Previously, we discussed non-equilibrium dynamics in tidal dwarf galaxies. These are the result of interactions between giant galaxies that are manifestly a departure from equilibrium, a circumstance that makes TDGs potentially a decisive test to distinguish between dark matter and MOND, and simultaneously precludes confident application of that test. There are other galaxies for which I suspect non-equilibrium dynamics may play a role, among them some (not all) of the so-called ultradiffuse galaxies (UDGs).
UDGs
The term UDG has been adopted for galaxies below a certain surface brightness threshold with a size (half-light radius) in excess of 1.5 kpc (van Dokkum et al. 2015). I find the stipulation about the size to be redundant, as surface brightness* is already a measure of diffuseness. But OK, whatever, these things are really spread out. That means they should be good tests of MOND like low surface brightness galaxies before them: their low stellar surface densities mean** that they should be in the regime of low acceleration and evince large mass discrepancies when isolated. It also makes them susceptible to the external field effect (EFE) in MOND when they are not isolated, and perhaps also to tidal disruption.
To give some context, here is a plot of the size-mass relation for Local Group dwarf spheroidals. Typically they have masses comparable to globular clusters, but much large sizes – a few hundred parsecs instead of just a few. As with more massive galaxies, these pressure supported dwarfs are all over the place – at a give mass, some are large while others are relatively compact. All but the one most massive galaxy in this plot are in the MOND regime. For convenience, I’ll refer to the black points labelled with names as UDGs+.
The UDGs are big and diffuse. This makes them susceptible to the EFE and tidal effects. The lower the density of a system, the easier it is for external systems to mess with it. The ultimate example is something gets so close to a dominant central mass that it gets tidally disrupted. That can happen conventionally; the stronger effective force of MOND increases tidal effects. Indeed, there is only a fairly narrow regime between the isolated case and tidally-induced disequilibrium where the EFE modifies the internal dynamics in a quasi-static way.
The trouble is the s-word: static. In order to test theories, we assume that the dynamical systems we observe are in equilibrium. Though often a good assumption, it doesn’t always hold. If we forget we made the assumption, we might think we’ve falsified a theory when all we’ve done is discover a system that is out of equilibrium. The universe is a very dynamic place – the whole thing is expanding, after all – so we need to be wary of static thinking.
Equilibrium MOND formulae
That said, let’s indulge in some static thinking. An isolated, pressure supported galaxy in the MOND regime will have an equilibrium velocity dispersion
where M is the mass (the stellar mass in the case of a gas-free dwarf spheroidal), G is Newton’s constant, and a0 is Milgrom’s acceleration constant. The number 4/81 is a geometrical factor that assumes we’re observing a spherical system with isotropic orbits, neither of which is guaranteed even in the equilibrium case, and deviations from this idealized situation are noticeable. Still, this is as simple as it gets: if you know the mass, you can predict the characteristic speed at which stars move. Mass is all that matters: we don’t care about the radius as we must with Newton (v2 = GM/r); the only other quantities are constants of nature.
But what do we mean by isolated? In MOND, it is that the internal acceleration of the system, gin, exceeds that from external sources, gex: gin ≫ gex. For a pressure supported dwarf, gin ≈ 3σ2/r (so here the size of the dwarf does matter, as does the location of a star within it), while the external field from a giant host galaxy would be gex = Vf2/D where Vf is the flat rotation speed stipulated by the baryonic mass of the host and D is the distance from the host to the dwarf satellite. The distance is not a static quantity. As a dwarf orbits its host, D will vary by an amount that depends on the eccentricity of the orbit, and the external field will vary with it, so it is possible to have an orbit in which a dwarf satellite dips in and out of the EFE regime. Many Local Group dwarfs straddle the line gin ≈ gex, and it takes time to equilibrate, so static thinking can go awry.
It is possible to define a sample of Local Group dwarfs that have sufficiently high internal accelerations (but also in the MOND regime with gex ≪ gin ≪ a0) that we can pretend they are isolated, and the above equation applies. Such dwarfs should& fall on the BTFR, which they do:
In contrast, three of the four the UDGs considered here do not fall on the BTFR. Should they?
Conventionally, in terms of dark matter, probably they should. There is no reason for them to deviate from whatever story we make up to explain the BTFR for everything else. That they do means we have to make up a separate story for them. I don’t want to go deeply into this here since the cold dark matter model doesn’t really explain the observed BTFR in the first place. But even accepting that it does so after invoking feedback (or whatever), does it tolerate deviants? In a broad sense, yes: since it doesn’t require the particular form of the BTFR that’s observed, it is no problem to deviate from it. In a more serious sense, no: if one comes up with a model that explains the small scatter of the BTFR, it is hard to make that same model defy said small scatter. I know, I’ve tried. Lots. One winds up with some form of special pleading in pretty much any flavor of dark matter theory on top of whatever special pleading we invoked to explain the BTFR in the first place. This is bad, but perhaps not as bad as it seems once one realizes that not everything has to be in equilibrium all the time.
In MOND, the BTFR is absolute – for isolated systems in equilibrium. In the EFE regime, galaxies can and should deviate from it even if they are in equilibrium. This always goes in the sense of having a lower characteristic velocity for a given mass, so below the line in the plot. To get above the line would require being out of equilibrium through some process that inflates velocities (if systematic errors are not to blame, which also sometimes happens.)
The velocity dispersion in the EFE regime (gin ≪ gex ≪ a0) is slightly more complicated than this isolated case:
This is just like Newton except the effective value of the gravitational constant is modified. It gets a boost^ by how far the system is in the MOND regime: Geff ≈ G(a0/gex). An easy way to tell which regime an object is in is to calculate both velocity dispersions σiso and σefe: the smaller one is the one that applies#. An upshot of this is that systems in the EFE regime should deviate from the BTFR to the low velocity side. The amplitude of the deviation depends on the system and the EFE: both the size and mass matter, as does gex. Indeed, if an object is on an eccentric orbit, then the velocity dispersion can vary with the EFE as the distance of the satellite from its host varies, so over time the object would trace out some variable path in the BTFR plane.
