. He is an expert on low surface brightness galaxies, a class of objects in which the stars are spread thin compared to bright galaxies like our own Milky Way. He demonstrated that these dim galaxies appear to be dark matter dominated, providing unique tests of theories of galaxy formation and modified gravity.
Professor McGaugh is currently the chair of the Department of Astronomy at Case Western Reserve University in Cleveland, Ohio, and director of the Warner and Swasey Observatory. Previously he was a member of the faculty at the University of Maryland, having also held research fellowships at Rutgers, the Department of Terrestrial Magnetism of the Carnegie Institution of Washington, and the Institute of Astronomy at the University of Cambridge after earning his Ph.D. from the University of Michigan.
I am mostly finished with a post following on from what would be required to make a convincing dark matter detection that explains why it will never happen. I’ve been nearly done for several weeks, but haven’t had time to complete it. Time carries on, and during that time a story appeared in Quanta magazine about observations claiming to be the final nail in the coffin for MOND, all the usual talking points we’ve coveredhere before, blah blah blah. But since it was in the news, I felt obliged to hop on social media (ug) and push back against the tidal wave of confirmation bias. So I’m just gonna reproduce that thread here, maybe adding further commentary if I can find some copious spare time.
First, some context: we’re talking about ultradiffuse galaxies in the vicinity of NGC 1052. These objects are abnormally large for their luminosity, so very diffuse. (Very low surface brightness in my 1996 terminology, ultradiffuse in that of van Dokkum et al. 2015.) I’m not a fan of describing them as “ghostly,” but low surface brightness galaxies have been described as ghostly before. Maybe “ghostly” is not entirely misplaced here, as one possibility is that they have been tidally whacked out of equilibrium and are slowly expanding towards an ultimate dissolution, fading like ghosts into the intergalactic graveyard.
But that’s getting ahead of the story. Let’s cut to the thread:
Story out in Quanta about the diffuse galaxies DF2, DF4, & DF9. It conveys the message that these objects are the nail in the coffin for MOND. Is it so? www.quantamagazine.org/triplet-of-g…
When DF2 was first reported, it was portrayed as being fatal for MOND. And so it appeared at first glance, with an observed velocity dispersion of 3 km/s where MOND predicted 20 km/s for an isolate object. Pretty bad! But a casual glance showed that it wasn’t an isolated object, so the EFE matters.
A more careful analysis yields a MOND prediction of 13.4 +4.8/-3.7 km/s. Contemporary reanalyses of the data estimated an observed velocity dispersion close to 12 km/s. That sounds more like corroboration than refutation: arxiv.org/abs/1804.04167
Lots of observational resources have been applied since then; I won’t attempt to review it all here. The new paper offers two independent measurements, which are 6.3+3.1/-3.6 and 9.2+3.8/-4.5 km/s. These are both consistent with 10 km/s, which is within the 1 sigma uncertainty of the prediction.
So MOND is falsified at one sigma.
That’s assuming these systems are in equilibrium, which is not obvious. We have to invoke some pretty dramatic non-equilibrium events to strip them of dark matter (in the conventional picture, galaxies cannot form without dark matter so we have to get rid of it. Difficult but not impossible.)
To be fair, we should worry about non-equilibrium situations in MOND as well. Fortunately, simulations have been done that illustrate what happens to a dwarf galaxy that gets tidally disrupted by a close passage with a giant galaxy. arxiv.org/abs/astro-ph…
The velocity dispersion departs from the equilibrium prediction and the scale size of the dwarf slowly expands. This actually provides a natural explanation for the large half-light radii of these DF dwarfs. tritonstation.com/2025/09/15/n…
I’ve no idea if this is what happened in these systems as I have not+ been around several Gyr to watch it happen. But there are certainly giant galaxies in the neighborhood, so it’s no less plausible than the bullet collision hypothesis in the DM context.
DF2 has consistently been portrayed as fatal for MOND despite consistently being shown not to be a problem. I’ve given up writing formal papers showing this, as nobody cares. They just hear what they want to hear, which is that MOND is wrong. Indeed, after the 2018& incident I swore off doing so.
It’s sad, but there is no point in correcting the scientific record while the relevant scientific community suffers a catastrophic failure of objectivity.
Strong words? Are you a scientist working in the field? Have you heard of DF2? If so, did you know its interpretation was in dispute? Or only that it falsifies MOND?
If you have heard of DF2, have you also heard of Crater 2? It has a tiny velocity dispersion (2.7 km/s) that makes no sense in LCDM but was correctly predicted a priori by MOND. arxiv.org/abs/1610.06189
Crater 2 and DF2 are similar in a number of respects. Why should we obsess about one while ignoring the other? That’d be an example of the failure of objectivity.
Or how about And XXVIII? You know that one? That was the golden test we proposed in arxiv.org/abs/1301.0822
Does MOND have problems? You bet! But it has lots and lots of predictive successes that people seem to be oblivious to. Many of these are predictions that cannot even be made in the context of dark matter. DF2 is a good example: we didn’t anticipate it was DM-free, we inferred that from the data.
If instead DF2 had a large velocity dispersion (like DF44), then we would happily report that it had a surplus of dark matter. There’s no prediction to test; it’s just whatever the observations dictate.
