Non-equilibrium dynamics in galaxies that appear to lack dark matter: ultradiffuse galaxies

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 size (radius encompassing half of the total light) and stellar mass of Local Group dwarf spheroidals (green points selected by McGaugh et al. 2021 to be relatively safe from external perturbation) along with two more Local Group dwarfs that are subject to the EFE (Crater 2 and Antlia 2) and the two UDGs NGC 1052-DF2 and DF4. Dotted lines show loci of constant surface density. For reference, the solar neighborhood has ~40 M_☉ pc^-2; the centers of high surface brightness galaxies frequently exceed 1,000 M_☉ pc^-2.

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, g_in, exceeds that from external sources, g_ex: g_in ≫ g_ex. For a pressure supported dwarf, g_in ≈ 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 g_ex = V_f²/D where V_f 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 g_in ≈ g_ex, 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 g_ex ≪ g_in ≪ a₀) that we can pretend they are isolated, and the above equation applies. Such dwarfs should^& fall on the BTFR, which they do:

The baryonic Tully-Fisher relation (BTFR) including pressure supported dwarfs (green points) with their measured velocity dispersions matched to the flat rotation speeds of rotationally supported galaxies (blue points) via the prescription of *McGaugh et al. (2021)*. The large blue points are rotators in the Local Group (with Andromeda and the Milky Way up near the top); smaller points are spirals with direct distance measurements (Schombert et al. 2020). The Local Group dwarfs assessed to be safe from external perturbation are on the BTFR (for V_f = 2σ); Crater 2 and the UDGs near NGC 1052 are not.

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 (g_in ≪ g_ex ≪ 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: G_eff ≈ G(a₀/g_ex). An easy way to tell which regime an object is in is to calculate both velocity dispersions σ_iso and σ_efe: 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 g_ex. 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 σ_efe 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 M_halo ~ V_halo³ that differs from the observed M_b ~ V_f⁴. 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 σ_efe applies rather than σ_iso, and no reason to expect via feedback that this distinction has anything to do with the dynamical accelerations g_in and g_ex.

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:

**Fig. 2** of Brada & Milgrom (2000): showing the numerically calculated (dotted line) variation of radius (left) and characteristic velocity (right) for a dwarf on a mildly eccentric orbit (peri- and apocenter of roughly 60 and 90 kpc, respectively, for a Milky Way-like host). Also shown is the variation in the EFE as the dwarf’s distance from the host varies (solid line). Dwarfs go through a breathing mode of increasing/decreasing size and decreasing/increasing velocity dispersion in phase with the orbit. If this process is adiabatic, it tracks the solid line and the static EFE approximation holds. This is not always the case in the simulation, so applying our usual assumption of dynamical equilibrium will result in an error stipulated by the difference between the dotted and solid lines. The amplitude of this error depends on the size, mass, and orbital history of each and every dwarf satellite.

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):

**Fig. 4** of Brada & Milgrom (2000): The variation of the size and characteristic velocity of dwarf models of different mass. The more massive models approximate the adiabatic limit, which gradually breaks down for the lowest mass models. In this example, the m = 1 and 2 models explode, with the scale size growing gradually without recovering.

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.

**Fig. 5** of Brada & Milgrom (2000): AQUAL numerical simulations dwarf satellites orbiting a more massive host galaxy. The parameter m describes the mass and effective surface density of the satellite; all the satellites are in the MOND regime and subject to the external field of the host galaxy, which exceeds their internal accelerations. In dimensionless simulation units, m = 5 x 10^-5, which for a satellite of the Milky Way corresponds roughly to a stellar mass of 3 x 10⁶ *M_☉*. For real dwarf satellite galaxies, the scale size is also relevant, but the sequence of m above suffices to illustrate the increasingly severe effects of the external field as m decreases.

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.

The dotted line shows a possible evolutionary track for Crater 2 as it expands after pericenter passage. Its initial condition would have been amongst the other dwarf spheroidals. It could also have lost some mass in the process, so any of the green low-mass dwarfs might be similar to the progenitor.

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.

**Fig. 9** from Ji et al. (2021) showing tidal features in Antlia 2 considering the effects of the Milky Way alone (left panel) and of the Milky Way and the Large Magellanic Cloud together (central panel) along with the position-velocity diagram from individual stars (right panel). The object is clearly not the isotropic, spherical cow presumed by the static equation for the velocity dispersion. Indeed, it is elongated as would be expected from tidal effects, with individual member stars apparently leaking out.

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 G_eff = G/μ(|g_ex+g_in|/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 σ_efe 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.

Non-equilibrium dynamics in galaxies that appear to lack dark matter: tidal dwarf galaxies

There are a number of galaxies that have been reported to lack dark matter. This is weird in a universe made of dark matter. It is also weird in MOND, which (if true) is what causes the inference of dark matter. So how can this happen?

In most cases, it doesn’t. These claims not only don’t make sense in either context, they are simply wrong. I don’t want to sound too harsh, as I’ve come close to making the same mistake myself. The root cause of this mistake is often a form of static thinking in dynamic situations that the here and now is always a representative test. The basic assumption we have to make to interpret observed velocities in terms of mass is that systems are in (or close to) gravitational equilibrium so that the kinetic energy is a measure of the gravitational potential. In most places, this is a good assumption, so we tend to forget we even made it.

However, no assumption is ever perfect. For example, Gaia has revealed a wealth of subtle non-equilibrium effects in the Milky Way. These are not so large as to invalidate the basic inference of the mass discrepancy, but neither can they be entirely ignored. Even maintaining the assumption of a symmetric but non-smooth mass profile in equilibrium complicates the analysis.

Since the apparent absence of dark matter is unexpected in either theory, one needs to question the assumptions whenever this inference is made. There is one situation in which it is expected, so let’s consider that special case:

Tidal dwarf galaxies

Most dwarf galaxies are primordial – they are the way they are because they formed that way. However, it is conceivable that some dwarfs may form in the tidal debris of collisions between large galaxies. These are tidal dwarf galaxies (TDGs). Here are some examples of interacting systems containing candidate TDGs:

**Fig. 1** from Lelli et al. (2015): *images of interacting systems with TDG candidates noted in yellow.*

I say candidate TDGs because it is hard to be sure a particular object is indeed tidal in origin. A good argument can be made that TDGs require such special conditions to form that perhaps they should not be able to form at all. As debris in tidal arms is being flung about in the (~ 200 km/s) potential well of a larger system, it is rather challenging for material to condense into a knot with a much smaller potential well (< 50 km/s). It can perhaps happen if the material in the tidal stream is both lumpy (to provide a seed to condense on) and sufficiently comoving (i.e., the tidal shear of the larger system isn’t too great), so maybe it happens on rare occasions. One way to distinguish TDGs from primordial dwarfs is metallicity: typical primordial dwarfs have low metallicity while TDGs have the higher metallicity of the giant system that is the source of the parent material.

A clean test of hypotheses

TDGs provide an interesting test of dark matter and MOND. In the vast majority of dark matter models, dark matter halos are dynamically hot, quasi-spherical systems with the particles that compose the dark matter (whatever it is) on eccentric, randomly oriented orbits that sum to a big, messy blob. Arguably it has to be this way in order to stabilize the disks of spiral galaxies. In contrast, the material that composes the tidal tails in which TDGs form originates in the baryonic material of the dynamically cold spiral disks where orbits are nearly circular in the same direction in the same thin plane. The phase space – the combination of position x,y,z and momentum v_x,v_y,v_z – of disk and halo couldn’t be more different. This means that when two big galaxies collide or have a close interaction, everything gets whacked and the two components go their separate ways. Starting in orderly disks, the stars and gas make long, coherent tidal tails. The dark matter does not. The expectation from these basic phase space considerations is consistent with detailed numerical simulations.

We now have a situation in which the dark matter has been neatly segregated from the luminous matter. Consequently, if TDGs are able to form, they must do it only* with baryonic mass. The ironic prediction of a universe dominated by dark matter is that TDGs should be devoid of dark matter.

In contrast, one cannot “turn off” the force law in MOND. MOND can boost the formation of TDGs in the first place, but if said TDGs wind up in the low acceleration regime, they must evince a mass discrepancy. So the ironic prediction here is that, in ignorance of MOND, MOND means that we would infer that TDGs do have dark matter.

Got that? Dark matter predicts TDGs with no dark matter. MOND predicts TDGs that look like they do have dark matter. That’s not confusing at all.

Clean in principle, messy in practice

Tests of these predictions have a colorful history. Bournaud et al. (2007) did a lovely job of combining simulations with observations of the Seashell system (NGC 5291 above) and came to a striking conclusion: the rotation curves of TDGs exceeded that expected for the baryons alone:

**Fig. 2** from Bournaud et al. (2007) *showing the rotation curves for the three TDGs identified in the image above.*

This was a strange, intermediary result. TDGs had more dark matter than the practically zero expected in LCDM, but less than comparable primordial dwarfs as expected in MOND. That didn’t make sense in either theory. They concluded that there must be a component of some other kind of dark matter that was not the traditional dark halo, but rather part of the spiral disk to begin with, perhaps unseen baryons in the form of very cold molecular gas.

Gentile et al. (2007) reexamined the situation, and concluded that the inclinations could be better constrained. When this was done, the result was more consistent with the prediction of MOND and the baryonic Tully-Fisher relation (BTFR. See their Fig. 2).

**Fig. 1** from Gentile et al. (2007): Rotation curve data (full circles) of the 3 tidal dwarf galaxies (Bournaud et al. 2007). The lower (red) curves are the Newtonian contribution V_bar of the baryons (and its uncertainty, indicated as dotted lines). The upper (black) curves are the MOND prediction and its uncertainty (dotted lines). The top panels have as an implicit assumption (following Bournaud et al.) an inclination angle of 45 degrees. In the middle panels the inclination is a free parameter, and the bottom panels show the fits made with the first estimate for the external field effect (EFE).

Clearly there was room for improvement, both in data quality and quantity. We decided to have a go at it ourselves, ultimately leading to Lelli et al. (2015), which is the source of the pretty image above. We reanalyzed the Seashell system, along with some new TDG candidates.

Making sense of these data is not easy. TDG candidates are embedded in tidal features. It is hard to know where the dwarf ends and the tidal stream begins, or even to be sure there is a clear distinction. Here is an example of the northern knot in the Seashell system:

**Fig. 5** from Lelli et al. (2015): *Top panels*: optical image (*left*), total H I map (*middle*), and H I velocity field (*right*). The dashed ellipse corresponds to the disc model described in Sect. 5.1. The cross and dashed line illustrate the kinematical centre and major axis, respectively. In the bottom-left corner, we show the linear scale (optical image) and the H I beam (total H I map and velocity field) as given in Table 6. In the total H I map, contours are at ~4.5, 9, 13.5, 18, and 22.5 M_⊙ pc^-2. *Bottom panels*: position-velocity diagrams obtained from the observed cube (*left*), model cube (*middle*), and residual cube (*right*) along the major and minor axes. Solid contours range from 2σ to 8σ in steps of 1σ. Dashed contours range from −2σ to −4σ in steps of −1σ. The horizontal and vertical lines correspond to the systemic velocity and dynamical centre, respectively.

Both the distribution of gas and the velocities along the tidal tail often blend smoothly across TDG candidates, making it hard to be sure they have formed a separate system. In the case above, I can see what we think is the velocity field of the TDG alone (contained by the ellipse in the upper right panel), but is that really an independent system that has completely decoupled from the tidal material from which it formed? Definite maybe!

Federico Lelli did amazing work to sort through these difficult-to-interpret data. At the end of the day, he found that there was no need for dark matter in any of these TDG candidates. The amplitude of the apparent circular speed was consistent with the enclosed mass of baryons.

**Figs. 11 and 13** from Lelli et al. (2015): the enclosed dynamical-to-baryonic mass ratio (left) and baryonic Tully-Fisher relation (right). TDGs (red points) are consistent with a mass ratio of unity: the observed baryons suffice; no dark matter is inferred. Contrary to Gentile et al., this manifests as a clear offset from the BTFR followed by normal galaxies.

Taken at face value, this absence of dark matter is a win for a universe made of dark matter and a falsification of MOND.

