Author: tritonstation

Stacy McGaugh is an astrophysicist and cosmologist who studies galaxies, dark matter, and theories of modified gravity. He is an expert on low surface brightness galaxies, a class of objects in which the stars are spread thin compared to bright galaxies like our own Milky Way. He demonstrated that these dim galaxies appear to be dark matter dominated, providing unique tests of theories of galaxy formation and modified gravity. Professor McGaugh is currently the chair of the Department of Astronomy at Case Western Reserve University in Cleveland, Ohio, and director of the Warner and Swasey Observatory. Previously he was a member of the faculty at the University of Maryland, having also held research fellowships at Rutgers, the Department of Terrestrial Magnetism of the Carnegie Institution of Washington, and the Institute of Astronomy at the University of Cambridge after earning his Ph.D. from the University of Michigan.

Recent Developments Concerning the Gravitational Potential of the Milky Way. II. A Closer Look at the Data

Continuing from last time, let’s compare recent rotation curve determinations from Gaia DR3:

Fig. 1 from Jiao et al. comparing three different realizations of the Galactic rotation curve from Gaia DR3. The vertical lines* mark the range of the Ou et al. data considered by Chan & Chung Law (2023).

These are different analyses of the same dataset. The Gaia data release is immense, with billions of stars. There are gazillions of ways to parse these data. So it is reasonable to have multiple realizations, and we shouldn’t expect them to necessarily agree perfectly: do we look exclusively at K giants? A stars? Only stars with proper motion and/or parallax data more accurate than some limit? etc. Of course we want to understand any differences, but that’s not going to happen here.

My first observation is that the various analyses are broadly consistent. They all show a steady decline over a large range of radii. Nothing shocking there; it is fairly typical for bright, compact galaxies like the Milky Way to have somewhat declining rotation curves. The issue here, of course, is how much, and what does it mean?

Looking more closely, not all of the data agree with each other, or even with themselves. There are offsets between the three at radii around the sun (we live just outside R = 8 kpc) where you’d naively think they would agree the best. They’re very consistent from 13 < R < 17 kpc, then they start to diverge a little. The Ou data have a curious uptick right around R = 17 kpc, which I wouldn’t put much stock in; weird kinks like that sometimes happen in astronomical data. But it can’t be consistent with a continuous mass distribution, and will come up again for other reasons.

As an astronomer, I’m happy with the level of agreement I see here. It is not perfect, in the sense that there are some points from one data set whose error bars do not overlap with those of other data sets in places. That’s normal in astronomy, and one of the reasons that we can never entirely trust the stated uncertainties. Jiao et al. make a thorough and yet still incomplete assessment of the systematic uncertainties, winding up with larger error bars on the Wang et al. realization of the data.

For example, one – just one of the issues we have to contend with – is the distance to each star in the sample. Distances to individual objects are hard, and subject to systematic uncertainties. The reason to choose A stars or K giants is because you think you know their luminosity, so can estimate their distance. That works, but aren’t necessarily consistent (let alone correct) among the different groups. That by itself could be the source of the modest difference we see between data sets.

Chan & Chung Law use the Ou et al. realization of the data to make some strong claims. One is that the gradient of the rotation curve is -5 km/s/kpc, and this excludes MOND at high confidence. Here is their plot.

You will notice that, as they say, these are the data of Ou et al, being identical to the same points in the plot from Jiao et al. above – provided you only look in the range between the lines, 17 < R < 23 kpc. This is where the kink at R = 17 kpc comes in. They appear to have truncated the data right where it needs to be truncated to ignore the point with a noticeably lower velocity, which would surely affect the determination of the slope and reduce its confidence level. They also exclude the point with a really big error bar that nominally is within their radial range. That’s OK, as it has little significance: it’s large error bar means it contributes little to the constraint. That is not the case for the datum just inside of R = 17 kpc, or the rest of the data at smaller radii for that matter. These have a manifestly shallower slope. Looking at the line boundaries added to Jiao’s plot, it appears that they selected the range of the data with the steepest gradient. This is called cherry-picking.