MOND makes actual predictions that mostly come true. This being astronomy, there are always some dodgy cases. So DM appears to be better in those cases by virtue of not making a testable prediction. Does that make it better science?
There still remains no satisfactory explanation in terms of dark matter for MOND’s predictive success. If you can come up with one, then perhaps dark matter can be viable. Without that, just complaining about what MOND gets wrong without engaging with what it gets right is intellectually dishonest.
+Apparently I omitted the “not” in the Bluesky thread. I may be old, but I’m not that old.
&Before that time, I was extremely conscientious about examining claims to falsify MOND. I’ve done this for literally hundreds of cases. The recurring theme had been that these claims are greatly exaggerated. Not all, of course, but more often that not, it was simply a matter of doing the analysis properly and including the errors. The current example is a good one: the claim is that these ultradiffuse galaxies have no dark matter (or MONDian mass discrepancy). The data are indeed consistent with that. But there are uncertainties in all data, and here they encompass MOND’s predictions as well.
At any rate, I no longer see correcting the mistakes of others as a constructive use of my time.
*Except And V. That one has a somewhat too-high velocity dispersion for MOND. If I were to follow the template offered by many examples in the literature, I should loudly advertise how this one case falsifies MOND while downplaying how MOND correctly predicted the velocity dispersions of the other three dozen dwarf satellites of Andromeda.
Here are the predicted and observed velocity dispersions for the dwarf satellites of Andromeda from McGaugh & Milgrom 2013a:
Figure 1 from McGaugh & Milgrom 2013a: The predicted and observed velocity dispersions of the dSph satellites of M31. Each bin along the x-axis represents the corresponding dwarf: And I is 1, And II is 2, and so on. Round symbols are the predictions of MOND. Large symbols assume M*/LV = 2 M☉/L☉. Small symbols represent M*/LV = 1 M☉/L☉ at the lower end of the predicted range and M*/LV = 4 M☉/L☉ at the upper end. The calculation in the isolated case is shown as a filled green circle; that when the EFE dominates is shown as an open circle. The appropriate case is shown for each mass-to-light ratio based on which predicts the lower velocity dispersion, as discussed in the text. Data from the SPLASH survey are shown as squares: open squares are from Kalirai et al. (2010); filled squares are from Tollerud et al. (2012). Triangles represent the data of Collins et al. (2010, 2011). In the case of And II, the early measurement of Côté et al. (1999) is shown as an ×, while the recent detection of rotation in this case is included in the measurement of Ho et al. (2012, star). The recent upper limit in the case of And XXII (Chapman et al. 2012) is shown as the downward pointing triangle.
Note that And IV is blank here because it turned out not to be a satellite of Andromeda as initially thought, but rather a background object. This was anticipated by MOND and later confirmed, so was in effect another successful prediction. (I do not remember the story behind the other blank cases offhand.)
This plot is from the follow-up paper, McGaugh & Milgrom 2013b. Note that these were almost all a priori predictions. (Some of the above were too, but observers were doing the follow-up work on newly discovered satellites so fast that these numbers appeared in the literature before we could finish writing the prediction paper. Not that it matters, as the predictions of MOND are not flexible beyond some range in the stellar mass-to-light ratio, so it predicts the same thing whether the answer is known or not.) And XXVIII was a golden case because it was far enough from Andromeda to be in the isolated rather than EFE regime, greatly reducing the uncertainty in the prediction (since it only depends on observations of the dwarf and not of the EFE from M31).
Figure 1 from McGaugh & Milgrom 2013b: Predicted and observed velocity dispersions of recently measured dSph satellites of M31. Each bin along the x-axis corresponds to the number of each dwarf for And XVII through And XXIX. The predictions from Paper I are shown without adjustment as circles for ϒ* = 1 (lower points), 2 (middle points), and 4 M☉/L☉ (upper points). Circles are filled if the dwarf is in the isolated regime for the assumed ϒ* and open if it is dominated by the external field of M31. The error bars on the predictions include both observational uncertainties in the photometry and a factor of two uncertainty in ϒ* around ϒ* = 2 M☉/L☉. Observed velocity dispersions are taken from Tollerud et al. (2012, small squares), Tollerud et al. (2013, large squares), and Collins et al. (2013, triangles or, in the case of upper limits, horizontal bars). The number of stars used to determine each velocity dispersion is shown next to the corresponding datum. Points appear in the chronological order of their appearance in the literature from left to right within each bin. Collins et al. (2013) find that And XXVII is not in equilibrium, in contradiction to a necessary assumption of the analysis.
Note the last line of the caption: we know there are cases that are out of equilibrium. There are certainly other systems in the universe that are out of equilibrium, like the tidal dwarfs for which we also found an apparent lack of dark matter. We haven’t been shouting that these falsify MOND (as we had hoped to be able to do! that’s why we did the experiment!) because we recognized that there was no way the equilibrium predictions could apply. Perhaps that is also the case for the ultradiffuse dwarfs discussed above.
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 Ωm = 1, and theorists chugged that Kool-Aid hard – Stacy
I’ve been trying to think more seriously about the entity that set off the LZ experiment. I was dismissive of the possibility of it being dark matter, perhaps unfairly so. On further reflection, I feel only fairly dismissive.
First, the formal significance that is claimed is 2.6σ. That seems to be a fair and fairly conservative statement. So it seems something did happen, the question is what. The odds of a 2.6σ event are about 1 in 200, so a fluke seems unlikely.