So we were prepared to say that, and did, but as Federico checked the numbers, it occurred to him to check the timescales. Mergers like this happen over the course of a few hundred million years, maybe a billion. The interactions we observe are ongoing; just how far into the process are they? Have the TDGs had time to settle down into dynamical equilibrium? That is the necessary assumption built into the mass ratio plotted above: the dynamical mass assumes the measured speed is that of a test particle in an equilibrium orbit. But these systems are manifestly not in equilibrium, at least on large scales. Maybe the TDGs have had time to settle down?

We can ask how long it takes to make an orbit at the observed speed, which is low by the standards of such systems (hence their offset from Tully-Fisher). To quote from the conclusions of the paper,

These [TDG] discs, however, have orbital times ranging from ~1 to ~3 Gyr, which are significantly longer than the TDG formation timescales (≲1 Gyr). This raises the question as to whether TDGs have had enough time to reach dynamical equilibrium.
Lelli et al. (2015)

So no, not really. We can’t be sure the velocities are measuring the local potential well as we want them to do. A particle should have had time to go around and around a few times to settle down in a new equilibrium configuration; here they’ve made 1/3, maybe 1/2 half of one orbit. Things have not had time to settle down, so there’s not really a good reason to expect that the dynamical mass calculation is reliable.

It would help to study older TDGs, as these would presumably have had time to settle down. We know of a few candidates, but as systems age, it becomes harder to gauge how likely they are to be legitimate TDGs. When you see a knot in a tidal arm, the odds seem good. If there has been time for the tidal stream to dissipate, it becomes less clear. So if such a thing turns out to need dark matter, is that because it is a TDG doing as MOND predicted, or just a primordial dwarf we mistakenly guessed was a TDG?

We gave one of these previously unexplored TDG candidates to a grad student. After much hard work combining observations from both radio and optical telescopes, she has demonstrated that it isn’t a TDG at all, in either paradigm. The metallicity is low, just as it should be for a primordial dwarf. Apparently it just happens to be projected along a tidal tail where it looks like a decent candidate TDG.

This further illustrates the trials and tribulations we encounter in trying to understand our vast universe.

*One expects cold dark matter halos to have subhalos, so it seems wise to suspect that perhaps TDGs condense onto these. Phase space says otherwise. It is not sufficient for tidal debris to intersect the location of a subhalo, the material must also “dock” in velocity space. Since tidal arms are being flung out at the speed that is characteristic of the giant system, the potential wells of the subhalos are barely speed bumps. They might perturb streams, but the probability of them being the seeds onto which TDGs condense is small: the phase space just doesn’t match up for the same reasons the baryonic and dark components get segregated in the first place. TDGs are one galaxy formation scenario the baryons have to pull off unassisted.

The minimum acceleration in intergalactic space

A strange and interesting aspect of MOND is the External Field Effect (EFE). If physics is strictly local, it doesn’t matter what happens outside of an experimental apparatus, only inside it. Examples of gravitational experiments include an Eötvös-style apparatus in a laboratory or a dwarf galaxy in space: in each case, test masses/stars respond to each other’s gravity.

The MOND force depends on the acceleration from all sources; it is not strictly local. Consequently, the results of a gravitational experiment depend on the environment in which it happens. An Eötvös experiment sitting in a laboratory on the surface of the Earth feels the one gee of acceleration due to the Earth and remains firmly in the Newtonian regime no matter how small an inter-particle acceleration experimenters achieve within the apparatus. This is the way in which MOND breaks the strong equivalence principle (but not the weak or Einstein equivalence principle).

A dwarf galaxy in the depths of intergalactic space behaves differently from an otherwise identical dwarf that is the satellite of a giant galaxy. In the isolated case, only the dwarf’s internal acceleration matters. For the typical low surface brightness dwarf galaxy, the internal acceleration g_in due to self-gravity is deep in the MOND regime (g_in < a₀). In contrast, a dwarf satellite with the same internal acceleration g_in is also subject to an external orbital acceleration g_ex around the host that may be comparable to or even greater than its internal acceleration. Both of those accelerations matter, so the isolated case (g_in < a₀) is deeper in the MOND regime and will evince a larger acceleration discrepancy than when the same dwarf is proximate to a giant galaxy and in the EFE regime (g_in < g_in+g_ex < a₀)*. This effect is observed in the dwarfs of the Local Group.

The same effect holds everywhere in the universe. There should be a minimum acceleration due to the net effect of everything: galaxies, clusters, filaments in the intergalactic medium (IGM); anything and everything add up to a nonzero acceleration everywhere. I first attempted to estimate this myself in McGaugh & de Blok (1998), obtaining ~0.026 Å s^-2, which is about 2% of a₀ (1.2 Å s^-2). This is a tiny fraction of a tiny number, but it is practically⁺ never zero: it’s as low as you can go, an effective minimum acceleration experienced even in the darkest depths of intergalactic space.

One can do better nowadays. The community has invested a lot in galaxy surveys; one can use those to construct a map of the acceleration the observed baryonic mass predicts in MOND. We did this in Chae et al. (2021) using a number of surveys. This gets us more than just a mean number as I guestimated in 1998, but also a measure of its variation.

Here is a map of the expected Newtonian acceleration across the sky for different ranges of distance from us. Blue is low acceleration; yellow higher. Glossing over some minor technical details, the corresponding MONDian acceleration is basically the square root (a0 g_N)^1/2, so 2% of a₀ corresponds to log(e_N) = -3.4 in the following plots, where e_N is the Newtonian environmental acceleration: what Newton would predict for the visible galaxies alone.

**Figure 4** from Chae et al. (2021): All-sky distributions of the environmental field acceleration g_Ne,env from 2M++ galaxies and MCXC clusters in Mollweide projection and equatorial coordinates averaged across various distance ranges. The locations of SPARC galaxies with independent estimates of g_Ne from RC fits are shown as points with color reflecting stronger (red) or weaker (blue) EFE and with the opacity of each point increasing with its accuracy.

The EFE imposes an effect on all objects, even giant galaxies, which we were trying to estimate in Chae et al. (2021) – hence the dots in the above maps. Each of those dots is a galaxy for which we had made an estimate of the EFE from its effect on the rotation curve. This is a subtle effect that is incredibly hard to constrain, but there is a signal when all galaxies are considered statistically in aggregate. It does look like the EFE is at work, but we can’t yet judge whether its variation from place to place matches the predicted map. Still, we obtained values for the acceleration in intergalactic space that are in the same realm as my crude early estimate.

Here’s another way to look at it. The acceleration is plotted as a function of distance, with the various colors corresponding to different directions on the sky. So where above we’re looking at maps of the sky in different distance bins, here we’re looking as a function of distance but relying on the color bar to indicate different directions. There is a fair amount of variation: some places have more structure and others less with a corresponding variation in the acceleration field.

**Figure 5** from Chae et al. (2021): Variation of e_N,env with distance for the galaxies in the NSA and Karachentsev catalogs. Individual galaxies are color-coded by right ascension (RA). The black lines show the mean trend (solid) and standard deviation (dashed) in bins of distance. This plot assumes the “max clustering” model for the missing baryons (see Figure 6, below).

Different catalogs have been used to map the structure here, but the answer comes out pretty much the same, but for one little (big) detail: how clustered are the baryons? The locations of the galaxies have been well mapped, so we can turn that into a map of their gravitational field. But we also know that galaxies are not the majority of the baryons. So where are the rest? Are they clustered like galaxies, or spread uniformly through intergalactic space?

When we did this, we knew a lot of baryons were in the IGM, but it really wasn’t clear how clustered they might be. So we took two limiting cases by assuming (1) all the baryons were as clustered as the galaxies or (2) not clustered at all, just a uniform background. This makes a difference since a uniform background, being uniform, doesn’t contribute. There’s as much force from this direction as that, and it cancels itself out, leading to a lower overall amplitude for the environmental acceleration field.

**Figure 6** from Chae et al. (2021): Variation of e_N,env with distance for the SPARC galaxies within the NSA footprint. The “max clustering” model (blue) assumes that missing baryons are effectively coincident with observed structures, while the “no clustering” model (orange) distributes them uniformly in space. See Section 3.2.1 for details.

That’s where the new result reported last time comes in. We now know that the missing baryons were all in the IGM. Indeed, the split is 1/4 clustered, 3/4 not. So something closer to (2), the “no clustering” limit above. That places the minimum acceleration in intergalactic space around log(e_N) = -3.5, which is very close to the 2% of a₀ that I estimated last century.

The diffuse IGM is presumably not perfectly uniform. There are large scale filaments and wall around giant voids. This structure will contribute variations in the local minimum acceleration, as visualized in this MOND structure formation simulation by Llinares (2008):

**Figures 7 & 9** of Llinares (2008): The simulated density field (left) and modulus (right) of the MONDian force |∇Φ_M | at z = 0 normalized by g²a₀. For values above 1 the particles are in the Newtonian regime whereas values below 1 indicate the MOND regime.

Surveys for fast radio bursts are very sensitive to variations in the free electron density along the line of sight. Consequently, they can be used to map out structure in the IGM. The trick is that we need to cover lots of the sky with them – the denser the tracers, the better. That means discovering lots of them all over the sky, a task the DSA-110 was built to do.

I sure hope NSF continues to fund it.

*I write g_in < g_in+g_ex for simplicity, but strictly speaking the acceleration is a vector quantity so it is possible for the orientation of g_in and g_ex to oppose one another so that their vector sum cancels out. This doesn’t happen often, but in periodic orbits it will always happen at some moment, with further interesting consequences. The more basic point is that the amplitude of the discrepancy scales with the ratio a₀/g: the lower the acceleration g, the bigger the discrepancy from Newton – or, equivalently, the more dark matter we appear to need. The discrepancy of the isolated case a₀/g_in is larger than the discrepancy of the non-isolated case a₀/(g_in+g_ex) just because g_in+g_ex > g_in.

⁺A test of MOND using Lyman-alpha clouds was proposed by Aguirre et al (2001). These tiny puffs of intergalactic gas have very low internal accelerations, so should evince much larger discrepancies than observed. Or at least that was their initial interpretation, until I pointed out that the EFE from large scale structures would be the dominant effect. They argued it was still a problem, albeit a much smaller one than initially estimated. I don’t think it is a problem at all, because the amplitude of the EFE is so uncertain. Indeed, they made an estimate of the EFE at the relevant redshifts that depended on the rate of structure formation being conventional, which it is not in MOND. Lyman-alpha clouds are entirely consistent with MOND when one takes into account the more rapid growth of structure.

NEIU debate: dark matter or modified gravity

As promised, the folks at NEIU have posted the video of my discussion with Scott Dodelson last week, so here you go:

I am in the midst of writing a related post on cosmic tensions, so hopefully I can post that soon as well.

Dark Matter or Modified Gravity? A virtual panel discussion

This is a quick post to announce that on Monday, April 7 there will be a virtual panel discussion about dark matter and MOND involving Scott Dodelson and myself. It will be moderated by Orin Harris at Northeastern Illinois University starting at 3pm US Central time*. I asked Orin if I should advertise it more widely, and he said yes – apparently their Zoom set up has a capacity for a thousand attendees.

See their website for further details. If you wish to attend, you need to register in advance.

*That’s 4PM EDT to me, which is when I’m usually ready for a nap.

Things I don’t understand in modified dynamics (it’s cosmology)

I’ve been busy, and a bit exhausted, since the long series of posts on structure formation in the early universe. The thing I like about MOND is that it helps me understand – and successfully predict – the dynamics of galaxies. Specific galaxies that are real objects: one can observe this particular galaxy and predict that it should have this rotation speed or velocity dispersion. In contrast, LCDM simulations can only make statistical statements about populations of galaxy-like numerical abstractions, they can never be equated to real-universe objects. Worse, they obfuscate rather than illuminate. In MOND, the observed centripetal acceleration follows directly from that predicted by the observed distribution of stars and gas. In simulations, this fundamental observation is left unaddressed, and we are left grasping at straws trying to comprehend how the observed kinematics follow from an invisible, massive dark matter halo that starts with the NFW form but somehow gets redistributed just so by inadequately modeled feedback processes.