Three of the four UDGs fall off the BTFR, so that sounds mostly right, qualitatively. Is it? Yes, for Crater 2, but but not really for the others. Even for Crater 2 it is only a partial answer, as non-equilibrium effects may play a role. This gets involved for Crater 2, then more so for the others, so let’s start with Crater 2.
Crater 2 – the velocity dispersion
The velocity dispersion of Crater 2 was correctly predicted a priori by the formula for σefe above. It is a tiny number, 2 km/s, and that’s what was subsequently observed. Crater 2 is very low mass, ~3 x 105 M☉, which is barely a globular cluster, but it is even more spread out than the typical dwarf spheroidal, having an effective surface density of only ~0.05 M☉pc-2. If it were isolated, MOND predicts that it would have a higher velocity dispersion – all of 4 km/s. That’s what it would take to put it on the BTFR above. The seemingly modest difference between 2 and 4 km/s makes for a clear offset. But despite its substantial current distance from the Milky Way (~ 120 kpc), Crater 2 is so low surface density that it is still subject to the external field effect, which lowers its equilibrium velocity dispersion. Unlike isolated galaxies, it should be offset from the BTFR according to MOND.
LCDM struggles to explain the low mass end of the BTFR because it predicts a halo mass-circular speed relation Mhalo ~ Vhalo3 that differs from the observed Mb ~ Vf4. A couple of decades ago, it looked like massive galaxies might be consistent with the lower power-law, but that anticipates higher velocities for small systems. The low velocity dispersion of Crater 2 is thus doubly weird in LCDM. It’s internal velocities are too small not just once – the BTFR is already lower than was expected – but twice, being below even that.
An object with a large radial extent like Crater 2 probes far out into its notional dark matter halo, making the nominal prediction$ of LCDM around ~17 km/s, albeit with a huge expected scatter. Even if we can explain the low mass end of the BTFR and its unnaturally low scatter in LCDM, we now have to explain this exception to it – an exception that is natural in MOND, but is on the wrong side of the probability distribution for LCDM. That’s one of the troubles with tuning LCDM to mimic MOND: if you succeed in explaining the first thing, you still fail to anticipate the other. There is no EFE% in LCDM, no reason to anticipate that σefe applies rather than σiso, and no reason to expect via feedback that this distinction has anything to do with the dynamical accelerations gin and gex.
But wait – this is a post about non-equilibrium dynamics. That can happen in LCDM too. Indeed, one expects that satellite galaxies suffer tidal effects in the field of their giant host. The primary effect is that the dark matter subhalos in which dwarf satellites reside are stripped from the outside in. Their dark matter becomes part of the large halo of the host. But the stars are well-cocooned in the inner cusp of the NFW halo which is more robust than the outskirts of the subhalo, so the observable velocity dispersion barely evolves until most of the dark mass has been stripped away. Eventually, the stars too get stripped, forming tidal streams. Most of the damage occurs during pericenter passage when satellites are closest to their host. What’s left is no longer in equilibrium, with the details depending on the initial conditions of the dwarf on infall, the orbit, the number of pericenter passages, etc., etc.
What does not come out of this process is Crater 2 – at least not naturally. It has stars very far out – these should get stripped outright if the subhalo has been eviscerated to the point where its velocity dispersion is only 2 km/s. This tidal limitation has been noted by Errani et al.: “the large size of kinematically cold ‘feeble giant’ satellites like Crater 2 or Antlia 2 cannot be explained as due to tidal effects alone in the Lambda Cold Dark Matter scenario.” To save LCDM, we need something extra, some additional special pleading on top of non-equilibrium tidal effects, which is why I previously referred to Crater 2 as the Bullet Cluster of LCDM: an observation so problematic that it amounts to a falsification.
Crater 2 – the orbit
We held a workshop on dwarf galaxies on CWRU’s campus in 2017 where issues pertaining to both dark matter and MOND discussed. The case of Crater 2 was one of the things discussed, and it was included in the list of further tests for both theories (see above links). Basically the expectation in LCDM is that most subhalo orbits are radial (highly eccentric), so that is likely to be the case for Crater 2. In contrast, the ultradiffuse blob that is Crater 2 would not survive a close passage by the Milky Way given the strong tidal force exerted by MOND, so the expectation was for a more tangential (quasi-circular) orbit that keeps it at a safe distance.
Subsequently, it became possible to constrain orbits with Gaia data. The exact orbit depends on the gravitational potential of the Milky Way, which isn’t perfectly known. However, several plausible choices of the global potential give an an eccentricity around 0.6. That’s not exactly radial, but it’s pretty far from circular, placing the pericenter around 30 kpc. That’s much closer than its current distance, and well into the regime where it should be tidally disrupted in MOND. No way it survives such a close passage!
So which is it? MOND predicted the correct velocity dispersion, which LCDM struggles to explain. Yet the orbit is reasonable in LCDM, but incompatible with MOND.
Simulations of dwarf satellites
It occurs to me that we might be falling victim to static thinking somewhere. We talked about the impact of tides on dark matter halos a bit above. What should we expect in MOND?
The first numerical simulations of dwarf galaxies orbiting a giant host were conducted by Brada & Milgrom (2000). Their work is specific to the Aquadratic Lagrangian (AQUAL) theory proposed by Bekenstein & Milgrom (1984). This was the first demonstration that it was possible to write a version of MOND that conserved momentum and energy. Since then, a number of different approaches have been demonstrated. These can be subtly different, so it is challenging to know which (if any) is correct. Sorting that out is well beyond the scope of this post, so let’s stick to what we can learn from Brada & Milgrom.