I can think of three generic possibilities:
It’s a fluke.
It’s dark matter (huzzah!)
It’s some strange new particle-entity that is not dark matter.
I remain of the opinion that the most likely possibility is that is is a fluke. I know I just said that seems unlikely, but “seems” is doing a lot of work here. I think it is fair to say that known explanations are excluded at this confidence, but that leaves a lot of unknowns. Most unknowns turn out to be boring.
We’ve been down this road many times. The LHC has been running for many years now. It detected the Higgs particle, which was the last missing piece of the Standard Model. That’s a great success. It was hoped that the LHC would also reveal new physics beyond the Standard Model. That it has not done. But there have been many hints during its operation. These appear as glitches (deviations from expectation); there have been multiple glitches that appeared to be significant at the 2.something σ level. Each time such a glitch was reported, it resulted in a flurry of theoretical activity. Within days, the arxiv preprint server would be inundated with hundreds (literally hundreds) of preprints staking out a claim as to what the latest glitch might be. Each time more data were acquired, the glitch went away. So this seems like another such occasion, making (1), a fluke, seem like the least improbable outcome.
This is why the [arbitrary] threshold for discovery is set at 5σ (1 in 3.5 million). Flukes at the 2.something σ level happen too easily, too often. This phenomenon is not restricted to particle physics. The same thing happens in astronomy all the time. All the gorram time. It’s almost as if nature places more probability in the tail of the distribution for “weird” events than the Gaussian statistics we use say there should be.
Given this experience, finding a new particle outside the bounds of the Standard Model of Particle Physics seems a lot less likely than 1 in 200. Seems is again doing a lot of work here. How likely 1 in 200 hundred seems depends on your priors. If you’ve been working on this for a long time in the hopes of a positive result, these odds seem pretty good:
“I have been waiting for a positive result for, oh, my god, 40 years. So you can bet I’m very, very excited by the LZ results.”
If, like me, you’ve already convinced yourself that dark matter cannot explain the kinematic data for galaxies, then it is just another opportunity for the eyes to roll up inside the skull. Obviously we can’t both be right, which is why we do these experiments.
So what about the other possibilities, (2) and (3)? If the event is real, it is tempting to believe that it is dark matter. That’s what the experiment was designed to detect, after all, so that’s a natural first assumption. But just because you’re looking for something doesn’t guarantee that when you find something you find what you’ve been looking for.
As I’ve reviewed before, we’re way past the regime of what we expected to find. So it’d be great to find something, but there’s little reason at this point to expect that something to be what we set out to find. So it might be dark matter (2) or it might be something completely different (3). I’ve no way to weigh the odds of those possibilities because they both seem very unlikely.
For WIMP-like dark matter we at least have some expectations, and this event does not conform to those expectations. It is too high energy. For every event like this one, the detector should have already recorded lots and lots of lower energy events.
That doesn’t mean it isn’t dark matter (2), just that it isn’t what we expected. But we already missed out on what we were expecting, so maybe? Or maybe it is something unrelated (3), or nothing at all (1). One event does not help override whatever assumption bias we might have had about these possibilities to start.
To be convincing…
It occurs to me that I am likely to hold the putative detection of dark matter to a higher standard than other scientists. So what would I find convincing?
First, the detection of a putative new particle has to cross the traditional 5σ threshold.
Second, the same result must be obtained by independent experiments.
Third, the new particle has to be demonstrated to have the right properties to be dark matter.
Fourth, the new particle has to exist with the required mass density to be the dark matter.
(Fifth, it should naturally explain the observed MONDian phenomenology.)
I put the fifth criterion in parentheses because it is a very high bar, and I can conceive of being persuaded to soften it. It is also more advanced tuition than most physicists have contemplated, so let’s start with the more basic requirements.
At present, none of the above criteria have been met by the new LZ result. There is an old claim to a significant detection by DAMA that satisfies the first criterion but was never independently reproduced, so fails the second criterion. Nobody outside the original experiment takes it seriously now for that reason. It is sobering to realize that while new and exciting, LZ has not yet progressed as far down this list as DAMA.
Supposing both the first and second requirements are met, then we have a new particle. That is certainly interesting, but by itself would not mean we’ve found dark matter. When we’re in the realm of imagining new particles, there is a vast landscape of possibilities. Only some features in that landscape could qualify as dark matter; there are other possibilities that could exist but have nothing whatever to do with dark matter. So the properties of the particle have to be right, and there is a long list to satisfy.
From a generic astronomical perspective, the fourth criterion is the most important. We have a pretty good idea of the mass density that dark matter needs to have in the solar neighborhood to explain Galactic kinematics. That’s what these experiments are trying to detect: the dark matter particles in our Galaxy’s dark matter halo that happen to be passing through just now. So for whatever we detect to qualify as the dark matter, there needs to be the right number of particles with the right mass to satisfy astronomical requirements.
This fourth criterion is a pretty high bar. It does not suffice merely to find a new particle, it’s gotta be the right kind of particle and these particles have to exist with the right density. That can be measured, but it requires lots of detections: enough to work out the properties of the particles and how many are whizzing by. This is part of the disconnect I had in the last post with people complaining about statistical significance: they were focused on the significance of this one event; I was worried about having enough detections to estimate the mass density, or at least see if it is anywhere near the right ballpark. I’m more concerned with the astrophysical relevance of a new particle than with the new particle itself.