Simply put, I do not understand galaxy dynamics in terms of dark matter, and not for want of trying. There are plenty of people who claim to do so, but they appear to be fooling themselves. Nevertheless, what I don’t like about MOND is the same thing that they don’t like about MOND which is that I don’t understand the basics of cosmology with it.

Specifically, what I don’t understand about cosmology in modified dynamics is the expansion history and the geometry. That’s a lot, but not everything. The early universe is fine: the expanding universe went through an early hot phase that bequeathed us with the relic radiation field and the abundances of the light elements through big bang nucleosynthesis. There’s nothing about MOND that contradicts that, and arguably MOND is in better agreement with BBN than LCDM, there being no tension with the lithium abundance – this tension was not present in the 1990s, and was only imposed by the need to fit the amplitude of the second peak in the CMB.

But we’re still missing some basics that are well understood in the standard cosmology, and which are in good agreement with many (if not all) of the observations that lead us to LCDM. So I understand the reluctance to admit that maybe we don’t know as much about the universe as we think we do. Indeed, it provokes strong emotional reactions.

*Screenshot from* Dr. Strangelove paraphrasing Major Kong (original quote at top).

So, what might the expansion history be in MOND? I don’t know. There are some obvious things to consider, but I don’t find them satisfactory.

The Age of the Universe

Before I address the expansion history, I want to highlight some observations that pertain to the age of the universe. These provide some context that informs my thinking on the subject, and why I think LCDM hits pretty close to the mark in some important respects, like the time-redshift relation. That’s not to say I think we need to slavishly obey every detail of the LCDM expansion history when constructing other theories, but it does get some things right that need to be respected in any such effort.

One big thing I think we should respect are constraints on the age of the universe. The universe can’t be younger than the objects in it. It could of course be older, but it doesn’t appear to be much older, as there are multiple, independent lines of evidence that all point to pretty much the same age.

Expansion Age: The first basic is that if the universe is expanding, it has a finite age. You can imagine running the expansion in reverse, looking back in time to when the universe was progressively smaller, until you reach an incomprehensibly dense initial phase. A very long time, to be sure, but not infinite.

To put an exact number on the age of the universe, we need to know its detailed expansion history. That is something LCDM provides that MOND does not pretend to do. Setting aside theory, a good ball park age is the Hubble time, which is the inverse of the Hubble constant. This is how long it takes for a linearly expanding, “coasting” universe to get where it is today. For the measured H₀ = 73 km/s/Mpc, the Hubble time is 13.4 Gyr. Keep that number in mind for later. This expansion age is the metric against which to compare the ages of measured objects, as discussed below.

Globular Clusters: The most famous of age constraints is provided by the ancient stars in globular clusters. One of the great accomplishments of 20th century astrophysics is a masterful understanding of the physics of stars as giant nuclear fusion reactors. This allows us to understand how stars of different mass and composition evolve. That, in turn, allows us to put an age on the stars in clusters. Globulars are the oldest of clusters, with a mean age of 13.5 Gyr (Valcin et al. 2021). Other estimates are similar, though I note that the age determinations depends on the distance scale, so keeping them rigorously separate from Hubble constant determinations has historically been a challenge. The covariance of age and distance renders the meaning of error bars rather suspect, but to give a flavor, the globular cluster M92 is estimated to have an age of 13.80±0.75 Gyr (Jiaqi et al. 2023).

Though globular clusters are the most famous in this regard, there are other constraints on the age of the contents of the universe.

White dwarfs: White dwarfs are the remnants of dead stars that were never massive enough to have exploded as supernova. The over/under line for that is about 8 solar mass; the oldest white dwarfs will be the remnants of the first stars that formed just below this threshold. Such stars don’t take long to evolve, around 100 Myr. That’s small compared to the age of the universe, so the first white dwarfs have just been cooling off ever since their progenitors burned out.

As the remnants of the incredibly hot cores of former stars, white dwarfs star off hot but cool quickly by radiating into space. The timescale to cool off can be crudely estimated from first principles just from the Stefan-Boltzmann law. As with so many situations in astrophysics, some detailed radiative transfer calculations are necessary to get the answer right in detail. But the ballpark of the back-of-the-envelope answer is not much different from the detailed calculation, giving some confidence in the procedure: we have a good idea of how long it takes white dwarfs to cool.

Since white dwarfs are not generating new energy but simply radiating into space, their luminosity fades over time as their surface temperature declines. This predicts that there will be a sharp drop in the numbers of white dwarfs corresponding to the oldest such objects: there simply hasn’t been enough time to cool further. The observational challenge then becomes finding the faint edge of the luminosity function for these intrinsically faint sources.

Despite the obvious challenges, people have done it, and after great effort, have found the expected edge. Translating that into an age, we get 12.5+1.4/-3.5 Gyr (Munn et al. 2017). This seems to hold up well now that we have Gaia data, which finds J1312-4728 to be the oldest known white dwarf at 12.41±0.22 Gyr (Torres et al. 2021). To get to the age of the universe, one does have to account for the time it takes to make a white dwarf in the first place, which is of order a Gyr or less, depending on the progenitor and when it formed in the early universe. This is pretty consistent with the ages of globular clusters, but comes from different physics: radiative cooling is the dominant effect rather than the hydrogen fusion budget of main sequence stars.

Radiochronometers: Some elements decay radioactively, so measuring their isotopic abundances provides a clock. Carbon-14 is a famous example: with a half-life of 5,730 years, its decay provides a great way to date the remains of prehistoric camp sites and bones. That’s great over some tens of thousands of years, but we need something with a half-life of order the age of the universe to constrain that. One such isotope is ²³²Thorium, with a half life of 14.05 Gyr.

Making this measurement requires that we first find stars that are both ancient and metal poor but with detectable Thorium and Europium (the latter providing a stable a reference). Then one has to obtain a high quality spectrum with which to do an abundance analysis. This is all hard work, but there are some examples known.

Sneden‘s star, CS 22892-052, fits the bill. Long story short, the measured Th/Eu ratio gives an age of 12.8±3 Gyr (Sneden et al. 2003). A similar result of ~13 Gyr (Frebel & Kratz 2009) is obtained from ²³⁸U (this “stable” isotope of uranium has a half-life of 4.5 Gyr, as opposed to the kind that can be provoked into exploding, ²³⁵U, which has a half-life of 700 Myr). While the search for the first stars and the secrets they may reveal is ongoing, the ages for individual stars estimated from radioactive decay are consistent with the ages of the oldest globular clusters indicated by stellar evolution.

Interstellar dust grains: The age of the solar system (4.56 Gyr) is well known from the analysis of isotopic abundances in meteorites. In addition to tracing the oldest material in the solar system, sometimes it is possible to identify dust grains of interstellar origin. One can do the same sort of analysis, and do the sum: how long did it take the star that made those elements to evolve, return them to the interstellar medium, get mixed in with the solar nebula, and lurk about in space until plunging to the ground as a meteorite that gets picked up by some scientifically-inclined human. This exercise has been done by Nittler et al. (2008), who estimate a total age of 13.7±1.3 Gyr

Taken in sum, all these different age indicators point to a similar, consistent age between 13 and 14 billion years. It might be 12, but not lower, nor is there reason to think it would be much higher: 15 is right out. I say that flippantly because I couldn’t resist the Monty Python reference, but the point is serious: you could in principle have a much older universe, but then why are all the oldest things pretty much the same age? Why would the universe sit around doing nothing for billions of years then suddenly decide to make lots of stars all at once? The more obvious interpretation is that the age of the universe is indeed in the ballpark of 13.something Gyr.

Expansion history

The expansion history in the standard FLRW universe is governed by the Friedmann equation, which we can write* as

H²(z) = H₀² [Ω_m(1+z)³+Ω_k(1+z)²+Ω_Λ]

where z is the redshift, H(z) is the Hubble parameter, H₀ is its current value, and the various Ω are the mass-energy density of stuff relative to the critical density: the mass density Ω_m, the geometry Ω_k, and the cosmological constant Ω_Λ. I’ve neglected radiation for clarity. One can make up other stuff X and add a term for it as Ω_X which will have an associated (1+z) term that depends on the equation of state of X. For our purposes, both normal matter and non-baryonic cold dark matter (CDM) share the same equation of state (cold meaning non-relativisitic motions meaning rest-mass density but negligible pressure), so both contribute to the mass density Ω_m = Ω_b+Ω_CDM.

Note that since H(z=0)=H₀, the various Ω’s have to sum to unity. Thus a cosmology is geometrically flat with the curvature term Ω_k = 0 if Ω_m+Ω_Λ = 1. Vanilla LCDM has Ω_m = 0.3 and Ω_Λ = 0.7. As a community, we’ve become very sure of this, but that the Friedmann equation is sufficient to describe the expansion history of the universe is an assumption based on (1) General Relativity providing a complete description, and (2) the cosmological principle (homogeneity and isotropy) holds. These seem like incredibly reasonable assumptions, but let’s bear in mind that we only know directly about 5% of the sum of Ω’s, the baryons. Ω_CDM = 0.25 and Ω_Λ = 0.7 are effectively fudge factors we need to make things works out given the stated assumptions. LCDM is viable if and only if cold dark matter actually exists.

Gravity is an attractive force, so the mass term Ω_m acts to retard the expansion. Early on, we expected this to be the dominant term due to the (1+z)³ dependence. In the long-presumed⁺ absence of a cosmological constant, cosmology was the search for two numbers: once H₀ and Ω_m are specified, the entire expansion history is known. Such a universe can only decelerate, so only the region below the straight line in the graph below is accessible; an expansion history like the red one representing LCDM should be impossible. That lots of different data seemed to want this is what led us kicking and screaming to rehabilitate the cosmological constant, which acts as a form of anti-gravity to accelerate an expansion that ought to be decelerating.

*The expansion factor maps how the universe has grown over time; it corresponds to 1/(1+z) in redshift so that z → ∞ as t → 0.* The “coasting” limit of an empty universe *(H₀ = 73, Ω_m = Ω_Λ = 0)* that expands linearly is shown as the straight line. The *red line* is the expansion history of vanilla LCDM (H₀ = 70, Ω_m = 0.3, Ω_Λ = 0.7).

The over/under between acceleration/deceleration of the cosmic expansion rate is the coasting universe. This is the conceptually useful limit of a completely empty universe with Ω_m = Ω_Λ = 0. It expands at a steady rate that neither accelerates nor decelerates. The Hubble time is exactly equal to the age of such a universe, i.e., 13.4 Gyr for H₀ = 73.

LCDM has a more complicated expansion history. The mass density dominates early on, so there is an early phase of deceleration – the red curve bends to the right. At late times, the cosmological constant begins to dominate, reversing the deceleration and transforming it into an acceleration. The inflection point when it switches from decelerating to accelerating is not too far in the past, which is a curious coincidence given that the entire future of such a universe will be spent accelerating towards the exponential expansion of the de Sitter limit. Why do we live anywhen close to this special time?

Lots of ink has been spilled on this subject, and the answer seems to boil down to the anthropic principle. I find this lame and won’t entertain it further. I do, however, want to point out a related strange coincidence: the current age of vanilla LCDM (13.5 Gyr) is the same as that of a coasting universe with the locally measured Hubble constant (13.4 Gyr). Why should these very different models be so close in age? LCDM decelerates, then accelerates; there’s only one moment in the expansion history of LCDM when the age is equal to the Hubble time, and we happen to be living just then.

This coincidence problem holds for any viable set of LCDM parameters, as they all have nearly the same age. Planck LCDM has an age of 13.7 Gyr, still basically the same as the Hubble time for the locally measured Hubble constant. The lower Planck Hubble value is balanced by a larger amount of early-time deceleration. The universe reaches its current point after 13.something Gyr in all of these models. That’s in good agreement with the ages of the oldest observed stars, which is encouraging, but it does nothing to help us resolve the Hubble tension, much less constrain alternative cosmologies.

Cosmic expansion in MOND

There is no equivalent to the Friedmann equation is in MOND. This is not satisfactory. As an extension of Newtonian theory, MOND doesn’t claim to encompass cosmic phenomena^$ – hence the search for a deeper underlying theory. Lacking this, what can we try?

