In the last few posts we’ve discussed the local missing baryon problem in extragalactic objects spanning over ten orders of magnitude in mass from tiny dwarfs to rich clusters of galaxies. This discussion has so far been entirely in the context of LCDM. So – how does LCDM compare with MOND?
As a refresher, these are the data we’re trying to understand:
The 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:
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
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:
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 is composed of two clusters that collided and passed through one another. The collision segregated the gas of the ICM (pink above) from the galaxies. This happens because gas is diffuse and collisional. The gas of the two clusters can’t help smacking into each other, slowing down and forming the shock front visible in the shape of the gas of the smaller cluster on the right. Galaxies, on the other hand, have lots of empty space between them. They are collisionless and pass right by each other. In doing so, they are slowed less than the gas, getting ahead of it, leading to the separation that we observe.
OK, cool. The argument one usually hears against MOND based on this is that the baryonic mass in gas outweighs that in galaxies, so the lensing signal should be centered on the gas: the blue should align with the pink, not with the galaxies. Instead, we see the opposite, so the mass has to be dark matter.
This would be a good argument if the gas were all of the baryonic mass. This is a common assumption that makes sense in LCDM, where the baryon fraction checks out, so most people seem to stop thinking at that point. But each theory needs to be considered in its own context, and it cannot be the case in pure# MOND that we see all the baryons## in the picture above. That’s what we learned above. It may be unsatisfactory, but we knew this already before the bullet cluster was discovered (e.g., Sanders & McGaugh 2002). So the only new thing we learn from this aspect of the bullet cluster is that if there is an additional reservoir of baryonic mass, it is collisionless. It didn’t collide like the gas, it passed through like the galaxies. There are lots of candidate baryonic objects that fit that requirement: brown dwarfs, neutron stars, black holes, very small rocks^. There is no requirement that the unseen mass be non-baryonic; we do not need the new physics of a new dark matter particle from beyond the Standard Model of particle physics on top of the new physics of MOND.
Now, as I think I’ve made clear, I am very uncomfortable with the apparent requirement that there is lots of undetected baryonic mass in clusters. If I were the MOND partisan that lots of people seem to assume I am, then I guess I’d portray this as a bold prediction. The dark baryons have to be there, and we should be turning all possible resources to detecting them, rather like we have for WIMPs. But I’m not that person. I am also not a person who sees this missing baryon problem for MOND as automatically worse than the missing baryon problem for LCDM. There is a much bigger deficit to be made up in LCDM, in many more systems### of very different types over a larger dynamic range in mass. The missing baryon problem in LCDM looks worse to me than that in MOND. Yet the community attitude seems to be largely unaware of it. Those who are seem mostly to presume that it’ll work out. Maybe, but this should not be accepted by assumption, it needs to be demonstrated. It has yet to be.
If you think the missing baryon problem in clusters is a terrible problem for MOND, then you should be similarly worried that LCDM evinces the same kind of problem – one that is objectively larger in amplitude. It seems that, having accepted that there is dark matter, people don’t much care what it is. I do. The dark matter paradigm has obliged us to abandon parsimony. Not only does LCDM need two novel substances, dark matter and dark energy, it requires two kinds of dark matter: baryonic dark matter and non-baryonic dark matter.
There is a communal failure of objectivity about this. The thought process is both transparent and simple: MOND doesn’t explain clusters; it requires dark matter. Therefore dark matter#### exists and it is silly to think about MOND. That would make sense if it weren’t a logical fallacy. Instead, it provides a permission structure to remain ignorant of what MOND gets right. I get that; there’s a lot to know. But I would also suggest that ignorance does not provide a strong basis for drawing scientific conclusions, especially for a subject so rife with confirmation bias and cognitive dissonance.
&The normalization is related to Newton’s constant and Milgrom’s constant through A = ζ/(a0G) where ζ is a factor of order unity that depends on the geometry of the system. It is one for spheres, and always approaches the limit ζ → 1 at sufficiently large radii, but observations are usually obtained at radii where the flattened geometry of disk galaxies is relevant, so in practice ζ ≈ 0.8. This can be derived from the geometry (all purely conventional; nothing to do with MOND) or one can obtain it empirically by comparing A = 50 M☉ km-4 s4 from fitting the BTFR to data for galaxies with known a0; for a0 = 1.2 x 10-10 m s-2, (a0G)-1 = 63 M☉ km-4 s4, so ζ =A(a0G) = 50/63 = 0.8.
*There remains room for improvement for poor clusters (here I call 1013 < 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.
“In effect, LCDM requires two kinds of dark matter: dark baryons mixed in with each and every dark matter halo, and some entirely novel form of particle to be the dark matter halo” – can you elaborate on this? It’s not very clear where it comes from.