Brada & Milgrom followed the evolution of low surface density dwarfs of a range of masses as they orbited a giant host galaxy. One thing they found was that the behavior of the numerical model could deviate from the analytic expectation of quasi-equilibrium enshrined in the equations above. For an eccentric orbit, the external field varies with distance from the host. If there is enough time to respond to this, the change can be adiabatic (reversible), and the static approximation may be close enough. However, as the external field varies more rapidly and/or the dwarf is more fragile, the numerical solution departs from the simple analytic approximation. For example:
As long as the behavior is adiabatic, the dwarf can be stable indefinitely even as it goes through periodic expansion and contraction in phase with the orbit. Departure from adiabaticity means that every passage will be different. Some damage will be done on the first passage, more on the second, and so on. As a consequence, reality will depart from our simple analytic expectations.
I was aware of this when I made the prediction for the velocity dispersion of Crater 2, and hedged appropriately. Indeed, I worried that Crater 2 should already be out of equilibrium. Nevertheless, I took solace in two things: first, the orbital timescale is long, over a Gyr, so departures from the equilibrium prediction might not have had time to make a dramatic difference. Second, this expectation is consistent with the slow evolution of the characteristic velocity for the most Crater 2-like, m=1 model of Brada & Milgrom (bottom track in the right panel below):
What about the size? That is not constant except for the most massive (m=16) model. The m=3 and 4 models recover, albeit not adiabatically. The m=4 model almost returns to its original size, but the m=3 model has puffed up after one orbit. The m=1 and 2 models explode.
One can see this by eye. The continuous growth in radii of the lower mass models is obvious. If one looks closely, one can also see the expansion then contraction of the heavier models.
The current size of Crater 2 is unusual. It is very extended for its mass. If the current version of Crater 2 has a close passage with the Milky Way, it won’t survive. But we know it already had a close passage, so it should be expanding now as a result. (I did discuss the potential for non-equilibrium effects.) Knowing now that there was a pericenter passage in the (not exactly recent) past, we need to imagine running back the clock on the simulations. It would have been smaller in the past, so maybe it started with a normal size, and now appears so large because of its pericenter passage. The dynamics predict something like that; it is static thinking to assume it was always thus.
This is a good example of a phenomena I’ve encountered repeatedly with MOND. It predicts something right, but seems to get something else wrong. If we’re already sure it is wrong, we stop there and never think further. But when one bothers to follow through on what the theory really predicts, more often than not the apparently problematic observation is in fact what we should have expected in the first place.
DF2 and DF4
DF2 and DF4 are two UDGs in the vicinity of the giant galaxy NGC 1052. They have very similar properties, and are practically identical in terms of having the same size and mass within the errors. They are similar to Crater 2 in that they are larger than other galaxies of the same mass.
When it was first discovered, NGC 1052-DF2 was portrayed as a falsification of MOND. On closer examination, had I known about it, I could have used MOND to correctly predict its velocity dispersion, just like the dwarfs of Andromeda. This seemed like yet another case where the initial interpretation contrary to MOND melted away to actually be a confirmation. At this point, I’ve seen literally hundreds^^of cases like that. Indeed, this particular incident made me realize that there would always be new cases like that, so I decided to stop spending my time addressing every single case.
Since then, DF2 has been the target of many intensive observing campaigns. Apparently it is easier to get lots of telescope time to observe a single object that might have the capacity to falsify MOND than it is to get a more modest amount to study everything else in the universe. That speaks volumes about community priorities and the biases that inform them. At any rate, there is now lots more data on this one object. In some sense there is too much – there has been an active debate in the literature over the best distance determination (which affects the mass) and the most accurate velocity dispersion. Some of these combinations are fine with MOND, but others are not. Let’s consider the worst case scenario.
In the worst case scenario, both DF2 and DF4 are too far from NGC 1052 for its current EFE to have much impact, and they have relatively low velocity dispersions for their luminosity, around 8 km/s, so they fall below the BTFR. Worse for MOND is that this is about what one expects from Newton for the stars alone. Consequently, these galaxies are sometimes referred to as being “dark matter free.” That’s a problem for MOND, which predicts a larger velocity dispersion for systems in equilibrium.
Perhaps we are falling prey to static thinking, and these objects are not in equilibrium. While their proximity to neighboring galaxies and the EFE to which they are presently exposed depends on the distance, which is disputed, it is clear that they live in a rough neighborhood with lots of more massive galaxies that could have bullied them in a close passage at some point in the past. Looking at Fig. 4 of Brada & Milgrom above, I see that galaxies whacked out of equilibrium not only expand in radius, potentially explaining the unusually large sizes of these UDGs, but they also experience a period during which their velocity dispersion is below the equilibrium value. The amplitude of the dip in these simulations is about right to explain the appearance of being dark-matter-free.
It is thus conceivable that DF2 and DF4 (the two are nearly identical in the relevant respects) suffered some sort of interaction that perturbed them into their current state. Their apparent absence of a mass discrepancy and the apparent falsification of MOND that follows therefrom might simply be a chimera of static thinking.
Make no mistake: this is a form of special pleading. The period of depressed velocity dispersion does not last indefinitely, so we have to catch them at a somewhat special time. How special depends on the nature of the interaction and its timescale. This can be long in intergalactic space (Gyrs), so it may not be crazy special, but we don’t really know how special. To say more, we would have to do detailed simulations to map out the large parameter space of possibilities for these objects.
I’d be embarrassed for MOND to have to make this kind of special pleading if we didn’t also have to do it for LCDM. A dwarf galaxy being dark matter free in LCDM shouldn’t happen. Galaxies form in dark matter halos; it is very hard to get rid of the dark matter while keeping the galaxy. The most obvious way to do it, in rare cases, is through tidal disruption, though one can come up with other possibilities. These amount to the same sort of special pleading we’re contemplating on behalf of MOND.
Recently, Tang et al. (2024) argue that DF2 and DF4 are “part of a large linear substructure of dwarf galaxies that could have been formed from a high-velocity head-on encounter of two gas-rich galaxies” which might have stripped the dark matter while leaving the galactic material. That sounds… unlikely. Whether it is more or less unlikely than what it would take to preserve MOND is hard to judge. It appears that we have to indulge in some sort of special pleading no matter what: it simply isn’t natural for galaxies to lack dark matter in a universe made of dark matter, just as it is unnatural for low acceleration systems to not manifest a mass discrepancy in MOND. There is no world model in which these objects make sense.