Felten (1984) tried to derive an equivalent to the Friedmann equation using the same trick that can be used with Newtonian theory to recover the expansion dynamics in the absence of a cosmological constant. This did not work. The result was unsatisfactory^& for application to the whole universe because the presence of a₀ in the equations makes the result scale-dependent. So how big the universe is matters in a way that the standard cosmology does not; there’s no way to generalize is to describe the whole enchilada.

In retrospect, what Felten had really obtained was a solution for the evolution of a top-hat over-density: the dynamics of a spherical region embedded in an expanding universe. This result is the basis for the successful prediction of early structure formation in MOND. But once again it only tells us about the dynamics of an object within the universe, not the universe itself.

In the absence of a complete theory, one makes an ansatz to proceed. If there is a grander theory that encompasses both General Relativity and MOND, then it must approach both in the appropriate limit, so an obvious ansatz to make is that the entire universe obeys the conventional Friedmann equation while the dynamics of smaller regions in the low acceleration regime obey MOND. Both Bob Sanders and I independently adopted this approach, and explicitly showed that it was consistent with the constraints that were known at the time. The first obvious guess for the mass density of such a cosmology is Ω_m = Ω_b = 0.04. (This was the high end of BBN estimates at the time, so back then we also considered lower values.) The expansion history of this low density, baryon-only universe is shown as the blue line below:

*As above, but with the addition of a low density, baryon-dominated, no-CDM universe *(H₀ = 73, Ω_m = *Ω_b =* 0.04, Ω_Λ = 0; blue line)*.*

As before, there is not much to choose between these models in terms of age. The small but non-zero mass density does cause some early deceleration before the model approaches the coasting limit, so the current age is a bit lower: 12.6 Gyr. This is on the small side, but not problematically so, or even particularly concerning given the history of the subject. (I’m old enough to remember when we were pretty sure that globular clusters were 18 Gyr old.)

The time-redshift relation for the no-CDM, baryon-only universe is somewhat different from that of LCDM. If we adopt it, then we find that MOND-driven structure forms at somewhat higher redshift than in with the LCDM time-redshift relation. The benchmark time of 500 Myr for L* galaxy formation is reached at z = 15 rather than z = 9.5 as in LCDM. This isn’t a huge difference, but it does mean that an L* galaxy could in principle appear even earlier than so far seen. I’ve stuck with LCDM as the more conservative estimate of the time-redshift relation, but the plain fact is we don’t really know what the universe is doing at those early times, or if the ansatz we’ve made holds well enough to do this. Surely it must fail at some point, and it seems likely that we’re past that point.

There is a bigger problem with the no-CDM model above. Even if it is close to the right expansion history, it has a very large negative curvature. The geometry is nowhere close to the flat Robertson-Walker metric indicated by the angular diameter distance to the surface of last scattering (the CMB).

Geometry

Much of cosmology is obsessed with geometry, so I will not attempt to do the subject justice. Each set of FLRW parameters has a specific geometry that comes hand in hand with its expansion history. The most sensitive probe we have of the geometry is the CMB. The a priori prediction of LCDM was that its flat geometry required the first acoustic peak to have a maximum near one degree on the sky. That’s exactly what we observe.

**Fig. 45** from Famaey & McGaugh (21012): The acoustic power spectrum of the cosmic microwave background as observed by WMAP [229] together with the a priori predictions of ΛCDM (red line) and no-CDM (blue line) as they existed in 1999 [265] prior to observation of the acoustic peaks. ΛCDM correctly predicted the position of the first peak (the geometry is very nearly flat) but over-predicted the amplitude of both the second and third peak. The most favorable a priori case is shown; other plausible ΛCDM parameters [468] predicted an even larger second peak. The most important parameter adjustment necessary to obtain an a posteriori fit is an increase in the baryon density Ω_b, above what had previously been expected from BBN. In contrast, the no-CDM model *ansatz* made as a proxy for MOND successfully predicted the correct amplitude ratio of the first to second peak with no parameter adjustment [268, 269]. The no-CDM model was subsequently shown to under-predict the amplitude of the third peak [442], so no model can explain these data without post-hoc adjustment.

In contrast, no-CDM made the correct prediction for the first-to-second peak amplitude ratio, but it is entirely ambivalent about the geometry. FLRW cosmology and MOND dynamics care about incommensurate things in the CMB data. That said, the naive prediction of the baryon-only model outlined above is that the first peak should occur around where the third peak is observed. That is obviously wrong.

Since the geometry is not a fundamental prediction of MOND, the position of the first peak is easily fit by invoking the same fudge factor used to fit it conventionally: the cosmological constant. We need a larger Ω_Λ = 0.96, but so what? This parameter merely encodes our ignorance: we make no pretense to understand it, let alone vesting deep meaning in it. It is one of the things that a deeper theory must explain, and can be considered as a clue in its development.

So instead of a baryon-only universe, our FLRW proxy becomes a Lambda-baryon universe. That fits the geometry, and for an optical depth to the surface of last scattering of τ = 0.17, matches the amplitude of the CMB power spectrum and correctly predicts the cosmic dawn signal that EDGES claimed to detect. Sounds good, right? Well, not entirely. It doesn’t fit the CMB data at L > 600, but I expected to only get so far with the no-CDM, so it doesn’t bother me that you need a better underlying theory to fit the entire CMB. Worse, to my mind, is that the Lambda-baryon proxy universe is much, much older than everything in it: 22 Gyr instead of 13.something.

*As above, but now with the addition of a low density, Lambda-dominated universe *(H₀ = 73, Ω_m = *Ω_b =* 0.04, Ω_Λ = 0.96; dashed line)*.*

This just don’t seem right. Or even close to right. Like, not even pointing in a direction that might lead to something that had a hope of being right.

Moreover, we have a weird tension between the baryon-only proxy and the Lambda-baryon proxy cosmology. The baryon-only proxy has a plausible expansion history but an unacceptable geometry. The Lambda-baryon proxy has a plausible geometry by an implausible expansion history. Technically, yes, it is OK for the universe to be much older than all of its contents, but it doesn’t make much sense. Why would the universe do nothing for 8 or 9 Gyr, then burst into a sudden frenzy of activity? It’s as if Genesis read “for the first 6 Gyr, God was a complete slacker and did nothing. In the seventh Gyr, he tried to pull an all-nighter only to discover it took a long time to build cosmic structure. Then He said ‘Screw it’ and fudged Creation with MOND.”

In the beginning the Universe was created.
This has made a lot of people very angry and been widely regarded as a bad move.
Douglas Adams, The Restaurant at the End of the Universe

So we can have a plausible geometry or we can have a plausible expansion history with a proxy FLRW model, but not both. That’s unpleasant, but not tragic: we know this approach has to fail somehow. But I had hoped for FLRW to be a more coherent first approximation to the underlying theory, whatever it may be. If there is such a theory, then both General Relativity and MOND are its limits in their respective regimes. As such, FLRW ought to be a good approximation to the underlying entity up to some point. That we have to invoke both non-baryonic dark matter and a cosmological constant is a hint that we’ve crossed that point. But I would have hoped that we crossed it in a more coherent fashion. Instead, we seem to get a little of this for the expansion history and a little of that for the geometry.

I really don’t know what the solution is here, or even if there is one. At least I’m not fooling myself into presuming it must work out.

*There are other ways to write the Friedmann equation, but this is a useful form here. For the mathematically keen, the Hubble parameter is the time derivative of the expansion factor normalized by the expansion factor, which in terms of redshift is

H(z) = -(dz/dt)/(1+z)².

This quantity evolves, leading us to expect evolution in Milgrom’s constant if we associate it with the numerical coincidence

2π a₀ = cH₀

If the Hubble parameter evolves, as it appears to do, it would seem to follow that so should a(z) ~ H(z) – otherwise the coincidence is just that: a coincidence that applies only now. There is, at present, no persuasive evidence that a₀ evolves with redshift.

A similar order-of-magnitude association can be made with the cosmological constant,

2π a₀ = c²Λ^1/2

so conceivably the MOND acceleration scale appears as the result of vacuum effects. It is a matter of judgement whether these numerical coincidences are mere coincidences or profound clues towards a deeper theory. That the proportionality constant is very nearly 2π is certainly intriguing, but the constancy of any of these parameters (including Newton’s G) depends on how they emerge from the deeper theory.

⁺In January 2019, I was attending a workshop at Princeton when I had a chance encounter with Jim Peebles. He was not attending the workshop, but happened to be walking across campus at the same time I was. We got to talking, and he affirmed my recollection of just how incredibly unpopular the cosmological constant used to be. Unprompted, he went on to make the analogy of how similar that seemed to how unpopular MOND is now.

Peebles was awarded a long-overdue Nobel Prize later that year.

^$This is one of the things that makes it tricky to compare LCDM and MOND. MOND is a theory of dynamics in the limit of low acceleration. It makes no pretense to be a cosmological theory. LCDM starts as a cosmological theory, but it also makes predictions about the dynamics of systems within it (or at least the dark matter halos in which visible galaxies are presumed to form). So if one starts by putting on a cosmology hat, there is nothing to talk about: LCDM is the only game in town. But from the perspective of dynamics, it’s the other way around, with LCDM repeatedly failing to satisfactorily explain, much less anticipate, phenomena that MOND predicted correctly in advance.

^&An intriguing thing about Felten’s MOND universe is that it eventually recollapses irrespective of the mass density. There is no critical value of Ω_m, hence no coincidence problem. MOND is strong enough to eventually reverse the expansion of the universe, it just takes a very long time to do so, depending on the density.

I’m surprised this aspect of the issue was overlooked. The coincidence problem (then mostly called the flatness problem) obsessed people at the time, so much so that its solution by Cosmic Inflation led to its widespread acceptance. That only works if Ω_m = 1; LCDM makes the coincidence worse. I guess the timing was off, as Inflation had already captured the community’s imagination by that time, likely making it hard to recognize that MOND was a more natural solution. We’d already accepted the craziness that was Inflation and dark matter; MOND craziness was a bridge too far.

I guess. I’m not quite that old; I was still an undergraduate at the time. I did hear about Inflation then, in glowing terms, but not a thing about MOND.

Kinematics suggest large masses for high redshift galaxies

This is what I hope will be the final installment in a series of posts describing the results published in McGaugh et al. (2024). I started by discussing the timescale for galaxy formation in LCDM and MOND which leads to different and distinct predictions. I then discussed the observations that constrain the growth of stellar mass over cosmic time and the related observation of stellar populations that are mature for the age of the universe. I then put on an LCDM hat to try to figure out ways to wriggle out of the obvious conclusion that galaxies grew too massive too fast. Exploring all the arguments that will be made is the hardest part, not because they are difficult to anticipate, but because there are so many* options to consider. This leads to many pages of minutiae that no one ever seems to read⁺, so one of the options I’ve discussed (e.g., super-efficient star formation) will likely emerge as the standard picture even if it comes pre-debunked.

The emphasis so far has been on the evolution of the stellar masses of galaxies because that is observationally most accessible. That gives us the opportunity to wriggle, because what we really want to measure to test LCDM is the growth of [dark] mass. This is well-predicted but invisible, so we can always play games to relate light to mass.

Mass assembly in LCDM from the IllustrisTNG50 simulation. The dark matter mass assembles hierarchically in the merger tree depicted at left; the size of the circles illustrates the dark matter halo mass. The corresponding stellar mass of the largest progenitor is shown at right as the red band. This does not keep pace with the apparent assembly of stellar mass (data points), but what is the underlying mass really doing?

Galaxy Kinematics

What we really want to know is the underlying mass. It is reasonable to expect that the light traces this mass, but is there another way to assess it? Yes: kinematics. The orbital speeds of objects in galaxies trace the total potential, including the dark matter. So, how massive were early galaxies? How does that evolve with redshift?

The rotation curve of NGC 6946 traced by stars at small radii and gas farther out. This is a typical flat rotation curve (data points) that exceeds what can be explained by the observed baryonic mass (red line deduced from the stars and gas pictured at right), leading to the inference of dark matter.