This is a generic problem in the field of dark matter, being as it is at the interface of two different fields. Astronomers need dark matter; we don’t care so much what it is if it performs as required. Particle physicists heard we astronomers needed dark matter, so are more than happy to suggest possibilities, but are more interested in the particle itself than what it does for galaxies. I don’t say what it does for the universe because that’s a lower bar where any dynamically cold, non-baryonic mass will do. The cosmic density needs to be right, but there’s no way to measure the cosmic average, only the local density of the Galactic dark matter halo.
The highest bar is the fifth criterion, which I think warrants its own post.
There is a press release from the LZ collaboration claiming to have detected one WIMP.
LZ is an amazing experiment. They’ve amassed ten metric tons of super-pure liquid xenon to be the target nuclei for WIMP detection. That’s xenon nuclei waiting for a WIMP to scatter off them. The xenon in LZ is super-purified to exclude radioactive contaminants to the extent possible and shielded by multiple layers of material to exclude non-WIMPs from entering the active part of the detector. The biggest shield is the Earth itself, as the experiment is conducted over a kilometer deep in the Sanford mine. The technology is amazing. The results, not so much.
Here is the apparent detection from Fig. 4 of their just-released preprint+.
Fig. 4 from Akerib et al. (2026). The apparent detection is the lone black point near the upper right corner of the dashed box. The one near the lower left corner is consistent with an accidental background, but the apparent WIMP is not.
LZ is an incredibly sensitive detector, so a huge part of the program is to reject uninteresting events. Many can be shielded, but there’s always something that gets through, including radioactive isotopes of xenon* in the detector itself. Hence there are many events that fall into an uninteresting part of parameters space (those that track the blue lines): these are expected backgrounds. To give a flavor for all the many things that still show up, here is their Table 1 listing them all:
Table 1 from Akerib et al. (2026). The expected and fitted numbers of events, by source.
All the fitted events are nicely consisted with those expected. Then, there at the bottom, is the one () event consistent with being a WIMP.
Do I find this convincing?
Note added: this part is wrong. See addendum.
I mean, come on. One event? I don’t know what statistics they’re smoking to come up with their error bars, but simple counting statistics makes it the square root of the number counted. You can’t beat that. So they have one () detection. A result. Hardly the that is the usual standard to claim a detection. That’d take 25 WIMPs, even if we believe all the background rejections.
We’ve been down this road before. Back in 2008-2009, there were months of speculation surrounding the planned unblinding of the CDMS experiment. People were sure we’d finally detect WIMPs. Excitement mounted until the day of the announcement. And lo and behold! On that occasion, there were two (2) WIMPs. WIMPs with properties that have not been reproduced, so that result went away, as low significance results usually do. I expect this “detection” will follow the same trajectory. That’s what experience teaches.
It is customary to acknowledge grants that support one’s research, so I recount here part of what they list as going into this:
Funding for this work is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Contract Numbers DE-AC02-05CH11231, DE-AC02-76SF00515, DE-AC52-07NA27344, DE-SC0008475, DE-SC0010004, DE-SC0010010, DE-SC0010072, DE-SC0011702, DE-SC0012447, DE-SC0012704, DE-SC0014223, DE-SC0015535, DE-SC0015910, DE-SC0018982, DE-SC0019066, DE-SC0019193, DE-SC0020216, DE-SC0025446, DE-SC0025629, DE-SC0026014, DE-SC0026544. This research was also supported by U.S. National Science Foundation (NSF) ; the UKRI’s Science UKRI2842, UKRI2843, UKRI2844, UKRI2847, UKRI2851, UKRI2852; the Portuguese Foundation for Science and Technology (FCT) under award number PTDC/FIS-PAR/2831/2020; the Institute for Basic Science, Korea (budget number IBS-R016-D1); the Swiss National Science Foundation (SNSF) under award number 10001549; the European Union’s research and innovation programme Horizon Europe under the Marie Sklodowska-Curie grant agreement No 101209429. This research was supported by the Australian Government through the Australian Research Council Centre of Excellence for Dark Matter Particle Physics under award number CE200100008.
That’s a lot of grants.
The list continues for a while beyond this, but the acknowledgements trend towards more general entities beyond just LZ, like support for Fermilab. You get the idea: this is a large, well-funded, international collaboration. That’s great; WIMPs were an obvious thing to look for. But these experiments have run their course; they’ve succeeded in failing. WIMPS are not there. At what point do we admit that and stop throwing$ good money after bad?
ADDENDUM
A lot of folks are giving me grief in the comments for my cavalier treatment of the statistics for this experiment. And you know what? Fair. I was not taking it seriously.
They’ve done a very careful analysis, and I think it is correct to say that the anomalous event is highly significant, at least in that nothing should be where it is. I’m convinced that the odds of this happening as a result of an unfiltered background are tiny. The huge bulk of xenon is excellent at shielding the central, active part of the detector; nothing like this should get in. So there’s something weird there, and it is wrong of me to simply quote Poisson statistics+.