Tang et al. (2024) also consider a number of other possibilities, which they conveniently tabulate:
There are many variations on awkward hypotheses for how these particular UDGs came to be in LCDM. They’re all forms of special pleading. Even putting on my dark matter hat, most sound like crazy talk to me. (Stellar feedback? Really? Is there anything it cannot do?) It feels like special pleading on top of special pleading; it’s special pleading all the way down. All we have left to debate is which form of special pleading seems less unlikely than the others.
I don’t find this debate particularly engaging. Something weird happened here. What that might be is certainly of interest, but I don’t see how we can hope to extract from it a definitive test of world models.
Antlia 2
The last of the UDGs in the first plot above is Antlia 2, which I now regret including – not because it isn’t interesting, but because this post is getting exhausting. Certainly to write, perhaps to read.
Antlia 2 is on the BTFR, which is ordinarily normal. In this case it is weird in MOND, as the EFE should put it off the BTFR. The observed velocity dispersion is 6 km/s, but the static EFE formula predicts it should only be 3 km/s. This case should be like Crater 2.
First, I’d like to point out that, as an observer, it is amazing to me that we can seriously discuss the difference between 3 and 6 km/s. These are tiny numbers by the standard of the field. The more strident advocates of cold dark matter used to routinely assume that our rotation curve observations suffered much larger systematic errors than that in order to (often blithely) assert that everything was OK with cuspy halos so who are you going to believe, our big, beautiful simulations or those lying data?
I’m not like that, so I do take the difference seriously. My next question, whenever MOND is a bit off like this, is what does LCDM predict?
I’ll wait.
Well, no, I won’t, because I’ve been waiting for thirty years, and the answer, when there is one, keeps changing. The nominal answer, as best I can tell, is ~20 km/s. As with Crater 2, the large scale size of this dwarf means it should sample a large portion of its dark matter halo, so the expected characteristic speed is much higher than 6 km/s. So while the static MOND prediction may be somewhat off here, the static LCDM expectation fares even worse.
This happens a lot. Whenever I come across a case that doesn’t make sense in MOND, it usually doesn’t make sense in dark matter either.
In this case, the failure of the static-case prediction is apparently caused by tidal perturbation. Like Crater 2, Antlia 2 may have a large half-light radius because it is expanding in the way seen in the simulations of Brada & Milgrom. But it appears to be a bit further down that path, with member stars stretched out along the orbital path. They start to trace a small portion of a much deeper gravitational potential, so the apparent velocity dispersion goes up in excess of the static prediction.
This is essentially what I inferred must be happening in the ultrafaint dwarfs of the Milky Way. There is no way that these tiny objects deep in the potential well of the Milky Way escape tidal perturbation%% in MOND. They may be stripped of their stars and their velocity dispersions mage get tidally stirred up. Indeed, Antlia 2 looks very much like the MOND prediction for the formation of tidal streams from such dwarfs made by McGaugh & Wolf (2010). Unlike dark matter models in which stars are first protected, then lost in pulses during pericenter passages, the stronger tides of MOND combined with the absence of a protective dark matter cocoon means that stars leak out gradually all along the orbit of the dwarf. The rate is faster when the external field is stronger at pericenter passage, but the mass loss is more continuous. This is a good way to make long stellar streams, which are ubiquitous in the stellar halo of the Milky Way.
So… so what?
It appears that aspects of the observations of the UDGs discussed here that seem problematic for MOND may not be as bad for the theory as they at first seem. Indeed, it appears that the noted problems may instead be a consequence of the static assumptions we usually adopt to do the analysis. The universe is a dynamic place, so we know this assumption does not always hold. One has to judge each case individually to assess whether this is reasonable or not.
In the cases of Crater 2 and Antlia 2, yes, the stranger aspects of the observations fit well with non-equilibrium effects. Indeed, the unusually large half-light radii of these low mass dwarfs may well be a result of expansion after tidal perturbation. That this might happen was specifically anticipated for Crater 2, and Antlia 2 fits the bill described by McGaugh & Wolf (2010) as anticipated by the simulations of Brada & Milgrom (2000) even though it was unknown at the time.
In the cases of DF2 and DF4, it is less clear what is going on. I’m not sure which data to believe, and I want to refrain from cherry-picking, so I’ve discussed the worst-case scenario above. But the data don’t make a heck of a lot of sense in any world view; the many hypotheses made in the dark matter context seem just as contrived and unlikely as a tidally-induced, temporary dip in the velocity dispersion that might happen in MOND. I don’t find any of these scenarios to be satisfactory.
This is a long post, and we have only discussed four galaxies. We should bear in mind that the vast majority of galaxies do as predicted by MOND; a few discrepant cases are always to be expected in astronomy. That MOND works at all is a problem for the dark matter paradigm: that it would do so was not anticipated by any flavor of dark matter theory, and there remains no satisfactory explanation of why MOND appears to happen in a universe made of dark matter. These four galaxies are interesting cases, but they may be an example of missing the forest for the trees.
*As it happens, the surface brightness threshold adopted in the definition of UDGs is exactly the same as I suggested for VLSBGs (very low surface brightness galaxies: McGaugh 1996), once the filter conversions have been made. At the time, this was the threshold of our knowledge, and I and other early pioneers of LSB galaxies were struggling to convince the community that such things might exist. Up until that time, the balance of opinion was that they did not, so it is gratifying to see that they do.
**This expectation is specific to MOND; it doesn’t necessarily hold in dark matter where the acceleration in the central regions of diffuse galaxies can be dominated by the cusp of the dark matter halo. These were predicted to exceed what is observed, hence the cusp-core problem.
+Measuring by surface brightness, Crater 2 and Antlia 2 are two orders of magnitude more diffuse than the prototypical ultradiffuse galaxies DF2 and DF4. Crater 2 is not quite large enough to count as a UDG by the adopted size definition, but Antlia 2 is. So does that make it super-ultra diffuse? Would it even be astronomy without terrible nomenclature?