The rotation curve for NGC 6946 shows a number of well-established characteristics for nearby galaxies, including the dominance of baryons at small radii in high surface brightness galaxies and the famous flat outer portion of the rotation curve. Even when stars contribute as much mass as allowed by the inner rotation curve (“maximum disk“), there is a need for something extra further out (i.e., dark matter or MOND). In the case of dark matter, the amplitude of flat rotation is typically interpreted as being indicative^& of halo mass.

So far, the rotation curves of high redshift galaxies look very much like those of low redshift galaxies. There are some fast rotators at high redshift as well. Here is an example observed by Neeleman et al. (2020), who measure a flat rotation speed of 272 km/s for DLA0817g at z = 4.26. That’s more massive than either the Milky Way (~200 km/s) or Andromeda (~230 km/s), if not quite as big as local heavyweight champion UGC 2885 (300 km/s). DLA0817g looks to be a disk galaxy that formed early and is sedately rotating only 1.4 Gyr after the Big Bang. It is already massive at this time: not at all the little nuggets we expect from the CDM merger tree above.

**Fig. 1** from Neeleman et al. (2020): the velocity field (left) and position-velocity diagram (right) of DLA0817g. The velocity field looks like that of a rotating disk with the raw *position-velocity diagram* shows motions of ~200 km/s on either side of the center. When corrected for inclination, the flat rotation speed is 272 km/s, corresponding to a massive galaxy near the top of the Tully-Fisher relation.

This is anecdotal, of course, but there are a good number of similar cases that are already known. For example, the kinematics of ALESS 073.1 at z ≈ 5 indicate the presence of a massive stellar bulge as well as a rapidly rotating disk (Lelli et al. 2021). A similar case has been observed at z ≈ 6 (Tripodi et al. 2023). These kinematic observations indicate the presence of mature, massive disk galaxies well before they were expected to be in place (Pillepich et al. 2019; Wardlow 2021). The high rotation speeds observed in early disk galaxies sometimes exceed 250 (Neeleman et al. 2020) or even 300 km s⁻¹ (Nestor Shachar et al. 2023; Wang et al. 2024), comparable to the most massive local spirals (Noordermeer et al. 2007; Di Teodoro et al. 2021, 2023). That such rapidly rotating galaxies exist at high redshift indicates that there is a lot of mass present, not just light. We can’t just tweak the mass-to-light ratio of the stars to explain the photometry and also explain the kinematics.

In a seminal galaxy formation paper, Mo, Mao, & White (1998) predicted that “present-day disks were assembled recently (at z ≤ 1).” Today, we see that spiral galaxies are ubiquitous in JWST images up to z ∼ 6 (Ferreira et al. 2022, 2023; Kuhn et al. 2024). The early appearance of massive, dynamically cold (Di Teodoro et al. 2016; Lelli et al. 2018, 2023; Rizzo et al. 2023) disks in the first few billion years after the Big Bang is contradictory the natural prediction of ΛCDM. Early disks are expected to be small and dynamically hot (Dekel & Burkert 2014; Zolotov et al. 2015; Krumholz et al. 2018; Pillepich et al. 2019), but they are observed to be massive and dynamically cold. (Hot or cold in this context means a high or low amplitude of the velocity dispersion relative to the rotation speed; the modern Milky Way is cold with σ ~ 20 km/s and V_c ~ 200 km/s.) Understanding the stability and longevity of dynamically cold spiral disks is foundational to the problem.

Kinematic Scaling Relations

Beyond anecdotal cases, we can check on kinematic scaling relations like Tully–Fisher. These are expected to emerge late and evolve significantly with redshift in LCDM (e.g., Glowacki et al. 2021). In MOND, the normalization of the baryonic Tully–Fisher relation is set by a₀, so is immutable for all time if a₀ is constant. Let’s see what the data say:

**Figure 9** from McGaugh et al (2024): The baryonic Tully–Fisher (left) and dark matter fraction–surface brightness (right) relations. Local galaxy data (circles) are from Lelli et al. (2019; left) and Lelli et al. (2016; right). Higher-redshift data (squares) are from Nestor Shachar et al. (2023) in bins with equal numbers of galaxies color coded by redshift: 0.6 < z < 1.22 (blue), 1.22 < z < 2.14 (green), and 2.14 < z < 2.53 (red). Open squares with error bars illustrate the typical uncertainties. The relations known at low redshift also appear at higher redshift with no clear indication of evolution over a lookback time up to 11 Gyr.

Not much to see: the data from Nestor Shachar et al. (2023) show no clear indication of evolution. The same can be said for the dark matter fraction-surface brightness relation. (Glad to see that being plotted after I pointed it out.) The local relations are coincident with those at higher redshift for both relations within any sober assessment of the uncertainties – exactly what we measure and how matters at this level, and I’m not going to attempt to disentangle all that here. Neither am I about to attempt to assess the consistency (or lack thereof) with either LCDM or MOND; the data simply aren’t good enough for that yet. It is also not clear to me that everyone agrees on what LCDM predicts.

What I can do is check empirically how much evolution there is within the 100-galaxy data set of Nestor Shachar et al. (2023). To do that, I fit a line to their data (the left panel above) and measure the residuals: for a given rotation speed, how far is each galaxy from the expected mass? To compare this with the stellar masses discussed previously, I normalize those residuals to the same M_*^* = 9 x 10¹⁰ M_☉. If there is no evolution, the data will scatter around a constant value as function of redshift:

This figure reproduces the stellar mass-redshift data for L* galaxies (black points) and the monolithic (purple line) and LCDM (red and green lines) models discussed previously. The blue squares illustrate deviations of the data of Nestor Shachar et al. (2023) from the baryonic Tully-Fisher relation (dashed line, normalized to the same mass as the monolithic model). There is no indication of evolution in the baryonic Tully-Fisher relation, which was apparently established within the first few billion years after the Big Bang (z = 2.5 corresponds to a cosmic age of about 2.6 Gyr). The data are consistent with a monolithic galaxy formation model in which all the mass had been assembled into a single object early on.

The data scatter around a constant value as function of redshift: there is no perceptible evolution.

The kinematic data for rotating galaxies tells much the same story as the photometric data for galaxies in clusters. The are both consistent with a monolithic model that gathered together the bulk of the baryonic mass early on, and evolved as an island universe for most of the history of the cosmos. There is no hint of the decline in mass with redshift predicted by the LCDM simulations. Moreover, the kinematics trace mass, not just light. So while I am careful to consider the options for LCDM, I don’t know how we’re gonna get out of this one.

Empirically, it is an important observation that there is no apparent evolution in the baryonic Tully-Fisher relation out to z ~ 2.5. That’s a lookback time of ~11 Gyr, so most of cosmic history. That means that whatever physics sets the relation did so early. If the physics is MOND, this absence of evolution implies that a₀ is constant. There is some wiggle room in that given all the uncertainties, but this already excludes the picture in which a₀ evolves with the expansion rate through the coincidence a₀ ~ cH₀. That much evolution would be readily perceptible if H(z) evolves as it appears to do. In contrast, the coincidence a₀ ~ c²Λ^1/2 remains interesting since the cosmological constant is constant. Perhaps this is just a coincidence, or perhaps it is a hint that the anomalous acceleration of the expansion of the universe is somehow connected with the anomalous acceleration in galaxy dynamics.

Though I see no clear evidence for evolution in Tully-Fisher to date, it remains early days. For example, a very recent paper by Amvrosiadis et al. (2025) does show a hint of evolution in the sense of an offset in the normalization of the baryonic Tully-Fisher relation. This isn’t very significant, being different by less than 2σ; and again we find ourselves in a situation where we need to take a hard look at all the assumptions and population modeling and velocity measurements just to see if we’re talking about the same quantities before we even begin to assess consistency or the lack thereof. Nevertheless, it is an intriguing result. There is also another interesting anecdotal case: one of their highest redshift objects, ALESS 071.1 at z = 3.7, is also the most massive in the sample, with an estimated stellar mass of 2 x 10¹² M_☉. That is a crazy large number, comparable to or maybe larger than the entire dark matter halo of the Milky Way. It falls off the top of any of the graphs of stellar mass we discussed before. If correct, this one galaxy is an enormous problem for LCDM regardless of any other consideration. It is of course possible that this case will turn out to be wrong for some reason, so it remains early days for kinematics at high redshift.

Cluster Kinematics

It is even earlier days for cluster kinematics. First we have to find them, which was the focus of Jay Franck’s thesis. Once identified, we have to estimate their masses with the available data, which may or may not be up to the task. And of course we have to figure out what theory predicts.

LCDM makes a clear prediction for the growth of cluster mass. This work out OK at low redshift, in the sense that the cluster X-ray mass function is in good agreement with LCDM. Where the theory struggles is in the proclivity for the most massive clusters to appear sooner in cosmic history than anticipated. Like individual galaxies, they appear too big too soon. This trend persisted in Jay’s analysis, which identified candidate protoclusters at higher redshifts than expected. It also measured velocity dispersions that were consistently higher than found in simulations. That is, when Jay applied the search algorithm he used on the data to mock data from the Millennium simulation, the structures identified there had velocity dispersions on average a factor of two lower than seen in the data. That’s a big difference in terms of mass.

**Figure 11** from McGaugh et al. (2024): Measured velocity dispersions of protocluster candidates (Franck & McGaugh 2016a, 2016b) as a function of redshift. Point size grows with the assessed probability that the identified overdensities correspond to a real structure: all objects are shown as small points, candidates with P > 50% are shown as light blue midsize points, and the large dark blue points meet this criterion and additionally have at least 10 spectroscopically confirmed members. The MOND mass for an equilibrium system in the low-acceleration regime is noted at right; these are comparable to cluster masses at low redshift.

At this juncture, there is no way to know if the protocluster candidates Jay identified are or will become bound structures. We made some probability estimates that can be summed up as “some are probably real, but some probably are not.” The relative probability is illustrated by the size of the points in the plot above; the big blue points are the most likely to be real clusters, having at least ten galaxies at the same place on the sky at the same redshift, all with spectroscopically measured redshifts. Here the spectra are critical; photometric redshifts typically are not accurate enough to indicate that galaxies that happen to be nearby to each other on the sky are also that close in redshift space.

The net upshot is that there are at least some good candidate clusters at high redshift, and these have higher velocity dispersions than expected in LCDM. I did the exercise of working out what the equivalent mass in MOND would be, and it is about the same as what we find for clusters at low redshift. This estimate assumes dynamical equilibrium, which is very far from guaranteed. But the time at which these structures appear is consistent with the timescale for cluster formation in MOND (a couple Gyr; z ~ 3), so maybe? Certainly there shouldn’t be lots of massive clusters in LCDM at z ~ 3.

Kinematic Takeaways

While it remains early days for kinematic observations at high redshift, so far these data do nothing to contradict the obvious interpretation of the photometric data. There are mature, dynamically cold, fast rotating spiral galaxies in the early universe that were predicted not to be there by LCDM. Moreover, kinematics traces mass, not just light, so all the wriggling we might try to explain the latter doesn’t help with the former. The most obvious interpretation of the kinematic data to date is the same as that for the photometric data: galaxies formed early and grew massive quickly, as predicted a priori by MOND.

*The papers I write that cover both theories always seem to wind up lopsided in favor of LCDM in terms of the bulk of their content. That happens because it takes many pages to discuss all the ins and outs. In contrast, MOND just gets it right the first time, so that section is short: there’s not much more to say than “Yep, that’s what it predicted.”

⁺I’ve yet not heard directly any criticisms of our paper. The criticisms that I’ve heard second or third hand so far almost all fall in the category of things we explicitly discussed. That’s a pretty clear tell that the person leveling the critique hasn’t bothered to read it. I don’t expect everyone to agree with our take on this or that, but a competent critic would at least evince awareness that we had addressed their concern, even if not to their satisfaction. We rarely seem to reach that level: it is much easier to libel and slander than engage with the issues.