The cosmic dark matter has to be non-baryonic since the gravitating mass density is larger than the baryon density allowed by big bang nucleosynthesis. Some new particle like WIMPs. That’s what people usually mean when they say dark matter. Here, we’ve accounted for that, and find that we still need unseen mass – mass than needs to be normal baryons for the check sum of the cosmic baryon fraction to be complete. So we need two kinds of dark matter: WIMPs (or whatever) and dark baryons.
Ah so that and the last paragraph relates to the entire post as a conclusion, I see. That makes sense, thanks. It’s a good point!
If you put on your LCDM hat, would you assume that the missing baryons in the low mass systems are scattered out into dark matter voids or along filaments?
Now if cluster collisions separate dark matter from baryons (as claimed in the Bullet Cluster observation), then they must be particularly rare in order for the cosmic baryon fraction to be best preserved in clusters. Or is it just that only a very rare high velocity collision can separate the two?
Conversely, why wouldn’t the less rare slower clusters pull in more “missing” baryons that were ejected from the lower mass systems along the filaments?
Curious how that is all balanced out in a way that matches both the distribution of the observations and also the LCDM model expectations.
If I put on my LCDM hat, I can think of many answers. The baryons could be hung up in the halo: there but not readily detectable or the could be expelled entirely. In the latter case I don’t expect they should follow the filaments any more or less than their source galaxies do, but I have no good intuition for that: who knows? Some simulations spew them all over the place, so conceivably they could all be way out in the void, but it certainly doesn’t have to work that way. In principle, I would expect a combination of effects – some (but not huge amounts of) baryons expelled, some still lurking about. There is probably some combination of coming and going, and we see examples of both, but not on the needed scales (trickles not super-winds). Either way, it makes the fine tuning worse, as there are multiple components to manage.
Interesting point about the frequency of bullet clusters. Implicit, as always, is that we’re talking about equilibrium systems, which the bullet manifestly is not. One imagines such things eventually settle down to normalcy, and catching them in this phase is rare. Whether it is as rare as all that is less clear – clusters are pretty messy places.
The bullet itself is moving too fast for LCDM, but that’s another story.
The pattern here looks less like a choice between “missing matter” and “modified gravity” and more like evidence that the gravitational field we infer is a constitutive response of a real spacetime medium: a Noether sea whose effective metric changes with density, stress, and delay structure. The key fact in the data is that the observed condensed baryons, $M_b=M_\star+M_g$, track the flat velocity with very little scatter, even when the stellar and gas fractions differ. In the low-acceleration galaxy regime, those baryons load the medium and select the response, so the extra acceleration stays locked to $M_b$ and the relation wants the $V_f^4$ law. In rich clusters and mergers, the medium is in a different state: hot gas is collisional, galaxies and any cold neutral reservoir pass through, and the lensing map follows the collisionless/medium-response record rather than the X-ray gas alone. So the real test is not $\Lambda$CDM vs MOND as labels. The test is whether one shared medium state can recover BTFR, the $M_b/M_{200}$ trend, cluster gas/lensing offsets, CMB/BBN baryon accounting, and the poor-cluster transition without changing assumptions per regime.
The double dark matter observation deserves more attention than it usually gets. ΛCDM doesn’t just require non-baryonic dark matter to form halos, it separately requires dark baryons to fill the galaxy-scale and group-scale deficit. These are different entities solving different problems at different regimes. When a framework needs structurally distinct invisible additions at each new scale it enters, that’s not one prediction being extended, that’s the auxiliary hypothesis count growing with the problem count.
But the deeper issue is the implicit assumption shared by both sides of this debate: that a framework validated in one complexity regime carries its validity when extended into a higher one. GR, validated in the solar system, gets extended to galactic scales, and requires dark matter to patch the failure. MOND, validated in spiral galaxies, gets extended to clusters, and requires missing baryons to patch the failure.
These are not two different problems. They are the same methodological error committed in two different directions. Each framework is meeting its complexity ceiling, and each community responds by invoking an invisible supplement rather than examining the extension itself.
What you document here is that the tolerance for this response is asymmetric. MOND’s cluster residual, roughly 60% missing mass in one regime, is treated as disqualifying. ΛCDM’s order-of-magnitude deficit across seven decades of baryonic mass is treated as a budget problem to be resolved later. The asymmetry isn’t methodological. It’s sociological. And the Bullet Cluster sharpens this: the “smoking gun” framing only holds if you’ve already assumed the gas accounts for all baryons. Grant MOND’s known cluster shortfall and the Bullet Cluster tells you the missing component is collisionless, consistent with entirely mundane objects. That’s a much narrower inference than the standard narrative implies.
It has been twenty years since the Bullet Cluster was hailed as the ‘smoking bullet that killed MOND’. At that time, I found myself raising my hand and saying, “Evidence for the need for dark Matter and/or MOND arises in systems that are in dynamic equilibrium. The Bullet Cluster(s) are a mess. This is like looking at the aftermath of a city that has been leveled by a nuclear weapon and arguing whether it was the ants or the roaches that leveled the city.”