Does it have anything to to do with dark matter? Almost certainly not, but that is the obvious leap to make; they even call the one anomalous event a WIMP in the table reproduced above. That leap of faith is what I’m reacting to, not the merits of the experiment itself. Why is there a press release? The reasonable claims made in the science paper are rather limited. They do not rise to the level of a warranting press release. So why are we having this conversation? Because it wasn’t adequate to simply write a paper reporting this to the scientific community; the implication is made to the public that maybe we’re on the brink of detecting dark matter, with the further implication that with just a little more time and a lot more funding we’ll finally get there. That’s what earns the harsh reaction.
I’ve seen this movie over and over and over and I know how it ends. Someone pedaling scientific hopium at me was giving examples of all the times we found something after many failures with just the last bit of improvement in sensitivity, with the clear implication that it would happen again this time. It will not. This situation is not analogous because the existence of WIMP dark matter has already been practically falsified. I’ve patiently been trying to explain that for thirty years, and yet the people who are most offended by my skepticism of this particular narrow claim either evince no awareness that this is the case or are aggressively derisive themselves. WIMPs are a dead end, and I guess I’m finally running out of the patience required to be polite about it.
+Just-released probably means unrefereed, but I see no reason to doubt what they say – just what it means.
*Xenon is one of my favorite nuclei, and not just because it was the subject of my first science paper. It has seven stable isotopes, so its sort of a noble nucleus as well as a noble gas. 124Xe and 136Xe are not stable but practically so, with half-lives in excess of 1021 years. That’s 1011 Hubble times. And yet, LZ is so sensitive it saw over a hundred decays by these isotopes during its active run. Amazing. If only that xenon were used for good instead of nothing.
(My early work on Na-Xe coupling, which I did not enjoy, was one tiny step among many on the path that made its use in medical imaging possible. LZ by itself uses about a quarter of global annual Xe production, and it’s not the only such experiment. I’ve heard discussions of plans to scale up dark matter detection experiments to use practically all of the xenon in the world.)
$We’re close to hitting the neutrino background, so if funding for WIMP searches sours, look for these experiments to seek to continue operating by rebranding themselves as neutrino observatories.
+I confess to agreeing with Rutherford: “If your experiment needs statistics, you ought to have done a better experiment.” It is easy to conceive of a successful experiment in which there are so many detections that the issue is moot.
The effective force law in galaxies is MOND. At high accelerations, this is the same as Newton’s inverse square law: . At low accelerations, the deep MOND limit is . The transition between the regimes occurs at .
The effective force law in galaxies (data) looks like MOND (line).
This is well established in galaxies. The question naturally arises whether this holds on larger scales. A recent paper by Gallardo et al. says no:
On scales from 30 – 230 megaparsecs, we constrain the gravitational acceleration between pairs of halos$ 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: . The boost factor 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, , 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 eN,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 .
$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 and in these units so corresponds to an EFE that is 3% of . 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.
It is straightforward to falsify theories that make clear predictions, should they happen to be wrong. It is impossible to falsify ideas that involve invisible components.
Specific theories of dark matter (e.g., WIMPs) can be seen to be increasingly unlikely, but can they be falsified outright? If we decide that WIMPs have been practically falsified – to all intents and purposes – that doesn’t mean dark matter is wrong, just that we’ve spent the past four decades building dozens of giant experiments costing billions of dollars and frustrating thousands of careers barking up the wrong tree. The unseen forest that is the ‘dark sector’ is vast; there are limitless opportunities to bark up other wrong trees.
How could we know if the entire dark sector is a non-entity? On many occasions, I’ve had colleagues say to me that they would “only consider MOND as a last resort.” OK, so when do we know we’ve reached that point? After an eternity of searching for unseen, undetected non-entities seems a little late.
This is the bind we’re in. Most of us (scientists working in the field) are unwilling to consider something as radical-seeming as MOND until dark matter has been falsified. But dark matter cannot be falsified. So we patch up our hypotheses accordingly, rinse and repeat with every new crisis#. This has been going on for nearly my entire career, to the point that I now see junior scientists who seem to think this is how science is supposed to work. Why wouldn’t they? They’ve known nothing else.
I empathize. I started from the same place: there has to be dark matter, it has to be non-baryonic, it is almost certainly a new particle and that new particle is almost certainly a WIMP. It was the hardest thing to realize and accept that we could be wrong, wrong, wrong, and wrong on all counts. I worked incredibly hard to avoid that conclusion. I take solace only in the fact that at every epoch in human history we’ve had a cosmology that we were absolutely sure$ was right but that later turned out not to be. Maybe we’re a blessed generation who finally got it right. Or maybe we’re just the latest in a long line of brilliant savants who fooled themselves into thinking we understand more than we actually do.
I remain unwilling to say that dark matter has been falsified because I don’t think it is falsifiable. That, in itself, is usually considered to be a bad thing for a scientific theory. Perhaps at some future point it will be portrayed that way for dark matter, but in the present I’ve heard plenty of scientists pretend like it is somehow a good thing. Certainly it makes a fertile playground for theorists, so I guess good/bad is a matter of perspective. I suppose an infinite forest of trees in a conveniently unobservable dark sector is an irresistible opportunity for a dog! with an insatiable bladder.