The one complaint I’ve heard so far that doesn’t fall in the category of things-we-already-discussed is that we didn’t do hydrodynamic simulations of star formation in molecular gas. That is a red herring. To predict the growth of stellar mass, all we need is a prescription for assembling mass and converting baryons into stars; this is essentially a bookkeeping exercise that can be done analytically. If this were a serious concern, it should be noted that most cosmological hydro-simulations also fail to meet this standard: they don’t resolve star formation, so they typically adopt some semi-empirical (i.e., data-informed) bookkeeping prescription for this “subgrid physics.”

Though I have not myself attempted to numerically simulate galaxy formation in MOND, Sanders (2008) did. More recently, Eappen et al. (2022) have done so, including molecular gas and feedback^$ and everything. They find a star formation history compatible with the analytic models we discuss in our paper.

^$Related detail: Eappen et al find that different feedback schemes make little difference to the end result. The deus ex machina invoked to solve all problems in LCDM is largely irrelevant in MOND. There’s a good physical reason for this: gravity in MOND is sourced by what you see; how it came to have its observed distribution is irrelevant. If 90% of the baryons are swept entirely out of the galaxy by some intense galactic wind, then they’re gone BYE BYE and don’t matter any more. In contrast, that is one of the scenarios sometimes invoked to form cores in dark matter halos that are initially cuspy: the departure of all those baryons perturbs the orbits of the dark matter particles and rearranges the structure of the halo. While that might work to alter halo structure, how it results in MOND-like phenomenology has never been satisfactorily explained. Mostly that is not seen as even necessary; converting cusp to core is close enough!

^&Though we typically associate the observed outer velocity with halo mass, an important caveat is that the radius also matters: M ~ RV², and most data for high redshift galaxies do not extend very far out in radius. Nevertheless, it takes a lot of mass to make rotation speeds of order 200 km/s within a few kpc, so it hardly matters if this is or is not representative of the dark matter halo: if it is all stars, then the kinematics directly corroborate the interpretation of the photometric data that the stellar mass is large. If it is representative of the dark matter halo, then we expect the halo radius to scale with the halo velocity (R₂₀₀ ~ V₂₀₀) so M₂₀₀~ V₂₀₀³ and again it appears that there is too much mass in place too early.

Old galaxies in the early universe

Continuing our discussion of galaxy formation and evolution in the age of JWST, we saw previously that there appears to be a population of galaxies that grew rapidly in the early universe, attaining stellar masses like those expected in a traditional monolithic model for a giant elliptical galaxy rather than a conventional hierarchical model that builds up gradually through many mergers. The formation of galaxies at incredibly high redshift, z > 10, implies the existence of a descendant population at intermediate redshift, 3 < z < 4, at which point they should have mature stellar populations. These galaxies should not only be massive, they should also have the spectral characteristics of old stellar populations – old, at least, for how old the universe itself is at this point.

*Theoretical predictions from* **Fig. 1** of McGaugh et al (2024) *combined with the data of* **Fig. 4**. The data follow the track of a monolithic model that forms early as a single galaxy rather than that of the largest progenitor of the hierarchical build-up expected in LCDM.

The data follow the track of stellar mass growth for an early-forming monolithic model. Do the ages of stars also look like that?

Here is a recent JWST spectrum published by de Graff et al. (2024). This appeared too recently for us to have cited in our paper, but it is a great example of what we’re talking about. This is an incredibly gorgeous spectrum of a galaxy at z = 4.9 when the universe was 1.2 Gyr old.

**Fig. 1** from de Graff et al. (2024): *JWST/NIRSpec PRISM spectrum (black line) of the massive quiescent galaxy RUBIES-EGS-QG-1 at a redshift of z = 4.8976.*

It is challenging to refrain from nerding out at great length over many of the details on display here. First, it is an incredible technical achievement. I’ve seen worse spectra of local galaxies. JWST was built to obtain images and spectra of galaxies so distant they approach the horizon of the observable universe. Its cameras are sensitive to the infrared part of the spectrum in order to capture familiar optical features that have been redshifted by a huge factor (compare the upper and lower x-axes). The telescope itself was launched into space well beyond the obscuring atmosphere of the earth, pointed precisely at a tiny, faint flicker of light in a vast, empty universe, captured photons that had been traveling for billions of years, and transmitted the data to Earth. That this is possible, and works, is an amazing feat of science, engineering, and societal commitment (it wasn’t exactly cheap).

In the raw 2D spectrum (at top) I can see by eye the basic features in the extracted, 1D spectrum (bottom). This is a useful and convincing reality check to an experienced observer even if at first glance it looks like a bug splot smeared by a windshield wiper. The essential result is apparent to the eye; the subsequent analysis simply fills in the precise numbers.

Looking from right to left, the spectrum runs from red to blue. It ramps up then crashes down around an observed wavelength of 2.3 microns. This is the 4000 Å break in the rest frame, a prominent feature of aging stellar populations. The amount of blue-to-red ramp-up and the subsequent depth of drop is a powerful diagnostic of stellar age.

In addition to the 4000 Å break, a number of prominent spectral lines are apparent. In particular, the Balmer absorption lines Hβ, Hγ, and Hδ are clear and deep. These are produced by A stars, which dominate the light of a stellar population after a few hundred million years. There’s the answer right there: the universe is only 1.2 Gyr old at this point, and the stars dominating the light aren’t much younger.

There are also some emission lines. These can be the sign of on-going star formation or an active galactic nucleus powered by a supermassive black hole. The authors attribute these to the latter, inferring that the star formation happened fast and furious early on, then basically stopped. That’s important to the rest of the spectrum; A stars only dominate for a while, and their lines are not so prominent if a population keeps making new stars. So this galaxy made a lot of stars, made them fast, then basically stopped. That is exactly the classical picture of a monolithic giant elliptical.

Here is the star formation history that de Graff et al. (2024) infer:

**Fig. 2** from de Graff et al. (2024): the star formation rate (top) and accumulated stellar mass (bottom) as a function of cosmic time (only the first 1.2 Gyr are shown). Results for stellar populations of two metallicities are shown (purple or blue lines). This affects the timing of the onset of star formation, but once going, an enormous mass of stars forms fast, in ~200 Myr.

There are all sorts of caveats about population modeling, but it is very hard to avoid the basic conclusion that lots of stars were assembled with incredible speed. A stellar mass a bit in excess of that of the Milky Way appears in the time it takes for the sun to orbit once. That number need not be exactly right to see that this is not a the gradual, linear, hierarchical assembly predicted by LCDM. The typical galaxy in LCDM is predicted to take ~7 Gyr to assemble half its stellar mass, not 0.1 Gyr. It’s as if the entire mass collapsed rapidly and experienced an intense burst of star formation during violent relaxation (Lynden-Bell 1967).

Collapse of shells within shells to form a massive galaxy rapidly in MOND (Sanders 2008). Note that the inner shells (inset) where most of the stars will be collapse even more rapidly than the overall monolith (dotted line).

Where MOND provides a natural explanation for this observation, the fiducial population model of de Graff et al. violates the LCDM baryon limit: there are more stars than there are baryons to make them from. It should be impossible to veer into the orange region above as the inferred star formation history does. The obvious solution is to adopt a higher metallicity (the blue model) even if that is a worse fit to the spectrum. Indeed, I find it hard to believe that so many stars could be made in such a small region of space without drastically increasing their metallicity, so there are surely things still to be worked out. But before we engage in too much excuse-making for the standard model, note that the orange region represents a double-impossibility. First, the star formation efficiency is 100%. Second, this is for an exceptionally rare, massive dark matter halo. The chances of spotting such an object in the area so far surveyed by JWST is small. So we not only need to convert all the baryons into stars, we also need to luck into seeing it happen in a halo so massive that it probably shouldn’t be there. And in the strictist reading, there still aren’t enough baryons. Does that look right to you?

Do these colors look right to you? Getting the color right is what stellar population modeling is all about.

OK, so I got carried away nerding out about this one object. There are other examples. Indeed, there are enough now to call them a population of old and massive quiescent galaxies at 3 < z < 4. These have the properties expected for the descendants of massive galaxies that form at z > 10.

Nanayakkara et al. (2024) model spectra for a dozen such galaxies. The spectra provide an estimate of the stellar mass at the redshift of observation. They also imply a star formation history from which we can estimate the age/redshift at which the galaxy had formed half of those stars, and when it quenched (stopped forming stars, or in practice here, when the 90% mark had been reached). There are, of course, large uncertainties in the modeling, but it is again hard to avoid the conclusion that lots of stars were formed early.

**Figure 7** from McGaugh et al. (2024): The stellar masses of quiescent galaxies from Nanayakkara et al. (2024). The inferred growth of stellar mass is shown for several cases, marking the time when half the stars were present (small green circles) to the quenching time when 90% of the stars were present (midsize orange circles) to the epoch of observation (large red circles). Illustrative star formation histories are shown as dotted lines with the time of formation t_i and the quenching timescale τ noted in Gyr. We omit the remaining lines for clarity, as many cross. There is a wide distribution of formation times from very early (t_i = 0.2 Gyr) to relatively late (>1 Gyr), but all of the galaxies in this sample are inferred to build their stellar mass rapidly and quench early (τ < 0.5 Gyr).

The dotted lines above are models I constructed in the spirit of monolithic models. The particular details aren’t important, but the inferred timescales are. To put galaxies in this part of the stellar mass-redshift plane, they have to start forming early (typically in the first billion years), form stars at a prolific rate, then quench rapidly (typically with e-folding timescales < 1 Gyr). I wouldn’t say any of these numbers are particularly well-measured, but they are indicative.

What is missing from this plot is the LCDM prediction. That’s not because I omitted it, it’s because the prediction for typical L* galaxies doesn’t fall within the plot limits. LCDM does not predict that typical galaxies should become this massive this early. I emphasize typical because there is always scatter, and some galaxies will grow ahead of the typical rate.

Not only are the observed galaxies massive, they have mature stellar populations that are pretty much done forming stars. This will sound normal to anyone who has studied the stellar populations of giant elliptical galaxies. But what does LCDM predict?

I searched through the Illustris TNG50 and TNG300 simulations for objects at redshift 3 that had stellar masses in the same range as the galaxies observed by Nanayakkara et al. (2024). The choice of z = 3 is constrained by the simulation output, which comes in increments of the expansion factor. To compare to real galaxies at 3 < z < 4 one can either look at the snapshot at z = 4 or the one at z = 3. I chose z = 3 to be conservative; this gives the simulation the maximum amount of time to produce quenched, massive galaxies.

These simulations do indeed produce some objects of the appropriate stellar mass. These are rare, as they are early adopters: galaxies that got big quicker than is typical. However, they are not quenched as observed: the simulated objects are still on the star forming main sequence (the correlation between star formation rate and stellar mass). The distribution of simulated objects does not appear to encompass that of real galaxies.

**Figure 8** from McGaugh et al. (2024): The stellar masses and star formation rates of galaxies from Nanayakkara et al. (2024; red symbols). Downward-pointing triangles are upper limits; some of these fall well below the edge of the plot and so are illustrated as the line of points along the bottom. Also shown are objects selected from the TNG50 (Pillepich et al. 2019; filled squares) and TNG300 (Pillepich et al. 2018; open squares) simulations at z = 3 to cover the same range of stellar mass. Unlike the observed galaxies, simulated objects with stellar masses comparable to real galaxies are mostly forming stars at a rapid pace. In the higher-resolution TNG50, none have quenched as observed.

If we want to hedge, we can note that TNG300 has a few objects that are kinda in the right ballpark. That’s a bit misleading, as the data are mostly upper limits. Moreover, these are the rare objects among a set of objects selected to be rare: it isn’t a resounding success if we have to scrape the bottom of the simulated barrel after cherry-picking which barrel. Worse, these few semi-quenched simulated objects are not present in TNG50. TNG50 is the higher resolution simulation, so presumably provides a better handle on the star formation in individual objects. It is conceivable that TNG300 “wins” by virtue of its larger volume, but that’s just saying we have more space in which to discover very rare entities. The prediction is that massive, quenched galaxies should be exceedingly rare, but in the real universe they seem mundane.