One problem common to all our gravitational models, those of Newton, Einstein and Milgrom, is that they do not scale. They work on the scales they work on and not so well elsewhere. They also share another common flaw, they describe the gravitational effects but they do not describe a causal mechanism for the gravitational effects they describe.
In science generally and physics in particular, it is a matter of of principle that observed physical effects have physical causes. Modern Theoretical Physics has spent the past century studiously ignoring the question of a causal gravitational mechanism; this despite the vast increase in our observational capabilities across all scales.
The fundamental theoretical error is that the correlation between mass and gravitational effects which varies by scale is somehow fundamentally a causal relationship rather than merely a variable correlation. In other words the qualitative analysis is wrong. There is a correlation between gravity and mass but it is not a causal relation.
An argument can be made that on all scales the observed gravitational effects track not the mass distribution but the electromagnetic radiation density/density-gradient that is associated with the mass distribution. The relationship between the mass distribution and radiation density varies in different mass configurations. That leads to the mistaken attribution of a seemingly excessive observed gravitational effect to “missing mass” when in fact the effect is tracking a radiation density distribution which is not factored into the gravitational models.
In the Solar System the Sun contains 98% of the mass and its emitted radiation field drops off as 1/r^2 – the same as the gravitational effect. Light from a distant source passing near the Sun behaves as if it were traversing a medium with a density gradient. The emitted radiation field of the Sun can be conceived of as a medium with a density gradient.
On the scale of a typical disk galaxy the overall matter distribution is planar, not spherical and the angular velocity remains almost constant for stars in the disk out to the edge of stellar disk field. Throughout the stellar portion of the disk the radiation density remains relatively constant because of the relatively constant stellar radiation in the plane of the disk. Beyond the extent of the stellar population the radiation density in the plane of the disk falls off at 1/r as does the gravitational effect – as should be expected from the planar geometry of the system.
In large galactic clusters 90% of the mass is in the form a diffuse, x-ray hot plasma. That plasma is radiating from its entire volume rather from its equivalent surface area if it were a compact body like a star. Our mass only gravitational models do not take account of that excess radiation and so determine that there must be “missing mass”.
At root, the “missing mass” problem represents an analytical error regarding the causal mechanism that produces the observed gravitational effects. The quantitative evidence is that gravitational effects do not consistently track the matter distribution of a gravitationally bound system. The qualitative evidence is that gravitational effects track the electromagnetic radiation distribution of a gravitationally bound system.
The geometrical effect of a planar mass distribution leading to a 1/r force has been considered and does not suffice. We know how to account for geometry and do so. Moreover, it is not true that the stellar radiation in the plane of the disk is relatively constant. It falls off as an exponential in both the radial direction and that perpendicular to the plane (the e-folding lengths are different is all). So I don’t think it can be a simple matter of tracking the E&M radiation.
“…it is not true that the stellar radiation in the plane of the disk is relatively constant. It falls off as an exponential in both the radial direction and that perpendicular to the plane…”
That can’t be correct. In the plane of the disk the radiation density has to track the luminous mass distribution and the luminous mass distribution does not fall off exponentially. In the plane of the disk – which is defined by the mass distribution – the emitted radiation that lies in the plane of the disk will be cumulative with respect to the radiation density within the disk.
The radiation density of the disk is a cumulative effect of all the radiation emitted in alignment with the plane by all the individual stars in the disk plus any unblocked radiation flowing radially outward from the central core. It is the radiation density within the disk, not the gravitational effect calculated for the mass distribution, which should fall off as 1/r in the plane of the disk – beyond the luminous mass distribution. That is, again, related to the geometric structure of the disk and the geometric alignment of the radiation density within the disk.
Only in the plane of the disk is the radiation from the individual stars of the disk cumulative (because that is how the luminous mass is distributed – in a radial plane) and it is that cumulative radiation density that other observations – such as the 1/r^2 fall off from the Sun – suggests tracks the gravitational effect.
If the observed 1/r gravitational fall off is tracked by the radiation density fall off, that would be consistent with other configurations such as the Solar System – that is the hypothesis – that the radiation density does fall off as 1/r. The only question then is how to determine the cumulative radiation density of the disk as it is determined by the luminous mass distribution – empirically or by indirect estimation. If the hypothesis is correct both methods should yield similar results.
The problem, of course, is that a direct empirical measurement of a disk’s radially aligned intensity would involve observations of edge-on galaxies which typically have a lot of non-luminous gas and dust obscuring the view. Still there may be some edge-on galaxies out there that might permit direct observations. There also does not appear to be a standard mathematical approach for dealing with radiation density except as an after effect.