I do, however, think that dark matter has been practically falsified. If I’m wrong about that, it is possible to demonstrate. To be clear, I don’t just mean WIMPs. I mean dark matter as an explanation for the mass discrepancies observed in extragalactic systems. No kind of dark matter* can suffice. I came to this conclusion before I really knew anything about MOND, which is why I was receptive to it. Others haven’t had that experience, so aren’t.
I don’t expect other scientists to accept that an entire paradigm is wrong because I say so. I do, however, expect them to acknowledge that the dark matter hypothesis should be falsifiable. It is incumbent on each scientist to establish for themselves criteria by which that conclusion could be reached, should it be appropriate.
For me, the problem is the contradiction present in the dynamical data for galaxies. We simultaneously require galaxy disks to be maximal and also for them not to be maximal. The only way to avoid this contradiction is to engage in fine-tuning: one must build a model in which everything works out just so. Since dark matter is not otherwise falsifiable, a requirement for fine-tuning& is pretty much the worst thing we can say about it.
The contradiction as framed by others
The contradiction that concerns me has certainly been noticed by others. Usually, they choose to come down on one side or the other. The problem is that they’re both right.
On the one hand, it is clear observationally that the luminous mass matters to the dynamics of galaxies. For example, Swaters et al. note that
the luminous mass dominates the gravitational potential in the central regions, even in low surface brightness dwarf galaxies
which is to say, disk galaxies are maximal. We’ll explore what that means below, but you can see the effect by eye:
Rotation curves color coded by galaxy surface brightness. There is an almost perfect rainbow variation that is plain to see. The gravitational potential traced by the rotation curve correlates well with the distribution of stellar mass. (Adapted from McGaugh 2020.)
On the other hand, there are no residuals from the Tully-Fisher relation.
Residuals from the Tully-Fisher relation as a function of size at a given mass. Blue points are star-dominated galaxies; cyan points are gas-dominated. Compact galaxies are to the left, diffuse ones to the right. The red dashed line [δlog(V)/δlog(R) = -1/2] is what should happen with maximal disks: more compact galaxies should rotate faster at a given mass. This is what Newton predicts, but galaxies didn’t get the memo. (Adapted from McGaugh 2005.)
The inference from this observation is that galaxy disks cannot be maximal. As Courteau & Rix put it,
The case of δlog(V)/δlog(R) = -0.5 expected% for a maximal disk is ruled out
which is to say, disk galaxies are not maximal – even the high surface brightness (HSB) disks that dominated their sample.
They say this because maximal disks – or even those that contribute noticeably to the mass budget at small radii – predict deviations from the Tully-Fisher relation. They did this by intentionally focusing on the point where the stars should contribute the most. I’ve shown similar things many times, so here let’s let LCDM advocate Frank van den Bosch demonstrate it:
Model galaxies (left) of the same mass but with different spin parameters (van den Bosch 2001). In this model, spin determines size and surface brightness: lower spin galaxies are more compact and have higher surface brightness. This causes them to evince higher velocities that should cause residuals from Tully-Fisher, but these are not observed (right, McGaugh 2021).
So galaxies cannot be maximal. Only they have to be to have the observed correlation between the luminous mass distribution and the kinematics, as seen in the first figure above. Well, which is it? Are all galaxies maximal? Or none? Or can they both^ be right?
Maximal disks
Maximal disk is a technical term that is well known to people who work on the subject but not outside that relatively narrow&& field. So what do we mean by this term?
The rotation curve of NGC 6946. The solid blue line shows the expected rotation curve for the visible mass including both stars (dashed line: stellar disk; dot-dashed line: bulge) and atomic gas (dotted line). The stars suffice to describe the inner rotation curve, illustrating the case of maximum disk. The red points are the rotation curve measured optically (Daigle et al. 2006; Epinat et al. 2008); the orange points that measured with radio interferometric observations of the gas (Boomsma et al. 2008; I’ve suppressed the overlap for clarity). The thin line is the prediction of MOND; the black line is the implied dark matter distribution.
In essence, a maximal disk is one in which the stars provide practically all the mass at small radii. The depiction of NGC 6946 above illustrates maximum disk. Sure, the rotation curve flattens out at large radii and we need to invoke dark matter. But the observed stars explain the amplitude and shape of the inner rotation curve quite well. In this case there is a compact bulge at the center of the galaxy that causes a sharp rise in the rotation curve right from R = 0. (This is an example of Renzo’s Rule.) The disk (plus bulge) is maximal in the sense that we cannot attribute any more mass to them without exceeding the observed rotation curve.
The amplitude of the portion of the rotation curve due to the stars depends on their mass-to-light ratio. While this cannot exceed maximum disk, it could be lower. So conceivably, the good match to the shape of the observed rotation curve is a chimera, and really this galaxy is dark matter dominated. That is a possibility many seem to have embraced, but while it might work for the disk, it does not work for the bulge. As we suppress the contribution of the stars as Courteau & Rix argue we must, then the rotation curve looks more and more like that of the dark matter halo alone. That goes up and flattens out (and ultimately must turn over again somewhere beyond the edge of the data) but it has no features.
The rotation curve of NGC 6946 as above but with the stellar mass reduced by a factor of two. This disk is submaximal, contributing less than the implied dark matter.