&I didn’t want to use a MOND-specific criterion in McGaugh et al. (2021) because I was making a more general point, so the green points are overly conservative from the perspective of the MOND isolation criterion: there are more dwarfs for which this works. Indeed, we had great success in predicting velocity dispersions in exactly this fashion in McGaugh & Milgrom (2013a, 2013b). And XXVIII was a case not included above that we highlighted as a great test of MOND, being low mass (~4×105 M☉) but still qualifying as isolated, and its dispersion came in (6.6+2.9-2.1 km/s in one measurement, 4.9 ± 1.6 km/s in another) as predicted a priori (4.3+0.8-0.7 km/s). Hopefully the Rubin Observatory will discover many more similar objects that are truly isolated; these will be great additional tests, though one wonders how much more piling-on needs to be done.
^This is an approximation that is reasonable for the small accelerations involved. More generally we have Geff = G/μ(|gex+gin|/a0) where μ is the MOND interpolation function and one takes the vector sum of all relevant accelerations.
#This follows because the boost from MOND is limited by how far into the low acceleration regime an object is in. If the EFE is important, the boost will be less than in the isolated case. As we said in 2013, “the case that reports the lower velocity dispersion is always the formally correct one.” I mention it again here because apparently people are good at scraping equations from papers without reading the associated instructions, so one gets statements like “the theory does not specify precisely when the EFE formula should replace the isolated MOND prediction.” Yes it does. We told you precisely when the EFE formula should replace the isolated formula. It is when it reports the lower velocity dispersion. We also noted this as the reason for not giving σefe in the tables in cases it didn’t apply, so there were multiple flags. It took half a dozen coauthors to not read that. I’d hate to see how their Ikea furniture turned out.
$As often happens with LCDM, there are many nominal predictions. One common theme is that “Despite spanning four decades in luminosity, dSphs appear to inhabit halos of comparable peak circular velocity.” So nominally, one would expect a faint galaxy like Crater 2 to have a similar velocity dispersion to a much brighter one like Fornax, and the luminosity would have practically no power to predict the velocity dispersion, contrary to what we observe in the BTFR.
%There is the 2-halo term – once you get far enough from the center of a dark matter halo (the 1-halo term), there are other halos out there. These provide additional unseen mass, so can boost the velocity. The EFE in MOND has the opposite effect, and occurs for completely different physical reasons, so they’re not at all the same.
^^For arbitrary reasons of human psychology, the threshold many physicists set for “always happens” is around 100 times. That is, if a phenomenon is repeated 100 times, it is widely presumed to be a general rule. That was the threshold Vera Rubin hit when convincing the community that flat rotation curves were the general rule, not just some peculiar cases. That threshold has also been hit and exceeded by detailed MOND fits to rotation curves, and it seems to be widely accepted that this is the general rule even if many people deny the obvious implications. By now, it is also the case for apparent exceptions to MOND ceasing to be exceptions as the data improve. Unfortunately, people tend to stop listening at what they want to hear (in this case, “falsifies MOND”) and fail to pay attention to further developments.
%%It is conceivable that the ultrafaint dwarfs might elude tidal disruption in dark matter models if they reside in sufficiently dense dark matter halos. This seems unlikely given the obvious tidal effects on much more massive systems like the Sagittarius dwarf and the Magellanic Clouds, but it could in principle happen. Indeed, if one calculates the mass density from the observed velocity dispersion, one infers that they do reside in dense dark matter halos. In order to do this calculation, we are obliged to assume that the objects are in equilibrium. This is, of course, a form of static thinking: the possibility of tidal stirring that enhances the velocity dispersion above the equilibrium value is excluded by assumption. The assumption of equilibrium is so basic that it is easy to unwittingly engage in circular reasoning. I know, as I did exactly that myself to begin with.
Ha. As soon as I posted that, I ran across “All Puffed Up: Exploring Ultra-diffuse Galaxy Origins through Galaxy Interactions” (https://arxiv.org/abs/2401.01931) wherein it is suggested that “The UDGs likely originated as normal dwarf galaxies that have been subjected to significant stripping and tidal heating, causing them to become more diffuse” so I guess I’m not alone in thinking that tidal effects may play a role in puffing up the half-light radii of some ultra diffuse galaxies.
We recently analyzed the SPARC database and identified a consistent mass–energy scaling regularity in disk galaxies. By applying a simple scaling correction to the baryonic contribution, we found a robust improvement in the fit to observed rotation curves for dynamically stable systems.
Our results suggest that part of the empirical evidence usually attributed to dark matter can instead be interpreted as a scale-dependent manifestation of baryonic mass–energy.
Full dataset, scripts, and plots are openly available here:
👉 https://doi.org/10.5281/zenodo.17063011
Detection of a dark matter sub-halo near the Sun from pulsar timing
Sukanya Chakrabarti, Philip Chang, Stefano Profumo, Peter Craig
https://arxiv.org/abs/2507.16932v1
your thoughts ?
Yes, I saw that. The data look more consistent with a point mass than a DM subhalo, as they say themselves: “The pulsar accelerations are however better fit by a compact object” so I think the say the found a DM subhalo because that’s what they were looking for. If instead they had been looking for intermediate mass black holes (which is its own subfield) they would claim a detection of one of those. So it’s intriguing but far from conclusive.
The “static thinking” vs non-equilibrium dynamics may demand further scrutiny with respect to reference frames. Perhaps the need for MOND/DM only appears in those cases where the observer and observable(s) are treated as being in the same inertial reference frame, analogous to how fictitious forces arise.
Would this not be consistent with the view that a0 can be seen as an acceleration assignable to the observer?
This could lead to the suggestion that a0 is a physical realization of the wormhole connecting rotating and inertial frames.