In either case, observed or estimated, the hypothetical radiation density argument with respect to the disk is, by assumption, consistent with the observed correlation between radiation density and gravitational effects in the Solar System. It is a qualitative argument that involves no hypothetical but unobserved forces or entities. It is also uniquely subject to direct empirical investigation.
Scale and context dependence isn’t a flaw shared by our gravitational models, it’s a universal feature of physical description. One could argue that when a framework genuinely is scale and context independent, it has ceased to be a physical theory and become a mathematical one. The physical and the measurable are inseparable, and measurement is always conducted within a regime.
This is why treating the universe itself as a physical object leads to ill-defined constructions. A universe wave function, a Wheeler-DeWitt equation, a Friedmann field equation, these are mathematically consistent objects, but the universe has no external reference frame, no boundary conditions set by something outside it, no regime of measurement it sits within. It can be treated as a mathematical object. As a physical object it is undefined. If extending General Relativity to galaxies led to the dark matter fiction it should not be surprising that extending it to cosmological scales led to dark energy and inflation.
The fact that Newton, Einstein, and Milgrom each work well within their validated regimes and less well outside them isn’t a shared flaw awaiting a unified fix. It’s the expected signature of effective theories, each fingerprinting the complexity scale at which it was fitted. The demand for a single framework that works at all scales isn’t a scientific requirement, it’s a philosophical preference, and one that neither the structure of physical description nor mathematical results obviously support.
Just wondering due to your phrase “scale dependence”: you know that MOND is scale invariant? In the deep-MOND regime, that is.
A framework can be scale invariant in its mathematical structure while remaining bounded in its empirical applicability. These are different claims. MOND’s deep regime exhibits scale invariance as a formal property, but the regime itself has boundaries, it was established from spiral galaxy dynamics and its validated range does not extend indefinitely in either direction. Scale invariance within a regime is not the same as universal applicability across regimes. No framework escapes that constraint, and none has.
The hope for universality is what produced the current situation. When a framework is assumed to be universally valid by construction, anomalies cannot be evidence of a complexity ceiling, they can only be evidence of missing entities. That is the logic that generated dark matter, dark energy, and inflation as a package. It is also why resistance to the missing mass as a fiction is structural rather than empirical. A significant portion of theoretical physics, string theory and the broader unification program included, is built on the unquestionable universality of GR.
Acknowledging a complexity ceiling for GR is not a local adjustment, it dissolves the foundation the entire unification program is built on.
Peter Woit, no friend of speculative physics, recently wrote that he is sympathetic to the idea that right-handed neutrinos are behind the dark matter mystery. That is a telling data point. Even sharp critics of the sociology of theoretical physics remain inside the ontological assumption, that the missing mass is a real entity awaiting identification, not a signal that the framework has met its ceiling.
If you’re looking for causes, as I do, and for “the causal mechanism that produces the observed gravitational effects”, it’s worth bearing in mind that we know space is flat at a large scale. Penrose has pointed out that inflation has a bigger fine tuning problem that the one it was brought in to remove – the fact that after 13.8 Gyr, space somehow remains flat. A lot of recent theories are set in flat space, which is also because curvature is the place where GR and QM are failing to join up.
And if you see that “light from a distant source passing near the Sun behaves as if it were traversing a medium with a density gradient”, then you’re near where I was in the early 2000s. To me the density gradient is caused by an emitted medium that dissipates – not the EM field, but a medium that can refract light. It led to an equation for the geodetic effect (often thought to be only explainable via curvature) I published in 2008, and an improved version in 2023. You just had to assume the inner edge of an orbiting gyroscope is slowed by the medium slightly more than the outer edge, due to a slight density difference – that turns the gyro through the same angle per orbit as GR to 16 decimal places.
If the emitted medium is at a very small scale (there’s no reliable picture of the Planck scale nowadays, particularly since string theory failed to get the expected support via supersymmetry), then matter can be refracted as well, behaving like light and travelling on helical paths, because it’s a rotating disturbance in the small-scale structure of space. And the medium can double up as DM at larger scales, where an excess builds up.
In the mass discrepancy, we have clues coming into focus at different scales, as the data gets better. The solution will inevitably include an element of explanation, it can’t be sorted out with mathematical adjustments alone. For instance, there’s a need to explain the close connection between dark and visible matter. (If one emits the other, that could explain it.) What’s needed is an explanation where a lot of things click into place, which is economical in assumptions.
PS It was 14 decimal places, not 16. And I meant to say, the medium is collisionless, behaves similarly to DM in lensing data, has a close connection with baryons, and is undetectable directly.
Can you give a link to a paper of yours? I’m wondering if it has to be a medium or that it might be something else. If the density scales from a point mass as 1/r, I might have an interesting idea.