That said, I don’t think this problem is fundamental. Hierarchical assembly is still ongoing at this epoch, bringing with it merger-induced star formation. There’s an easy fix for that: change the star formation prescription. Instead of “wet” mergers with gas that can turn into stars, we just need to form all the stars already early on so that the subsequent mergers are “dry” – at least, for those mergers that build this particular population. One winds up needing a new and different mode of star formation. In addition to what we observe locally, there needs to be a separate mode of super-efficient star formation that somehow turns all of the available baryons into stars as soon as possible. That’s basically what I advocate as the least unreasonable possibility for LCDM in our paper. This is a necessary but not sufficient condition; these early stellar nuggets also need to assemble speedy quick to make really big galaxies. While it is straightforward to mess with the star formation prescription in models (if not in nature), the merger trees dictating the assembly history are less flexible.

Putting all the data together in a single figure, we can get a sense for the evolutionary trajectory of the growth of stellar mass in galaxies across cosmic time. This figure extends from the earliest galaxies so-far known at z ~ 14 when the universe was just a few hundred million years old (of order on orbital time in a mature galaxy) to the present over thirteen billion years later. In addition to data discussed previously, it also shows recent data with spectroscopic redshifts from JWST. This is important, as the sense of the figure doesn’t change if we throw away all the photometric redshifts, it just gets a little sparse around z ~ 8.

**Figure 10** from McGaugh et al. (2024): The data from Figures 4 and 6 shown together using the same symbols. Additional JWST data with spectroscopic redshifts are shown from Xiao et al. (2023; green triangles) and Carnall et al. (2024). The data of Carnall et al. (2024) distinguish between star-forming galaxies (small blue circles) and quiescent galaxies (red squares); the latter are in good agreement with the typical stellar mass determined from Schechter fits in clusters (large circles). The dashed red lines show the median growth predicted by the Illustris ΛCDM simulation (Rodriguez-Gomez et al. 2016) for model galaxies that reach final stellar masses of M_* = 10¹⁰, 10¹¹, and 10¹² M_☉. The solid lines show monolithic models with a final stellar mass of 9 x 10¹⁰ M_☉ and t_i = τ = 0.3, 0.4, and 0.5 Gyr, as might be appropriate for giant elliptical galaxies. The dotted line shows a model appropriate to a monolithic spiral galaxy with t_i = 0.5 and τ = 13.5 Gyr.

The solid lines are monolithic models we built to represent classical giant elliptical galaxies that form early and quench rapidly. These capture nicely the upper envelope of the data. They form most of their stars at z > 4, producing appropriately old populations at lower redshifts. The individual galaxy data merge smoothly into those for typical galaxies in clusters.

The LCDM prediction as represented by the Illustris suite of simulations is shown as the dashed red lines for objects of several final masses. These are nearly linear in log(M_*)-linear z space. Objects that end up with a typical L* elliptical galaxy mass at z = 0 deviate from the data almost immediately at z > 1. They disappear above z > 6 as the largest progenitors become tiny.

What can we do to fix this? Massive galaxies get a head start, as it were, by being massive at all epochs. But the shape of the evolutionary trajectory remains wrong. The top red line (for a final stellar masses of 10¹² M_☉) corresponds to a typical galaxy at z ~ 2, but it continues to grow to be atypical locally. The data don’t do that. Even with this boost, the largest progenitor is still predicted to be too small at z > 3 where there are now many examples of massive, quiescent galaxies – known both from JWST observations and from Jay Franck’s thesis before it. Again, the distribution of the data do not look like the predictions of LCDM.

One can abandon Illustris as the exemplar of LCDM, but it doesn’t really help. Other models show similar things, differing only in minor details. That’s because the issue is the mass assembly history they all share, not the details of the star formation. The challenge now is to tweak models to make them look more monolithic; i.e., change those red dashed lines into the solid black lines. One will need super-efficient star formation, if it is even possible. I’ll leave discussion of this and other obvious fudges to a future post.

Finally, note that there are a bunch of galaxies with JWST spectroscopic redshifts from 3 < z < 4 that are not exceptionally high mass (the small blue points). These are expected in any paradigm. They can be galaxies that are intrinsically low mass and won’t grow much further, or galaxies that may still grow a lot, just with a longer fuse on their star formation timescale. Such objects are ubiquitous in the local universe as spiral and irregular galaxies. Their location in the diagram above is consistent with the LCDM predictions, but is also readily explained by monolithic models with long star formation timescales. The dotted line shows a monolithic model that forms early (t_i = 0.5) but converts gas into stars gradually (τ = 13.5 Gyr rather than < 1 Gyr). This is a boilerplate model for a spiral that has been around for as long as the short-τ model for giant ellipticals. So while these lower mass galaxies exist, their location in the M_*-z plane doesn’t really add much to this discussion as yet. It is the massive galaxies that form early and become quiescent rapidly that most challenge LCDM.

Measuring the growth of the stellar mass of galaxies over cosmic time

This post continues the series summarizing our ApJ paper on high redshift galaxies. To keep it finite, I will focus here on the growth of stellar mass. The earlier post discussed what we expect in theory. This depends both on mass assembly (slow in LCDM, fast in MOND), how the assembled mass is converted into stars, and how those stars shine in light we can detect. We know a lot about stars and their evolution, so for this post I will assume we know how to convert a given star formation history into the evolution of the light it produces. There are of course caveats to that which we discuss in the paper, and perhaps will get to in a future post. It’s exhausting to be exhaustive, so not today, Satan.

The principle assumption we are obliged to make, at least to start, is that light traces mass. As mass assembles, some of it turns into stars, and those stars produce light. The astrophysics of stars and the light they produce is the same in any structure formation theory, so with this basic assumption, we can test the build-up of mass. In another post we will discuss some of the ways in which we might break this obvious assumption in order to save a favored theory. For now, we assume the obvious assumption holds, and what we see at high redshift provides a picture of how mass assembles.

Before JWST

This is not a new project; people have been doing it fo for decades. We like to think in terms of individual galaxies, but there are lots out there, so an important concept is the luminosity function, which describes the number of galaxies as a function of how bright they are. Here are some examples:

**Figure 3.** from Franck & McGaugh (2017) showing the number of galaxies as a function of their brightness in the 4.5 micron band of the Spitzer Space Telescope in candidate protoclusters from z = 2 to 6. Each panel notes the number of galaxies contributing to the Schechter luminosity function⁺ fit (gray bands), the apparent magnitude m* corresponding to the typical luminosity L*, and the redshift range. The magnitude m* is *characteristic* of how bright typical galaxies are at each redshift.

One reason to construct these luminosity functions is to quantify what is typical. Hundreds of galaxies inform each fit. The luminosity L* is representative of the typical galaxy, not just anecdotal individual examples. At each redshift, L* corresponds to an observed apparent magnitude m*, which we plot here:

**Figure 3** from McGaugh et al. (2024): The redshift dependence of the Spitzer [4.5] apparent magnitude m* of Schechter function fits to populations of galaxies in clusters and candidate protoclusters; each point represents *the characteristic* brightness of the galaxies in each cluster. The apparent brightness of galaxies gets fainter with increasing redshift because galaxies are more distant, with the amount they dim depending also on their evolution (lines). The purple line is the monolithic exponential model we discussed last time. The orange line is the prediction of the Millennium simulation (the state of the art at the time Jay Franck wrote his thesis) and the Munich galaxy formation model based on it. The open squares are the result of applying the same algorithm to the simulation as used on the data; this is what we would have observed if the universe looked like LCDM as depicted by the Munich model. The real universe does not look like that.

We plot faint to bright going up the y-axis; the numbers get smaller because of the backwards definition of the magnitude scale (which dates to ancient times in which the stars that appeared brightest to the human eye were “of the first magnitude,” then the next brightest of the second magnitude, and so on). The x-axis shows redshift. The top axis shows the corresponding age of the universe for vanilla LCDM parameters. Each point shows the apparent magnitude that is typical as informed by observations of dozens to hundreds of individual galaxies. Each galaxy has a spectroscopic redshift, which we made a requirement for inclusion in the sample. These are very accurate; no photometric redshifts are used to make the plot above.

One thing that impressed me when Jay made the initial version of this plot is how well the models match the evolution of m* at z < 2, which is most of cosmic time (the past ten billion years). This encourages one that the assumption adopted above, that we understand the evolution of stars well enough to do this, might actually be correct. I was, and remain, especially impressed with how well the monolithic model with a simple exponential star formation history matches these data. It’s as if the inferences the community had made about the evolution of giant elliptical galaxies from local observations were correct.

The new thing that Jay’s work showed was that the evolution of typical cluster galaxies at z > 2 persists in tracking the monolithic model that formed early (z_f = 10). There is a lot of scatter in the higher redshift data even though there is little at lower redshift. This is to be expected for both observational reasons – the data get rattier at larger distances – and theoretical ones: the exponential star formation history we assume is at best a crude average; at early times when short-lived but bright massive stars are present there will inevitably be stochastic variation around this trend. At later times the law of averages takes over and the scatter should settle down. That’s pretty much what we see.

What we don’t see is the decline in typical brightness predicted by contemporaneous LCDM models. The specific example shown is the Munich galaxy formation model based on the Millennium simulation. However, the prediction is generic: galaxies get faint at high redshift because they haven’t finished assembling yet. This is not a problem of misunderstanding stellar evolution, it is a failure of the hierarchical assembly paradigm.

In order to identify [proto]clusters at high redshift, Jay devised an algorithm to identify galaxies in close proximity on the sky and in redshift space, in excess of the average density around them. One question we had was whether the trend predicted by the LCDM model (the orange line above) would be reproduced in the data when analyzed in this way. To check, Jay made mock observations of a simulated lookback cone using the same algorithm. The results (not previously published) are the open squares in the plot above. These track the “right” answer known directly in the form of the orange line. Consequently, if the universe had looked as predicted, we could tell. It doesn’t.

The above plot is in terms of apparent magnitude. It is interesting to turn this into the corresponding stellar mass. There has also been work done on the subject after Jay’s, so I wanted to include it. An early version of a plot mapping m* to stellar mass and redshift to cosmic time that I came up with was this:

*The stellar mass of L* galaxies as a function of cosmic age. Data as noted in the inset. The purple/orange lines represent the monolithic/hierarchical models, as above.*

The more recent data (which also predate JWST) follow the same trend as the preceding data. All the data follow the path of the monolithic model. Note that the bulk of the stars are formed in situ in the first few billion years; the stellar mass barely changes after that. There is quite a bit of stellar evolution during this time, which is why m* in the figure above changes in a complicated fashion while the stellar mass remains constant. This again provides some encouragement that we understand how to model stellar populations.

The data in the first billion years are not entirely self-consistent. For example, the yellow points are rather higher in mass than the cyan points. This difference is not one in population modeling, but rather in how much of a correction is made for non-stellar, nebular emission. So as not to go down that rabbit hole, I chose to adopt the lowest stellar mass estimates for the figure that appears in the paper (below). Note that this is the most conservative choice; I’m trying to be as favorable to LCDM as is reasonably plausible.

**Figure 4** from McGaugh et al. (2024): *The characteristic stellar mass as a function of time with the corresponding redshift noted at the top.*

There were more recent models as well as more recent data, so I wanted to include those. There are, in fact, way too many models to illustrate without creating a confusing forest of lines, so in the end I chose a couple of popular ones, Illustris and FIRE. Illustris is the descendant of Millennium, and shows identical behavior. FIRE has a different scheme for forming stars, and does so more rapidly than Illustris. However, its predictions still fall well short of the data. This is because both simulations share the same LCDM cosmology with the same merger tree assembly of structure. Assembling the mass promptly enough is the problem; it isn’t simply a matter of making stars faster.

I’ll show one more version of this plot to illustrate the predicted evolutionary trajectories. In the plots above, I only show models that end up with the mass of a typical local giant elliptical. Galaxies come in a variety of masses, so what does that look like?

*The stellar mass of galaxies as a function of cosmic age. Data as above. The orange lines represent the hierarchical models that result in different final masses at z = 0.*

The curves of stellar growth predicted by LCDM have pretty much the same shape, just different amplitude. The most massive case illustrated above is reasonable insofar as there are real galaxies that massive, but they are rare. They are also rare in simulations, which make the predicted curve a bit jagged as there aren’t enough examples to define a smooth trajectory as there are for lower mass objects. More importantly, the shape is wrong. One can imagine that the galaxies we see at high redshift are abnormally massive, but even the most massive galaxies don’t start out that big at high redshift. Moreover, they continue to grow hierarchically in LCDM, so they wind up too big. In contrast, the data look like the monolithic model that we made on a lark, no muss, no fuss, no need to adjust anything.