It is a strange form of cherry-picking, as there is no physical reason to expect a linear fit to be appropriate. A Keplerian downturn has velocity decline as the inverse square root of radius (see the dotted line above.) These data, over this limited range, may be consistent with a Keplerian downturn, but certainly do not establish that it is required.

Contrast the statements of Chan & Chung Law with the more measured statement from the paper where the data analysis is actually performed:

… a low mass for the Galaxy is driven by the functional forms tested, given that it probes beyond our measurements. It is found to be in tension with mass measurements from globular clusters, dwarf satellites, and streams.
Ou et al. (2023)

What this means is that the data do not go far enough out to measure the total mass. The low mass that is inferred from the data is a result of fitting some specific choice of halo form to it. They note that the result disagrees with other data, as I discussed last time.

Rather than cherry pick the data, we should look at all of it. Let’s see, I’ve done that before. We looked at the Wang et al. (2023) data via Jiao et al. previously, and just discussed the Ou et al. data. That leaves the new Zhao et al. data, so let’s look at those:

Milky Way rotation curve with RAR model (blue line from 2018) and the Gaia DR3 data as realized by Zhou et al. (2023: purple triangles). The dashed line shows the number of stars (right axis) informing each datum.

These data were the last of the current crop that I looked at. They look… pretty good in comparison with the pre-existing RAR model. Not exactly the falsification I had been led to expect.

So – the three different realizations of the Gaia DR3 data are largely consistent, yet one is being portrayed as a falsification of MOND while another is in good agreement with its prediction.

This is why you have to take astronomical error bars with a grain of salt. Three different groups are using data from the same source to obtain very nearly the same result. It isn’t quite the same result, as some of the data disagree at the formal limits of their uncertainty. No big deal – that’s what happens in astronomy. The number of stars per bin helps illustrate one reason why: we go from thousands of stars per bin near the sun to tens of stars in wider bins at R > 20 kpc. That’s not necessarily problematic, but it is emblematic of what we’re dealing with: great gobs of data up close, but only scarce scratches of it far away where systematic effects are more pernicious.

In the meantime, one realization of these data are being portrayed as a death knell for a theory that successfully predicts another realization of the same data. Well, which is it?

*Thanks to Moti Milgrom for pointing out the restricted range of radii considered by Chan & Chung Law and adding the vertical lines to this figure.

Recent Developments Concerning the Gravitational Potential of the Milky Way. I.

Recent results from the third data release (DR3) from Gaia has led to a flurry of papers. Some are good, some are great, some are neither of those. It is apparent from the comments last time that while I’ve kept my pledge to never dumb it down, I have perhaps been assuming more background knowledge on the part of readers than is adequate. I can’t cram a graduate education in astronomy into one web page, but will try to provide a little relevant context.

Galactic Astronomy is an ancient field, dating back at least to the Herschels. There is a lot that is known in the field. There have also been a lot of misleading observations, going back just as far to the Herschel’s map of the Milky Way, which was severely limited by extinction from interstellar dust. That’s easy to say now, but Herschel’s map was the standard for over a century – longer than our modern map has persisted.

So a lot has changed, including a lot that seemed certain, so I try to keep an open mind. The astronomers working with the Gaia data – the ones deriving the rotation curve – are simply following where those data take them, as they should. There are others using their analyses to less credible ends. A lot of context is required to distinguish the two.

The total mass of the Milky Way

There are a lot of constraints on the mass of the Milky Way that predate Gaia; it’s not like these are the first data that address the issue. Indeed, there are lots and lots and lots of other applicable data acquired using different methods over the course of many decades. Here is a summary plot of determinations of the mass of the Milky Way compiled by Wang et al. (2019).

This is an admirable compilation, and yet no such compilation can be complete. There are just so many determinations by lots of independent authors. Still, this is nice for listing multiple results from many distinct methodologies. They all consistently give numbers around 10¹² solar masses. (Cast in these terms, my own estimate is 1.4 x 10¹² albeit with a substantial systematic uncertainty.) I’ve added a point for the total mass according to the alleged Keplerian downturn seen in the Gaia data, 2 x 10¹¹ solar masses. One of these things is not like the others.