PSG theory is here: https://gwwsdk1.wixsite.com/link/newpreprint-pdf
But here’s the first part of the abstract of a paper that will be on a preprint site soon, and might be of interest, as it puts the interpretation for MOND into focus, and includes a new interpolation function:
A geometrical interpretation for MOND, from a hybrid theory
In a standard way of taking MOND, g = (gNa0)^1/2, the theory refers to Newtonian gravity as it would otherwise be beyond a0. However literally this aspect should be taken, an adjustment is made, as if tracing a comparatively superficial alteration to the outer field, that maps each point onto another. Here, in a partly geometrical interpretation (acceleration-based but emphasising radial distances), there is a deformation of the outer part of the field. A gravity theory, Planck scale gravity (PSG), with loose similarities to Eddington’s refractive medium interpretation for general relativity (GR), leads to a hybrid approach, from a basis involving the small-scale structure of space. An emitted medium that dissipates to create the gradient of the field allows an interpretation in which the Newtonian pattern becomes increasingly ‘compressed’ beyond the MOND radius, boosting accelerations by r/rM, which alone is enough to lead to flat rotation curves, and much of MOND. Newtonian gravity can be restored with a radial adjustment, via the effective radius r’ = (rMr)^1/2.
have you seen
Baryonic mass budgets in the central regions of the Bullet Cluster and their consistency with strong lensing in MOND
Dong Zhang1,*, Hosein Haghi2,1,3,†, Elena Asencio1, Indranil Banik4,‡, Akram Hasani Zonoozi1,2, Sangjun Cha5,§, Boseong Young Cho5,∥, Hyungjin Joo5,¶, Pavel Kroupa1,6 et al.
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Phys. Rev. D 114, 023001 – Published 1 July, 2026
DOI: https://doi.org/10.1103/6zrp-q7c4
Please see my reply to Jean who points out the same paper.
About the bullet cluster this might be interesring.
https://arxiv.org/pdf/2606.19454
Thanks for your blog
Jean
Maybe. I worry that this scenario would over-produce the mass of isolated elliptical galaxies, solving this problem but creating another. The remnants of massive stars need to be flung out of their birth galaxies into the intracluster medium, so maybe there is a path for a scenario in which these stick around in clusters and are lost from isolated ellipticals to the IGM.
One thing that strikes me about cluster mass lensing reconstructions is how closely they adhere to the galaxies. It’s not just that some dark mass passed through collisionlessly; it really seems to stick with the galaxies. Perhaps this is more likely in a scenario like this in which the galaxies are the source of the unseen mass.
Looking at how the unseen mass sticks with the galaxies, there may be a list of different possibilities, such as sterile neutrinos, to explain that. But perhaps the connection is more defined, you once said about the RAR ‘If you measure the distribution of star light, you know the rotation curve, and vice versa’. Can things be narrowed down via the details of the connection – does that affect the possibilities?
Yes, in the sense that knowing what the distribution of the hypothetical unseen mass needs to be limits what it can be.
Using a framework well beyond its validated context and then claiming that the resulting discrepancies imply missing ontology, rather than recognizing that the framework is being used beyond its limits, is a weak and dogmatic position.
This is exactly what happens when General Relativity is extended from simple gravitational systems to galaxies, and dark matter is invoked when discrepancies arise, using then GR plus dark matter to clusters is doubling down in the same mistake.
The closest historical analog is not Neptune, which was a localized, ordinary object proposed within a validated context. The closest historical analog is the luminiferous ether, an exotic, universe-wide substance invented to preserve an existing framework.
Beyond any arguments, one glaring fact remains. The dark matter approach has motivated increasingly sensitive searches, yet no independent detection of the proposed dark matter particle has emerged, decades of negative results. At some point, repeated negative results become methodologically decisive.
Like String Theory, the dark matter approach has been a decades long failed approach, and these failures are not disconnected.
Your idea that the MOND puzzle we’re up against is a result of applying a theory beyond its domain is right in one way. Gravity gets different at an acceleration scale we have no experience of in the solar system. Thanks to Milgrom we now have some rules for gravity at low accelerations, but no explanation.
But the boundary is very specific, even though it needs an interpolation function (incidentally, the IF I got out of r/rM makes a curve that lands between the simple and standard IFs). At the boundary, we find a weirdly specific set of clues. You get the EFE, which is a complex set of rules about how the boundary applies. That isn’t about trying to apply a theory way outside its domain, it’s about finding the edge of one thing, and the edge of something else beyond it.
And you can’t ignore the two-tone aspect of galaxies. Newton/GR works fine inside the MOND radius, so very near the boundary, standard theory is not outside its domain.
And there’s evidence for a collisionless medium in clusters, and other evidence for DM, so people are trying to make hybrid theories. Until this weird set of clues has a solution, you can’t just throw a blanket generalisation at it, about how theories should be applied. Although true in some cases, it’s not the kind of thing that makes a hard and fast rule anyway.