I won’t comment on the paper that claims to find a mass-energy scaling regularity, but it’s interesting that if there’s a pattern, it doesn’t work in disturbed systems, or bulge-dominated systems. At the galaxy scale the patterns we find work without DM, and tend to work far better than putting in DM. Stacy listed some laws of galactic rotation in a set of pages from way back, which I used to read. The RAR is the main one, but there also Renzo’s rule and others.
One way to interpret MOND is to take it as a part of basic gravity that we didn’t know about until we had better data. Whether in Newton’s theory, Einstein’s, or some UT, MOND turns out to be built into the basic equations. Alternatively, MOND could be an addition at some more superficial level – more like an effect, less like a universal gravity theory. That fits with the fact that MOND wins in some areas and not others, and it seems to win at some scales and not others. It would also fit with the detailed set of rules the EFE brings, and hints that MOND can vary its rules in different places.
If a pattern doesn’t work in disturbed systems, it’s as if the system ‘needs time to settle down’ (which also came up as a possibility with tidal dwarf galaxies). That fits with PSG’s emitted medium, and with the general idea that MOND doesn’t describe a modification to gravity, but some more superficial effect, apparently more to do with the disk than the bulge – a configuration a galaxy can go into over time. If this was a modification of gravity at a deep level, it wouldn’t be so sensitive.
“… the stronger effective force of MOND increases tidal effects.” Does the MONDian effective force indicate a new concept of force and/or a new concept of inertia? Are both Milgrom & Guendelman underappreciated geniuses? Is there a FUNDAMOND string theory?
Professor Milgrom asks, “What is wrong with existing theories?” There might be 2 different forms of spacetime: 4-dimensional Einstein spacetime (consisting entirely of tardyons) & infinite-dimensional spukhafte spacetime (consisting entirely of tachyons). TITLE String theory with spukhafte Fernwirkung
Abstract : Guendelman’s idea that “each string and each brane generates its own tension” might provide a basis for understanding the empirical successes of Milgrom’s Modified Newtonian Dynamics (MOND).
Professor Milgrom of the Weizmann Institute might be considered the Kepler of contemporary cosmology, based upon MOND’s empirical successes [1]. However, Green-Schwarz-Witten string theory [2] does not seem to readily yield a mathematical justification for MOND. String theory with spukhafte Fernwirkung is a somewhat mathematical theory consisting of a mathematical part, Guendelman’s version of string theory [3], and a non-mathematical part consisting of super-luminal time combined with various speculative hypotheses. String theory with spukhafte Fernwirkung might be necessary to provide an explanation for MOND’s empirical successes. Consider the following hypotheses: (1) Gravitational energy is conserved. (2) All gravitons have spin 2. (3) There is a FUNDAMOND [4] string theory. (4) There are 2 types of time: luminal time and super-luminal time. Luminal string vibrations occur at the speed of light in 1 dimension of luminal time and 15 dimensions of space; 9 of the spatial dimensions involve Green-Schwarz-Witten string tension and 6 of the spatial dimensions involve Guendelman string tension. Super-luminal string vibrations occur in an infinite-dimensional FUNDAMOND stringy wave.
(5) Time does not move forward continuously; instead, time moves in discrete jumps, each of which lasts precisely one Planck-time interval. (6) There are two fundamental types of inertia: Newton-Einstein inertia and quantum inertia, (7) Einstein’s equivalence principle is empirically valid, but quantum inertia creates inertial influences without corresponding gravitational mass-energy influences. (8) Measurement consists of the collapse of the FUNDAMOND stringy wave. (9) The successive collapses and re-formations of the FUNDAMOND stringy wave depend upon quantum disentanglements involving Green-Schwarz-Witten string vibrations based on Green-Schwarz-Witten string tension and, also, quantum entanglements involving Guendelman string vibrations based on Guendleman string tension. (10) Guendelman’s version of string theory does (A), (B), and (C): (A) solves string theory’s swampland problem, (B) revolutionizes astrophysics by providing a mathematical approach to MOND’s empirical successes, and (C) revolutionizes high energy physics by providing a mathematical basis for a 16-dimensional stringy uncertainly principle. (11) Between every pair of Green-Schwarz-Witten string vibrations, there is a virtual tachyonic link that can be suppressed by quantum disentanglement and can be expressed by quantum entanglement. (12) Dark matter is caused by quantum disentanglement involving Newton-Einstein inertia, while dark energy is caused by quantum entanglement involving quantum inertia. (13) The luminal string vibrations with Guendelman string tension create 6 dimensions of stringy topological defects near the Planck space at every point in 4-dimensional spacetime; these hypothetical stringy defects create 6-dimensional portals for the hypothetical tachyonic links among the pairs of Green-Schwarz-Witten string vibrations. Furthermore, the stringy defects might cause slight corrections to both the metric tensor and the energy-momentum tensor. (14) Quantum inertia is fairly well approximated by MOND inertia in the MOND regime postulated by Milgrom. (15) In the standard form of Einstein’s field equations, the –1/2 needs to be replaced by
–1/2 + dark-matter-data-function and the Λ needs to be replaced by Λ + dark-energy-data-function, where the two data functions are 1-dimensional real functions that somehow arise from the hypothetical super-luminal string vibrations. References:
[1] M. Milgrom., MOND vs. dark matter in light of historical parallels, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 71, (2020) 170-195
[2] Superstrings, John H. Schwarz, Vols- 1 and 2, World Scientific, 1985 ; M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, 1987
[3] E. Guendelman, Dynamical string tension theories with target space scale invariance SSB and restoration, European Physical Journal C 85(3) (2025), 276
[4] M. Milgrom, MOND as manifestation of modified inertia, arXiv preprint, (2023) arXiv.2310.14334
https://ui.adsabs.harvard.edu/abs/2025ApJ…978…45L/abstract
Interesting news. Dark matter can according to this study only be spherically distributed in the Milky Way.
That link is broken, so in case anyone else wants to follow it, hopefully this one works: https://iopscience.iop.org/article/10.3847/1538-4357/ad94f5#artAbst
This result conforms to my impression that there isn’t much extra gravity near to the disk. Will have to read it more closely to say more.
Looking forward to your summary. Many interesting things to learn on this blog!