One thing the rotation curve due to the dark matter halo cannot do is go up then down then up again. Yet that is exactly what it needs to do to explain the inner peak inside 1 kpc if the bulge component is not maximal. I suppose we could have a smaller dark matter halo inside the main dark matter halo that does this, but its mass distribution would have to be practically identical to that of the bulge. That’s insane. Why would we invoke a second dark matter halo when the stars are right there?
In this case, the stars have the right mass for what we expect from stellar populations. There was a long debate historically about whether the optical band mass-to-light ratios for maximum disk were consistent with those expected from stellar population synthesis models. For a long time they looked close but a bit high. This difference has pretty much gone away now that we have access to near-infrared data: the two are consistent, and having a stellar mass-to-light ratio much below the maximum disk value for high surface brightness disks becomes problematic from a population perspective.
Indeed, the 3.6 micron M*/L = 0.37 M☉/L☉ in the case depicted for NGC 6946 with a maximum disk. That’s reasonable but on the low side for what is plausible for the stars in a mature spiral galaxy like this. Halving that strains credulity, so there is no room for a second inner halo, or even for the cusp predicted** for the primary cold dark matter halo. Stellar mass really does seem to dominate in the inner parts, just as Swaters et al. said.
LSB galaxies
The issue that confounded me was whether the low surface brightness (LSB) galaxies I was working on were maximal or not. My inital expectation was that LSB galaxies would be stretched out versions of HSB galaxies. I expected them to shift off of the Tully-Fisher relation and follow the line δlog(V)/δlog(R) = -0.5. They did not do that. If I didn’t have them be maximal, I found that I could explain pretty much any slope other than the one observed (δlog(V)/δlog(R) = 0). That required fine-tuning to perfectly balance the lesser contribution of stars in LSB galaxies which we had to back fill with dark matter just so. I spent ages running around in circles trying to make that work. Every time I thought I had succeeded, I realized I had assumed something that made it so: tautologies abound.
If we want to explain the shapes of rotation curves as seen up top, we need the stars to contribute to the gravitational potential. For that to work for LSB galaxies, we have to turn maximum disk up to eleven:
The low surface brightness, gas rich dwarf DDO 154. The lines have the same meaning as above, with the case of maximum disk in the left panel and that natural for stellar populations at right. The difference in disk mass is a factor of ten.
A crazy-high stellar mass-to-light ratio is what happens if we just ignore what we know about stars and just focus on the kinematics. But we do know a lot about stars. Population models indicate stellar masses that are very submaximal. Even boosting the mass-to-light ratio doesn’t get us very far. LSB galaxies aren’t really maximal in the same sense as HSB galaxies, and there is even less room for the expected cuspy halos that are already problematic when the stellar contribution is small.
Fine tuning is unavoidable
Even if we ignore what we know about stars, we still have a fine-tuning problem. The lack of a shift in the Tully-Fisher relation with either surface brightness or radial size implies that disks are all the same mass surface density. So we observe a wide range of surface brightness, but the surface mass density is always the same. That makes no sense, and is just another example of squeezing the toothpaste tube: we can make a model look OK from one perspective as long as we don’t look from another.
Worse, we still need to explain the role of the luminous mass in LSB galaxies. These are dark matter dominated at almost all radii, and yet the distribution of the observed stars and gas is predictive of the kinematics. This is a contradiction to Newtonian dynamics. The only theory that does this right – and predicted it a priori – is MOND. But that’s too horrible to contemplate, so we shield our eyes and ignore%% one or the other set of inconvenient facts. As a result, the field has become moribund, and will remain so until we free ourselves of our invisible demons.
#We’ve experienced so many crises that we seem no longer able to recognize new ones. JWST observations of high redshift galaxies follows a well-worn trajectory: an observation that contradicts the standard model is made, much huffing and puffing ensues, the theorists get to work constructing implausible models, these are accepted as patching up the hypothesis (whether satisfactory or not), and the field moves on as if nothing happened.
$To give one historical example, prior to Hubble’s discoveries in the 1920s, it was thought that the Milky Way was the entire universe. Certainly there were no other galaxies comparable to the Milky Way:
“No competent thinker, with the whole of the available evidence before him, can now, it is safe to say, maintain any single nebula to be a star system of coordinate rank with the Milky Way. A practical certainty has been attained that the entire contents, stellar and nebular, of the sphere belong to one mighty aggregation.” [i.e., the Milky Way]
-Agnes Mary Clerke in The System of the Stars (1890)
!It used to be that one would not claim a detection of dark matter until all astrophysical alternatives had been exhausted. Now it seems to be the fad to claim a detection first on the off-chance it works out later. I already peed on that tree! It’s mine!
*Excepting some sort of hybrid “dark matter” that is invented to do what ordinary dark matter cannot. By ordinary I mean CDM, WDM, SIDM, and every other variation on particle physics that simply invents new mass with no consideration of how the observed galaxy dynamics comes about. That would include primordial black holes and various macroscopic DM ideas (e.g., MACHOS, strange nuggets). Coming up with half-baked ideas for new particle dark matter is big business these days, but any idea not informed by observed astrophysics (which are most of them) is doomed to fail.
Examples of hybrid dark matter that are informed by observed astrophysics include dipolar dark matter and superfluid dark matter. Regardless of whether these specific cases are viable, the point is that the observed dynamics are a fundamental aspect if nature and require a commiserate explanation. Simply throwing in some extra mass with some fine-tuned feedback models can never provide a satisfactory explanation. Note that coming up with extra mass is mostly done by particle phenomenologists while feedback models are built by numerical astrophysicists. There is very little overlap between these communities; they pretty much just take it on faith that since dark matter has to exist, the part they don’t know about will magically work out.