One thing that comes out of these points, to me, is that a0 doesn’t look universal. A change to the physics when a parameter reaches a certain value is not a characteristic of a universal law, as far as we know. And there are things that suggest limitations: vertical velocities, and the RAR in clusters, with g++ instead of a0, which would remove the missing baryons problem. So perhaps in clusters people are trying to apply a theory beyond its domain of applicability – perhaps MOND with a0 belongs in galaxies. But it may all come under one heading in the future, when we understand the RAR better.
That is exactly what we see throughout physics. Different effective descriptions can coexist within the same system. A nested hierchycal structure.
For example classical behavior emerges once a sufficient complexity threshold is crossed. You can successfully apply quantum mechanics to the microscopic constituents of a classical object, but not to the classical object itself. The classical context lies beyond quantum mechanics’ effective range of applicability.
Galaxies appear to behave the same way. Newtonian gravity and General Relativity work well in the high-acceleration inner regions, while the MOND regime becomes apparent in the low-acceleration outskirts. You cannot observe the MOND regime without the galactic context that gives rise to it.
I’d say QM is rather similar. There’s an unexplained transition, the physics changes suddenly at a particular point, and we don’t know what’s going on at all.
But there are all these clues about the transition that link one side of the boundary to the other, and that’s why you can’t yet make statements of that kind on either puzzle. In QM, that’s a bit out of date – forty years ago people thought there were two sets of rules, now we tend to think it’s somehow all quantum. Decoherence has shown state reduction takes a finite period of time, it can be predicted. We know when that happens a rapid series of interactions creates entanglements. That links the two states closely, classical and quantum, with a process that joins them. So you have to say what too few people say – simply that there’s more to be found out.
You idea works well when you look back at past theories, now well understood, and point out limitations – and to be fair you’ve done that to good effect sometimes. But wait till QM and gravity are understood to apply it there, or your idea will itself be outside its domain.
According to AI sterile neutrinos with a mass of 10-11 eV could account for the MOND’s missing mass problem in clusters. Without air-conditioning I’m going to wait few days till the humidity and heat diminish before researching this further. It’s too hot to do much thinking.
Yes, that’s one hypothesis, made originally by Angus (https://arxiv.org/abs/0805.4014). Part of the motivation was to explain the CMB, which I don’t think it can do, but it remains an active topic of research. Nevertheless, if sterile neutrinos exist and have cosmologically significant mass density, then they will contribute to the cluster mass budget but not to galaxies because the potential wells of the latter are too shallow to retain them. Heck, if ordinary neutrinos have a mass that exceeds 0.12 eV, that would invalidate the entire LCDM structure formation paradigm.
Perhaps the more interesting lesson from MOND is methodological.
MOND demonstrates that modifying the gravitational description can explain a large body of galactic phenomena.
Instead of asking whether this points to the limits of the underlying framework, Angus asks what additional unseen particle can be added so that MOND fits within the existing cosmological paradigm.
That misses the deeper implication. It keeps the assumption of a universal gravitational description intact, even with dark energy, and replaces one long standing, experimentally unvalidated component with another.
If the total angular momentum LT of the gravitationally bound system is considered for the total baryonic mass MT, one can derive an interesting relationship that the MOND acceleration a0 is proportional to MT^7/LT^4. So if the LT for the very massive galaxy clusters is less than expected for the MT, one can accommodate a factor of 5 or more for the “missing baryonic mass” issue. Possible reasons for low LT in a galaxy cluster: (1) IC gas rotation speed is less than for single galaxies, (2) the IC stars rotate slower, (3) L vector for individual galaxies could have a great variety of directions. So the question becomes: Why not consider the role LT might have in determining the ΔM of a cluster?
Angular momentum isn’t entirely independent of mass so I’m not sure there’s additional leverage here.
Do you think the missing baryon problem is fundamentally an inventory problem or an inference problem?
I don’t know. Wish I did.
This is great. I’ve got my reading material lined up; the reference paper by Angus that Dr. McGaugh provided on sterile neutrinos as a possible solution to MOND’s relatively small mass discrepancy in clusters, the very recent (July, 2026) paper by multiple authors “Baryonic Mass Budgets in the Central Regions of the Bullet Cluster…..”, as well as another paper I haven’t dialed up. And, the icing on the cake is the weather has finally cooled here in the US northeast.
With the Bullet cluster, the system dynamics have a particular relationship to us where the clusters are moving primarily in the plane of the sky, or normal to our Hubble flow. Correct me if I got that wrong please.
Recently in the press there is a lot of discussion about the connection between the Bullet Cluster and three dwarf galaxies which appear to be missing dark matter, and are all in a straight line (suggesting a similar collision mechanism for separating baryons from dark matter).