Am I correct in cautiously inferring the following about MOND based on this blog and on the data presented or referenced:
There is not much extra gravity near the disk (i.e. little to no DM in the plane), MOND and DM are often complementary descriptions of the data, MOND extends indefinitely radially outward, but may only exist for systems in dynamic equilibrium.
Is that about correct? What else would need to be added to summarize at a very general level the essence of the observed MOND properties?
There appears to be not much extra gravity near the disk, but there should be some in most flavors of MOND. MOND works very well as a description of the radial force. It is less clear how well it works in the vertical direction.
MOND and DM explain the same phenomena but are not complimentary as the predictions of MOND follow uniquely from the observed baryon distribution but DM is that plus an unseen component that could do pretty much anything. That’s the problem for DM: to match the observations, it has to mimic MOND. The reverse is not the case.
Ok, complementary is either incorrect or too general a term to be supported here. Maybe “mutually exclusive and noncommutative” is better?
What I find really interesting is that MOND can extend indefinitely radially and yet disappear perhaps entirely under tidal disruption. Is that a data-supported statement?
In general, I expect that tidal disruption stirs things up, increasing velocities. So I wouldn’t say it disappears entirely so much as temporarily giving the illusion of taking a break for a brief time. For an object like DF2, the pre-encounter velocity dispersion would be 13 km/s (depending on the distance yada yada) and the Newton-only v-disp is 8 km/s. These numbers are not much different to start with given the uncertainties, especially in M*/L which can be a factor of 2 (so sqrt(2) in velocity).
That there can be a temporary suppression of the velocity dispersion is based on simulations, which is pretty much the only way to assess this. The only data to test it are the data we’re talking about, but we don’t control the experiment so have a hard time knowing how applicable the available simulations are to any given dwarf.
That makes sense. Would there be any difference in simulations if MOND were treated as modified inertia vs modified gravity?
Yes, different implementations of MOND could give a different result. The differences between AQUAL and QUMOND (two different modifications of the Poisson equation for which simulation codes exist) are subtle but do exist. Modified inertia could be even more different, as it is non-local ergo trajectory-dependent. This property makes it extremely challenging to code.
“The universe is a very dynamic place – the whole thing is expanding, after all – so we need to be wary of static thinking.”
Thinking that the ‘whole thing is expanding’ is also static thinking. The Little Red Dots fit nicely into a universe that is not expanding; specifically the luminosity and the angular distances are consistent with garden variety galaxies when non-expanding cosmologies are used. (See Marmet https://ui.adsabs.harvard.edu/abs/2009ASPC..413..315M/abstract)
UDGs may be a good testing ground for theories that limit the inertial capacity of gravity in low mass systems. In these theories, systems radiate inertial energy when their is insufficient mass to facilitate momentum transfer. Normally, this keeps them aligned with the BTFR (actually contributing to the stable alignment). But in a diffused state, each star’s inertial capacity is even more limited (inertial capacity is a bulk effect); so they radiate like more massive stars, moving them off of the BTFR major axis. This would put the masses of DF2 and DF4 at about 10^7(solar).
Since we are getting into Final Jeopardy here, my answer is that the observer must be assigned an acceleration a0, likely resulting from the effective surface gravity of the Nariai black hole, and crucially that the observed frequency shift due to galactic rotation is represented in an effective inertial frame when the galaxy is in dynamic equilibrium. These are the conditions which could give rise to fictitious forces, which we then need MOND or DM to explain. This is because we don’t realize that we have to assign a special acceleration to the observer. However it was shown by Kudler-Flam et al. that this is indeed necessary: https://arxiv.org/abs/2309.15897
We don’t see MOND/DM in systems that are not in dynamic equilibrium because no inertial frame exists for the measured frequency shifts, hence no fictitious forces.
And my wager is . . . . $1.
This is a fascinating read! McGaughs insights into ultra-diffuse galaxies and the challenges they pose to both LCDM and MOND are incredibly thought-provoking. The idea that non-equilibrium dynamics could explain some of these anomalies is particularly intriguing.
Can the number of UDGs predicted by MOND be usefully compared to the number of UDGs predicted by LCDM?
Do the following findings significantly favor MOND?
“Early assembly of the most massive galaxies” by Chris A. Collins et al., published in “Nature”, 2nd April 2009
https://arxiv.org/abs/0904.0006
I had not been aware of this particular paper, but it is one of many to point out that there were already a difficulty posed for CDM structure formation prior to JWST pushing that to even higher redshift.
So yes, all these findings favor the rapid formation of massive galaxies predicted by Sanders (1998) using MOND.
Vertical velocities have looked Newtonian before, this paper seems to strengthen that possibility (you once mentioned that it’s hard to determine, but that they might be somewhere between Milgrom and Newton).
But what’s the geometry of the MOND prediction? Is this about the disk’s field in the vertical direction anywhere on the disk – it might cross over into the MOND regime somewhere above and below the plane.
Or is it about what happens beyond the galaxy’s overall MOND radius, which is a large circle. Does MOND predict 1/r gravity in all directions beyond that radius?
There have been hints of a directional aspect before, certainly in PSG, but someone else mentioned it – it would look more like some kind of effect if this was so.
The vertical force is a secondary effect that can be theory-specific. For example, QUMOND predicts that we would infer a thick disk of phantom dark matter near the Galactic plane as well as a more spherical distribution further out. Modified inertia theories don’t necessarily do that; for them the interpolation function (the shape of the RAR) is potentially specific to the orbital geometry (quasi-circular orbits in the thin cylinder of a galactic disk) – it settles down to that – but assuming the form of that RAR applies everywhere could be incorrect (one reason I’ve been careful to call the RAR the RADIAL acceleration relation) so assuming the vertical force gets amplified in the same way might not be right. So I’d expect an enhanced vertical force, but I’m not clear on exactly what form that should take. In principle, mapping it out observationally should help discriminate between flavors of MOND theory.