This really shouldn’t have come as a surprise. We already knew that galaxies were impossibly massive at z ~ 4 before JWST discovered that this was also true at z ~ 10. The a priori prediction that LCDM has made since its inception (earlier models show the same thing) fails. More recent models fail, though I have faith that they will eventually succeed. This is the path theorists has always taken, and the obvious path here, as I remarked previously, is to make star formation (or at least light production) artificially more efficient so that the hierarchical model looks like the monolithic model. For completeness, I indulge in this myself in the paper (section 6.3) as an exercise in what it takes to save the phenomenon.

A two year delay

Regular readers of this blog will recall that in addition to the predictions I emphasized when JWST was launched, I also made a number of posts about the JWST results as they started to come in back in 2022. I had also prepared the above as a science paper that is now sections 1 to 3 of McGaugh et al. (2024). The idea was to have it ready to go so I could add a brief section on the new JWST results and submit right away – back in 2022. The early results were much as expected, but I did not rush to publish. Instead, it has taken over two years since then to complete what turned into a much longer manuscript. There are many reasons for this, but the scientific reason is that I didn’t believe many of the initial reports.

JWST was new and exciting and people fell all over themselves to publish things quickly. Too quickly. To do so, they relied on a calibration of the telescope plus detector system made while it was on the ground prior to launch. This is not the same as calibrating it on the sky, which is essential but takes some time. Consequently, some of the initial estimates were off.

Stellar masses and redshifts of galaxies from Labbe et al. The pink squares are the initial estimates that appeared in their first preprint in July 2022. The black squares with error bars are from the version published in February 2023. The shaded regions represent where galaxies are too massive too early for LCDM. The lighter region is where galaxies shouldn’t exist; the darker region is a where they cannot exist.

In the example above, all of the galaxies had both their initial mass and redshift estimates change with the updated calibration. So I was right to be skeptical, and wait for an improved analysis. I was also right that while some cases would change, the basic interpretation would not. All that happened in the example above was that the galaxies moved from the “can’t exist in LCDM” region (dark blue) into the “really shouldn’t exist in LCDM” region (light blue). However, the widespread impression was that we couldn’t trust photometric redshifts at all, so I didn’t see what new I could justifiably add in 2022. This was, after all, the attitude Jay and I had taken in his CCPC survey where we required spectroscopic redshifts.

So I held off. But then it became impossible to keep up with the fire hose of data that ensued. Every time I got the chance to update the manuscript, I found some interesting new result had been published that I had to include. New things were being discovered faster than I could read the literature. I found myself stuck in the Red Queen’s dilemma, running as fast as possible just to stay in place.

Ultimately, I think the delay was worthwhile. Lots new was learned, and actual spectroscopic redshifts began to appear. (Spectroscopy takes more telescope time than photometry – spreading out the light reduces the signal-to-noise per pixel, necessitating longer exposure times, so it always lags behind. One also discovers the galaxies in the same images that are used for photometry, so it also gets a head start.) Consequently, there is a lot more in the paper than I had planned on. This is another long blog post, so I will end it where I had planned for the original paper to end, with the updated version of the plot above.

Massive galaxies at high redshift from JWST

The stellar masses of galaxies discovered by JWST as a function of redshift is shown below. Unlike most of the plots above, these are individual galaxies rather than typical L* galaxies. Many are based on photometric redshifts, but those in solid black have spectroscopic redshifts. There are many galaxies that reside in a region they should not, at least according to LCDM models: their mass is too large at the observed redshift.

**Figure 6** from McGaugh et al. (2024): Mass estimates for high-redshift galaxies from JWST. Colored points based on photometric redshifts are from Adams et al. (2023; dark blue triangles), Atek et al. (2023; green circles), Labbé et al. (2023; open squares), Naidu et al. (2022; open star), Harikane et al. (2023; yellow diamonds), Casey et al. (2024; light blue left-pointing triangles), and Robertson et al. (2024; orange right-pointing triangles). Black points from Wang et al. (2023; squares), Carniani et al. (2024; triangles), Harikane et al. (2024; circles) and Castellano et al. (2024; star) have spectroscopic redshifts. The upper limit for the most massive galaxy in TNG100 (Springel et al. 2018) as assessed by Keller et al. (2023) is shown by the light blue line. This is consistent with the maximum stellar mass expected from the stellar mass–halo mass relation of Behroozi et al. (2020; solid blue line). These merge smoothly into the trend predicted by Yung et al. (2019b) for galaxies with a space density of 10⁻⁵ dex⁻¹ Mpc⁻³ (dashed blue line), though L. Yung et al. (2023) have revised this upward by ∼0.4 dex (dotted blue line). This closely follows the most massive objects in TNG300 (Pillepich et al. 2018; red line). The light gray region represents the parameter space in which galaxies were not expected in LCDM. The dark gray area is excluded by the limit on the available baryon mass (Behroozi & Silk 2018; Boylan-Kolchin 2023). [Note added: I copied this from the caption in our paper, but the links all seem to go to that rather than to each of the cited papers. You can get to them from our reference list if you want, but it’ll take some extra clicks. It looks like AAS has set it up this way to combat trawling by bots.]

One can see what I mean about a fire hose of results from the number of references given here. Despite the challenges of keeping track of all this, I take heart in the fact that many different groups are finding similar results. Even the results that were initially wrong remain problematic for LCDM. Despite all the masses and redshifts changing when the calibration was updated, the bulk of the data (the white squares, which are the black squares in the preceding plot) remain in the problematic region. The same result is replicated many times over by others.

The challenge, as usual, is assessing what LCDM actually predicts. The entire region of this plot is well away from the region predicted for typical galaxies. To reside here, a galaxy must be an outlier. But how extreme an outlier?

The dark gray region is the no-go zone. This is where dark matter halos do not have enough baryons to make the observed mass of stars. It should be impossible for galaxies to be here. I can think of ways to get around this, but that’s material for a future post. For now, it suffices to know that there should be no galaxies in the dark gray region. Indeed, there are not. A few straddle the edge, but nothing is definitively in that region given the uncertainties. So LCDM is not outright falsified by these data. This bar is set very low, as the galaxies that do skirt the edge require that basically all of the available baryons have been converted into starts practically instantaneously. This is not a reasonable.

*Not with ten thousand simulations could you do this.*

So what is a reasonable expectation for this diagram? That’s hard to say, but that’s what the white and light gray region attempts to depict. Galaxies might plausibly be in the white region but should not be in the light gray region for any sensible star formation efficiency.

One problem with this statement is that it isn’t clear what a sensible star formation efficiency is. We have a good idea of what it needs to be, on average, at low redshift. There is no clear indication that it changes as a function of redshift – at least until we hit results like this. Then we have to be on guard for confirmation bias in which we simply make the star formation efficiency be what we need it to be. (This is essentially what I advocate as the least unreasonable option in section 6.3 of the ApJ paper.)

OK, but what should the limit be? Keller et al. (2023) made a meta-analysis of the available simulations; I have used his analysis and my own reading of the literature to establish the lower boundary of the light gray area. It is conceivable that you would get the occasional galaxy this massive (the white region is OK), but not more so (the light gray region is not OK). The boundary is the most extreme galaxy in each simulation, so as far from typical as possible. The light gray region is really not OK; the only question is where exactly it sets in.

The exact location of this boundary is not easy to define. Different simulations give different answers for different reasons. These are extremal statistics; we’re asking what the one most massive galaxy is in an entire simulation. Higher resolution simulations perceive the formation of small structures like galaxies sooner, but large simulations have more opportunity for extreme events to happen. Which “wins” in terms of making the rare big galaxy early is a competition between these effects that appears, in my reading, to depend on details of simulation implementation that are unlikely to be representative of physical reality (even assuming LCDM is the correct underlying physics).

To make my own assessment, I reviewed the accessible simulations (they don’t all provide the necessary information) to fine the very most massive simulated galaxy as a function of redshift. As ever, I am looking for the case that is most favorable to LCDM. The version I found comes from the large-box, next generation Illustris simulation TNG300. This is the red line a bit into the gray area above. Galaxies really, really should not exist above or to the right of that line. Not only have I adopted the most generous simulation estimate I could find, I have also chosen not to normalize to the area surveyed by JWST. One should do this, but the area so far surveyed is tiny, so the line slides down. Even if galaxies as massive as this exist in TNG300, we have to have been really lucky to point JWST at that spot on a first go. So the red line is doubly generous, and yet there are still galaxies that exceed this limit.

The bottom line is that yes, JWST data pose a real problem for LCDM. It has been amusing watching this break people’s brains. I’ve seen papers that say this is a problem for LCDM because you’d have to turn more than half of the available baryons into stars and that’s crazy talk, and others that say LCDM is absolutely OK because there are enough baryons. The observational result is the same – galaxies with very high stellar-to-dark halo mass ratios, but the interpretation appears to be different because one group of authors is treating the light gray region as forbidden while the other sets the bar at the dark gray region. So the difference in interpretation is not a conflict in the data, but an inconsistency in what [we think] LCDM predicts.

That’s enough for today. Galaxy data at high redshift are clearly in conflict with the a priori predictions of LCDM. This was true before JWST, and remains true with JWST. Whether the observations can be reconciled with LCDM I leave as an exercise for scientists in the field, or at least until another post.

⁺A minor technical note: the Schechter function is widely used to describe the luminosity function of galaxies, so it provides a common language with which to quantify both their characteristic luminosity L* and space density Φ*. I make use of it here to quantify the brightness of the typical galaxy. It is, of course, not perfect. As we go from low to high redshift, the luminosity function becomes less Schechter-like and more power law-like, an evolution that you can see in Jay Franck’s plot. We chose to use Schechter fits for consistency with the previous work of Mancone et al. (2010) and Wylezalek et al. (2014), and also to down-weight the influence of the few very bright galaxies should they be active galactic nuclei or some other form of contaminant. Long story short, plausible contaminants (no photometric redshifts were used; sample galaxies all have spectroscopic redshifts) cannot explain the bulk of the data; our estimates of m* are robust and, if anything, underestimate how bright galaxies typically are.

A few videos for the new year

Happy new year to those who observe the Gregorian calendar. I will write a post on the observations that test the predictions discussed last time. It has been over a quarter century since Bob Sanders correctly predicted that massive galaxies would form by z = 10, and three years since I reiterated that for what JWST would see on this blog. This is a testament to both the scientific method and the inefficiency of communication.

Here I provide links to some recent interviews on the subject. These are listed in chronological order, which happen to flow in order of increasing technical detail.

The first entry is from my colleague Federico Lelli. It is in Italian rather than English, but short and easy on the ears. If nothing else, appreciate that Dr. Lelli did this on the absence of sleep afforded a new father.

Next is an interview I did with EarthSky. I thought this went well, and should be reasonably accessible.

Next is Scientific Sense:

Most recently, there is the entry from the AAS Journal Author Series. These are based on papers published in the journals of the American Astronomical Society in which authors basically narrate their papers, so this goes through it at an appropriately high (ApJ) level.

We discuss the “little red dots” some, which touches on the issues of size evolution that were discussed in the comments previously. I won’t add to that here beyond noting again that the apparent size evolution is proportional to (1+z), in the sense that high redshift galaxies are apparently smaller than those of similar stellar mass locally. This (1+z) is the factor that relates the angular diameter distance of the Robsertson-Walker metric to that of Euclidean geometry. Consequently, we would not infer any size evolution if the geometry were Euclidean. It’s as if cosmology flunks the Tolman test. Weird.

There is a further element of mystery towards the end where the notion that “we don’t know why” comes up repeatedly. This is always true at some deep philosophical level, but it is also why we construct and test hypotheses. Why does MOND persistently make successful predictions that LCDM did not? Usually we say the reason why has to do with the successful hypothesis coming closer to the truth.

That’s it for now. There will be more to come as time permits.

Triton Station

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

Category: MOND