I understand from your earlier posts that the dynamic equilibrium conditions and external field effects are important considerations for these observations. However, I’m curious if this line that the dwarf galaxies are on has any interesting or particular relationship to us?
Yes – the motion of the bullet cluster is inferred to be almost exactly in the plane of the sky.
Yes, the dwarfs DF2, DF4, & DF9 seem to lack dark matter and appear in a line, suggesting a possible origin in a collision/interaction. I do not know what the vector is relative to our line of sight. It does seem that we have to invoke some kind of non-equilibrium event to make them so, which could occur in either paradigm, so all best are off. Like the bullet cluster, these are a good example of the community latching onto unusual objects that adhere to its confirmation bias whilst ignoring the hundreds of normal objects that don’t.
Yes, there is an interesting difference between latching on to a rare and unusual isolated object as opposed to addressing the wider unusualness, which you have done a great job of highlighting. The tightness of the BTFR and the systematic divergence from the cosmic baryon fraction across many orders of magnitude in mass scale has got to be an unusual result in a LCDM universe. It just doesn’t add up.
Perhaps the difference in resolution matters more than we think.
To solve missing baryons problem, you should use either the real matrix group SL(4,R) with signature (9,6), or the group SU(3,1) with signature (6,9).
“MOND … suffers a missing baryon problem in rich clusters … predicting only ~80% of the observed velocity translates to missing ~60% of the mass.” General relativity predicts twice as much gravitational lensing as Newton’s law of gravity combined with Newton’s corpuscular theory of light. CONJECTURE: FUNDAMOND string theory predicts twice as much gravitational lensing as MOND combined with Newton’s corpuscular theory of light. What evidence indicates that the preceding conjecture is wrong? Have any of the astronomers and astrophysicists who understand the importance of MOND considered Guendelman’s new version of string theory as a highly plausible mathematical approach to FUNDAMOND string theory? Google “guendelman ssb”.
“… uncomfortable with the apparent requirement that there there is lots of undetected baryonic mass in clusters.” Isn’t it fairly obvious that MOND’s (approximately) successful predictions require a drastic revision in the foundations of physics (i.e. some version of FUNDAMOND correcting general relativity theory)? Does the Bullet Cluster merely indicate that MOND needs to be upgraded to FUNDAMOND?
Could be a clue, yes.
With the Bullet Cluster, have radial velocity measurements been made of the individual galaxies and do these agree (using the virial theorem) with the mass determinations from gravitational lensing?
I don’t know if that has been done in this particular case. We know the redshift, so some must have been measured, but I’ve never heard the virial mass estimate discussed, so there probably aren’t enough galaxies measured to do it. This is a surprisingly common situation, and has been a recurring frustration when I’ve tried to work on clusters in general.
Pengfei Li did this for a sample of clusters (not including the bullet) and found that kinematics showed qualitative similarities to hydrostatics (e.g., an offset RAR) but that they didn’t agree in total mass. The kinematic indicated *more* mass than hydrostatics, making the problem for MOND worse and also pushing clusters well away from the cosmic baryon fraction (https://arxiv.org/abs/2303.10175). This inconsistency is one reason I say clusters ruin everything. https://tritonstation.com/2024/02/06/clusters-of-galaxies-ruin-everything/
“Could be a clue …” How might we assemble all of the clues needed to understand the importance of MOND and its relation to a possible FUNDAMOND?
In reply to a recent email inquiry, Professor Milgrom replied in part:
As to the number of people interested in MOND, it is hard to estimate. You can try to judge by the list of papers that have MOND in the abstract:
https://arxiv.org/search/advanced?advanced=1&terms-0-operator=AND&terms-0-term=MOND&terms-0-field=abstract&classification-physics_archives=all&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first
Stacy, do you think the compressed field interpretation I posted above is a possible explanation? It’s a way of taking MOND – one of a number of them, but Milgrom does seem to make an adjustment to Newton, and you find Newton’s pattern persists exactly, but with the radius term getting steadily shorter than expected. The idea might have had a lot more attention if more of our gravity theories had the kind of field that can easily get warped out of shape.
It is possible that there is a generalization needed; e.g., eMOND has a Lagrangian that depends on potential well depth in addition to acceleration. Having it be length-scale dependent contradicts my experience in galaxies where the dependence on acceleration but not size is key. But perhaps there is a way it crops up in clusters that doesn’t in galaxies.
It’s not length scale dependent, but acceleration-based (as it says in the abstract I quoted from). The length scale is just something that shows what may be happening to the Newtonian field, but the boost to accelerations by r/rM is the main element. Although it’s just one way of taking MOND, that change to accelerations alone can lead to flat rotation curves.
OK, but we need something extra in clusters if there is not any more mass there. So is it a compounding of accelerations? I guess I need to go look and do some math.