I don’t have anything specific for this, but if it’s of interest, looking at possible differences between the radial and vertical directions in PSG, the radial small-scale waves have travelled further, and are more aligned in their direction of travel. In the vertical direction most of the contribution is from the fields of nearby stars, a lot fewer of them.
I don’t know if there might be a distance cutoff point, it depends on wide binaries – Banik suggested something like that for if WBs are Newtonian, but it’s undecided.
In PSG the waves’ alignment seems a more likely cause, and perhaps the number of fields that combine. GM/r^2 = a0 can arise from a few stronger fields, or many weaker fields. In the radial direction there’s a wide range of different field strengths – they’ve been travelling a range of distances, and perhaps that mix affects the self-interaction of the medium.
“The universe is a very dynamic place …” How is Milgromian dynamics related to Einsteinian dynamics? How is MOND related to Hawking radiation and Bekenstein’s no-scalar-hair theorem?
https://en.wikipedia.org/wiki/Hawking_radiation
Bekenstein, J. D. (1995). Novel ‘‘no-scalar-hair’’theorem for black holes. Physical Review D, 51(12), R6608. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.51.R6608
I have suggested some hypotheses concerning a FUNDAMOND string theory. However, the suggested hypotheses fail to answer 2 questions: Why are there 6 basic quarks (generating 18 quarks and 18 antiquarks)? Why are there 3 generations of fermions? There might be a FUNDAMOND-AHAB (addendum-Hawking-and-Bekenstein) string theory in which there are 3 fundamental string tensions (Green-Schwarz-Witten string tension, Guendelman string tension, and AHAB string tension that yields a stringification of Calabi-Yau manifolds, justifying black-hole quantum entanglement. There might be 3 generations of fermions because there are 3 fundamental types of string tension. There might be 1 stringy temporal dimension and 21 stringy spatial dimensions. To the Standard Model of particle physics add the graviton, the axion, and the inflaton. The masses & mixings of the extended Standard Model of particle physics might require 20 parameters + 1 parameter for the axion mass + 1 parameter for the inflaton mass. The 22 parameters might correspond to 22 stringy dimensions. There might be 6 basic quarks because AHAB string tension allows 6-dimensional Calabi-Yau manifolds to explain how quark stars collapse into black holes.
https://en.wikipedia.org/wiki/Calabi–Yau_manifold
My dollar says MOND, the BTFR, starburst regions and the gradient in the absolute magnitude of supernovas within galaxies are inter-related in ways that will eventually tie everything together. I am optimistic that cosmology will emerge from the tired dark stuff that has us dancing on the ends of pins, or little red dots, if you prefer.
Stacy is a pioneer in this transition: His remarkable defense of MOND is an inspiration. The key is, and always has been, recognizing root false narratives when they are not longer supported by data.
I don’t see myself as a defender of MOND so much as a defender of the scientific method. A seemingly unlikely theory made a proiri predictions that came true. The standard paradigm did not.
Hi Stacy. Thank you for this post on “Non-equilibrium dynamics in galaxies that appear to lack dark matter: ultra-diffuse galaxies”. As you mention, it is indeed a long read.
You make use of two diagrams: (a) size vs baryonic mass; (b) velocity vs baryonic mass (BTFR: baryonic Tully-Fisher relation). My interpretation is that encounters with larger galaxies cause the ultra-diffuse galaxies (UFGs) to move up & then back down in the size vs mass diagram (i.e. they expand and then contract), and down & then back up in the BTFR diagram. The implication being the galaxies lying below the BTFR line are possibly not in equilibrium.
In this post it is only the few UFGs that lie below the line in the BTFR diagram that are plotted, whereas in your “McGaugh et al (2021)” paper there are lots of objects below the BTFR line making the UFGs not look that exceptional.
Separately, I am intrigued that the pressure supported dwarf spheroidal galaxies lie on the same BTFR line as the rotationally supported galaxies. I suspect that a closer reading would show that this is exactly what MOND predicts.
I also note that you work with the baryonic masses and that should any dark matter mass be included then the galaxies would move horizontally to higher masses and that this would not improve anything.
Yes, I think the ultrafaints are likely nearing the end of dissolution, so their measured v-disps have little to do with any equilibrium value. I had hoped to talk about this here but the post was already too long. I did talk about it before in https://tritonstation.com/2018/09/12/dwarf-satellite-galaxies-ii/
Every object is an individual. One really needs to simulate each particular case to see what is “should” be doing under the influence of the massive host given its orbital history. Then one has to worry about the many ways in which things can go amiss – there are zillions of ways in which the velocity dispersion can be overestimated even for a system in equilibrium. Even perfect observations may mistake the velocities of binary stars for part of the overall potential; that seems inevitably to be a potential source for an additional 2 or 3 km/s unless you re-observe each star many times to see if its velocity changes (because it is orbiting a partner) and by how much.
That’s why I mention that it would be better to find lots of examples of isolated low mass dwarfs. Of course we already have some, ranging from Leo P to And XXVIII to And IV and so far isolated MOND always works for those.
“… defense of MOND …” At this stage of evidence accumulation, I doubt that MOND needs defense. Instead, there should be aggressive pursuit of FUNDAMOND. In a quantum theory of gravity in which all gravitons have spin 2, there should be a concept of quantum event horizons (QEHs). Consider 6 conjectures: (1) There are 3 fundamental levels of physics: classical field theory, quantum field theory, and FUNDAMOND-QEH string theory. (2) There are 3 fundamental types of inertia: Newton-Einstein inertia, FUNDAMOND inertia, and QEH inertia. (3) FUNDAMOND inertia is essential for fully understanding how quantum entanglement involves particles. (4) QEH inertia is essential for fully understanding how quantum entanglement involves QEHs. (5) Very near to QEHs, the concepts of spacetime and energy fail and should be replaced by 32 dimensions of stringy uncertainty. (6) The group SO(32) operates on 32 dimensions of stringy uncertainty. Are the preceding conjectures wrong? How can we understand FUNDAMOND & convince string theorist to pursue an understanding of FUNDAMOND?