The difference from the bulk of the data has nearly every astronomer rolling our collective eyes. Most of us straight up don’t believe it. That’s not to say the Gaia data are wrong, but the interpretation of those data as indicative of such a small, finite total mass seems unlikely in the light of all other results.

As I discussed briefly last time, it is conceivable that previous results are wrong or misleading due to some systematic effect or bad assumption. For example, mass estimates based on “satellite phenomenon” require the assumption that the satellite galaxies are indeed satellites of the Milky Way on bound orbits. That seems like a really good assumption, as without it, their presence is an instantaneous coincidence particular to the most recent few percent of a Hubble time: they wouldn’t have been nearby more than a billion years ago, and won’t be around another for even a few hundred million more. That sounds like a long time to you and me, but it is not that long on a cosmic scale. Maybe they’re raining down all the time to give the appearance of a steady state? Where have I heard that before?

Even if we’re willing to dismiss satellite constraints, that doesn’t suffice. It isn’t good enough to find flaw with one set of determinations; one must question all distinct methods. I could probably do that; there’s always a systematic uncertainty that might be bigger than expected or an assumption that could go badly wrong. But it is asking a lot for all of them to conspire to be wrong at the same time by the same amount. (The assumption of Newtonian gravity is a catch-all.)

Some constraints are more difficult to dodge than others. For example, the escape velocity method merely notes that there are fast moving stars in the solar neighborhood. Those stars are many billions of years old, and wouldn’t be here if the gravitational potential couldn’t contain them. The mass implied by the Gaia quasi-Keplerian downturn doesn’t suffice.

That said, the total mass of the Milky Way as expressed above is a rather notional quantity. M₂₀₀ occurs roughly 200 kpc out for the Milky Way, give or take a lot. And the “200” in the subscript has nothing to do with that radius being 200 kpc for reasons too technical and silly to delve into. So my biggest concern about the compilation above is not that the data are wrong so much as they are being extrapolated to an idealized radius that we don’t directly observe. This extrapolation is usually done by assuming the potential of an NFW halo, which makes perfect sense in terms of LCDM but none whatsoever empirically, since NFW predicts the wrong density profile at small, intermediate, and large radii: where the density profile ρ ∝ r^-α is predicted to have α = (1,2,3), it is persistently observed to be more like (0,1,2). While the latter profile is empirically more realistic, it also fails to converge to a finite total mass, rendering the concept meaningless.

Rather than indulge yet again in a discussion of the virtues and vices of different dark matter halo profiles, let’s look at an observationally more robust quantity: the enclosed mass. Wang et al. also provide a tabulation of this quantity from many sources, as depicted here:

Rotation curve constraints implied by the enclosed mass measurements tabulated by Wang et al. (2019) combined with the halo stars and globular clusters previously discussed. The location of the Large Magellanic Cloud is also indicated; data beyond this radius (and perhaps even within it) are subject to perturbation by the passage of the LMC. The RAR-based model is shown as the blue line; the light blue line includes a very uncertain estimate of the effect of the coronal gas. This is very diffuse and extended, and only becomes significant at very large radii. The dotted line is the Keplerian curve for a mass of 2 x 10¹¹ M_☉.

Not all of the enclosed mass data are consistent with one another. The bulk of them are consistent with the RAR model Milky Way (blue line). None of them are consistent with the small mass indicated by recent Gaia analyses (dotted line). Hence the collective unwillingness of most astronomers to accept the low-mass interpretation.

An important thing to note when considering data at large radii, especially those beyond 50 kpc, is that 50 kpc is the current Galactocentric radius of the Large Magellanic Cloud. The LMC brings with it its own dark matter halo, which perturbs the outer regions of the Milky Way. This effect is surprisingly strong*, and leads to the inference that the mass ratio of the two is only 4 or 5:1 even though the luminosity ratio is more like 20:1. This makes the interpretation of the data beyond 50 kpc problematic. If we use that as a pretext to ignore it, then we infer that our low mass Milky Way is no more massive then the LMC – an apparently absurd situation.