One way to take clusters is the ‘distinct RAR’ they found, with a different acceleration scale. A good clue is that both RARs begin at the same place, 2e-9. Then the transition is quick in clusters, slow in galaxies, and the new pattern takes until a0 and beyond to settle in (in my interpretation, if self-interaction of the medium causes the transition, it may be slow in galaxies due to the medium’s waves being more aligned in their direction of travel).
It’s not a compounding of accelerations, g/gN = r/rM is mathematically identical to g = (gNa0)^1/2, it’s another way of saying it as rM = (GM/a0)^1/2. So the radial aspect is ‘built-in’ if you take MOND that way. In PSG it’s about a rate of change with radius that causes accelerations, and the inverse square law.
Btw, of the two papers I mentioned, the current one corrects an error in the earlier one, the ‘field compression factor’ (which is just a way of showing something) is r/r’, and the acceleration boost is its square, r/rM.
Oh, so the “smoking gun” of (NB)DM here is that lensing is exactly where the galaxies are. But the gas has more mass (and its not lensing (enough?)). And we are sure about this? And how?
The gas does outweigh the galaxies unless the IMF in the latter is abnormal and produces a lot of remnants (neutron stars/black holes). This is exactly what Kroupa and his collaborators are advocating in https://arxiv.org/abs/2602.06082. Seems like a stretch to me, but it does have virtues, like making the metals seen in clusters. The lensing of the gas in MOND is currently being debated; see https://arxiv.org/abs/2604.10811 and https://arxiv.org/abs/2605.10022
“… AQUAL … and .. QUMOND … must only be some approximate, effective theories, with limited validity. … the appearance, in all existing theories, of an “interpolating function” (IF), that is introduced “by hand” already at the level of the action of the theory, and that artificially interpolates …
This will surely not be the case in a more fundamental MOND theory – a FUNDAMOND. Quantum mechanics and relativity, also make predictions that involve various IFs. But these differ from phenomenon to phenomenon, and are not introduced at the fundamental level of the theory. We expect that also in a FUNDAMOND theory there will be different IFs appearing in different contexts, and they will all be derivable from the theory and not appear as some fundamental function of the theory.”
“Generalizations of quasilinear MOND (QUMOND)” (version 2, 2023) by Mordecai Milgrom https://arxiv.org/pdf/2305.01589
It seems to me that DMCC with FUNDAMOND-cutoff might be a extremely good candidate in terms of FUNDAMOND gravitational lensing.
Where general relativity (GR) has no MOND-type problems, use GR (i.e. FUNDAMOND-cutoff). Where GR does have MOND-type problems, simply use GR with the –1/2 replaced by –1/2 + DMCC, where DMCC (dark-matter-compensation-constant) = (3.9±.5) * (10^–5) — or some better number.
What is the status of Weyl Transverse Gravity (WTG) as an underlying description? How does it compare MOND?
I haven’t gotten through this paper yet on Weyl Transverse Gravity,
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.111.064019,
but it is already really striking a chord. Thoughts on this direction?
Interesting. I’ve been aware of Weyl gravity for a very long time – its conformal invariance was appealing before MOND came up in my data – but this is the first I’ve head of this particular flavor. The alleged relation to entropy puts in mind Verlinde’s entropic gravity (which gives approximately MOND-like behavior) but there does not appear to be any real connection on casual scrutiny, and the one citation to Verlinde is contained in assertion that their way is better than his. At any rate, I do not see MOND in this, nor would there seem to be much chance of it emerging from this approach as it claims to justify the strong equivalence principle which MOND violates.
Perhaps anticipating that problem?
Equivalence principles in Weyl transverse gravity
https://arxiv.org/html/2502.03888v1
“We further analyse how the weaker formulations of the equivalence principle are realised in Weyl transverse gravity (and its generalisations). The analysis sheds light on the behaviour of matter fields in this theory.”
They appear to take a different approach than Verlinde.
“In summary, we present a complete and self-contained
argument for the recovery of Weyl transverse gravity from
thermodynamics of local causal horizons. We do so without
assuming in any way that gravitational dynamics emerges
as a thermodynamic limit of the behavior of some quantum
degrees of freedom of the spacetime unrelated to the
metric [4,6,42]. We instead take a more modest position
that thermodynamics encodes all the relevant features of the
gravitational dynamics, regardless of whether it is ulti-
mately emergent or fundamental.”
My simple summary of the paper is that Weyl Transverse Gravity is a modification that behaves like a conformal version of GR, and it can be derived from entropy (specifically the thermodynamic equilibrium of local causal diamonds) and the Strong EP.
WTG may be a mechanism connecting observational entropy to the dark sector.
Does this general direction seem promising for finding a satisfactory explanation for the “dark matter”/MOND and “dark energy” phenomena?