&The classic example of fine-tuning in the sense that I mean is the Ptolemaic model of epicycles and deferents. If one adds enough of these and tunes them just so, anything can be fit. Note that epicycles are not explicitly falsifiable for this reason; we rejected them because they got ridiculously complicated and there turns out to be a more parsimonious explanation. The same thing holds now for dark matter and MOND.
%This slope is expected because Newton teaches us that V2 = GM/R. δlog(V)/δlog(R) = -0.5 follows from taking the logarithm of this at fixed mass. Galaxies are observed to span a large range of radius at a given mass, but not a corresponding range in circular velocity.
^Yes, they can both be right, but not with dark matter. Only MOND naturally explains both observations simultaneously.
Also, for the hyper vigilant, Courteu and Rix (1999) use a slightly different definition of velocity than I do in this Tully-Fisher residual plot. I went through all that in McGaugh & de Blok (1998) and in McGaugh (2005) and it makes no difference to the discussion here.
&&A vote we held at a conference on disk dynamics in Rome in 2000. The question of whether disks were maximal was posed; most people voted no based on the statistical lack of residuals from Tully-Fisher. After the vote, one of the dissenters noted that those who voted in favor of maximal disks were the people who actually worked in the subject. Those of us with other concerns were persuaded by the statistical evidence because we didn’t engage with the details of real, individual galaxies in the same way.
%%A common excuse I here for this behavior is that galaxies are “small” and nonlinear – complicated entities that we can never hope to understand, so whatever they do can be ignored as irrelevant. As a scientific argument, that’s pathetic. Galaxies should be complicated in LCDM, but in observational reality they’re kinematics are sufficiently simple that they obey a single effective force law. That’s one thing they should not do, just as a complicated set of epicycles and deferents shouldn’t always add up to the inverse square law.
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.
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:
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 σinst = 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 σv=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.
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 Vf 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, mb = Mb/M200, to be equal to the cosmic baryon fraction (fb = 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, mb = fb 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 fb = 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 (Mb > 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 Vf = fvV200. 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 Vf from the halo V200. This can’t work, but it is useful to think about and provides a direct comparison with MOND.
In MOND, Mb = AVf4 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, ΔM = Vf,obs/Vf,pred. If the mass is a good predictor of the flat velocity, then ΔM = 1. This leads to
Figure 8 fromMcGaugh 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 < Mb < 1014 M☉, so we’ll just ignore that here.)
LCDM is spot on for clusters over the range 1014 < Mb < 1015 M☉: fv = 1 suffices to explain the data. Outside of that range, fv 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 < fv < 1.4, but we need it to grow to fv = 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 < Mb < 5 x 1012 M☉: the data are consistent with ΔM = 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 (Vf3 in the top panel, Vf4 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 Vf. (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 Mb ~ Vf4 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 Mb ~ Vf4, 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 lensingsignal 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 < Mb < 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.
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 fv = Vf/V200 from fv= 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 fv, 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 < 1010.5 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.
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, fb = Mb/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
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 Vf has always been taken as the signature of the dark matter halo. One therefore expects V200 ~ Vf. 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 Vf 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, fv, such that:
If a rotation curve stays flat indefinitely (as our empirical experience suggests), fv = 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 fv > 1. How much greater?
One way to estimate the fudge factor fv 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 Vf 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 Vf = V200. The gray line in the right panel shows Equation (2a) of Katz et al. (2019), which corresponds roughly to fv ≈ 1.4.
The result for pseudo-isothermal halos is consistent with fv = 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 fv = 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 Vf. 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 Vf 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 Vf with a fixed fudge factor. So we’ll do that, leaving the exact value of fv as an open question, but noting that for most objects it almost certainly resides in the narrow range
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 Vf 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 Vf 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 Mb correlates with the flat rotation velocity Vf. 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 Mb ~ M200 and Vf ~ 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, mb, that relates the mass we see to the halo that spawned each galaxy:
The obvious assumption is that mb is a constant for all galaxies, in which case Tully-Fisher follows because Mb ~ M200 ~ V2003 and V200 ~ Vf. The wee problem is that this predicts a Tully-Fisher relation with slope 3: Mb ~ Vf3 when we observe one with slope 4: Mb ~ Vf4. 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: mb ~ Vf. 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 (mb ~ Vf ~ Mb1/4):
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 fb = 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 mb = fbtanh(Mb/M0)1/4 where fb is the cosmic baryon fraction (0.157) and M0 = 5 x 1013 M☉.
To be specific, I’ve computed the halo mass assuming fv = 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:
where fb = 0.157 is the cosmic baryon fraction and and M0 = 5 x 1013 M☉ is the scale where the function bends, transitioning from the Mb ~ Vf4 of the BTFR that holds over most of the mass range to the mb = fb 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 fv. 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 fv = 1.0 (solid blue circles) and again with fv = 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 fv = 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 fv ≈ 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 fv ≈ 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 (Mb = 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 simplescaling 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, MΔ, then the inferred baryon fraction of an objects scales by M200/MΔ. 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 rΔ 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.