There are many rabbit holes we could dig down here, but the basic message is that a small Milky Way mass violates a gazillion well-established constraints. That doesn’t mean the Gaia data are wrong, but it does call into question their interpretation. So next time we’ll look more closely at the data.

*This is not surprising in MOND. The LMC is in the right place at the right time to cause the Galactic warp. The LMC as a candidate perturber to excite the Galactic warp was recognized early, but the conventional mass was thought to be much too small to do the job. The small baryonic mass of the LMC in MOND is not a problem as the long range nature of the force law makes tidal effects more pronounced: it works out about right.

Is the Milky Way’s rotation curve declining?

Yes, some. That much is a step forward from a decade ago, when a common assumption was that the Milky Way’s rotation curve remained flat at the speed at which the sun orbited. This was a good guess based on empirical experience with other galaxies, but not all galaxies have rotation curves that are completely flat, nor can we be sure the sun is located where that is the case.

A bigger question whether the Milky Way’s rotation curve is declining in a Keplerian fashion. This would indicate that the total mass has been enclosed. That would be a remarkable result. If true, it would be the first time that the total mass of an individual galaxy has been measured. There have been claims to this effect before that have not panned out when the data have been extended to larger radii, so one might be inclined to be skeptical.

There are several claims now to see a distinctly declining rotation curve based on the third data release (DR3) from Gaia. The most recent, Jiao et al., has gained some note by virtue of putting “Keplerian decline” in the title, but very similar results have also been reported by Ou et al., Wang et al. and Sylos Labini et al. They all obtain basically the same answer using the same data, with minor differences in the error assessment and other details. There are also differences in interpretation*, which is always possible even when everyone agrees about what the data say.

Jiao et al. measure a total mass for the Milky Way of about 2 x 10¹¹ M_☉. Before looking at the data, let’s take a moment to think about that number. Most mass determinations – and there are lots, see Fig. 2 of Wang et al. – for the Milky Way have been in the neighborhood of 10¹² M_☉. Indeed, for most of my career, it was traditionally Known to be 2 x 10¹² M_☉. The new measurement is an order of magnitude smaller. That’s a lot to be off by, even in extragalactic astronomy. The difference, as we’ll see, has to do with what data we use.

The mass of stars and gas in the Milky Way is about 6 x 10¹⁰ M_☉, give or take ten billion. That means that nearly a third of the total mass is normal baryonic matter that we can readily see. So the ratio of dark-to-baryonic mass is only 2.3:1, well short of the cosmic ratio of about 6:1. That’s embarrassing – especially since much of the effort in galaxy formation theory has been to explain why the baryon fraction is much less than the cosmic fraction, not much more. And here our Galaxy is an outlier, having much less dark matter for its stellar mass than everything else. It is always a bad sign when the Galaxy appears to violate the Copernican Principle.

Nonetheless, this is what we find if we look at the Gaia DR3 data. Here is a model I’ve shown before, extrapolated to larger radii with some new data added. The orange circles are the Gaia DR3 rotation curve as given by Jiao et al. For radii greater than 18 kpc, they show a clear decline consistent with a Keplerian curve for a 1.95 x 10¹¹ M_☉ point mass (dotted line), as per Fig. 9 of Jiao et al.

*Milky Way model (blue line) compared with various data.*

This is the first time we’ve been able to trace the rotation curve so far out with stars in the disk of the Milky Way, and the Keplerian line is a good match. If that’s all we know, then a total mass of only 2 x 10¹¹ M_☉ is a reasonable inference. That’s not all we know.

As I alluded above, a halo mass this small makes no sense in the context of cosmology. Not only is 2 x 10¹¹ M_☉ too small, the more commonly inferred dynamical mass of 10¹² M_☉ is also too small. According to abundance matching, which has become an important aspect of LCDM, the Milky Way should reside in a 3 or 4 x 10¹² M_☉ halo. So the new mass makes a factor of 2 or 3 problem into a factor a ten problem. That is too large to attribute to scatter in the stellar mass-halo mass relation. Worse, there is no evidence that the Milky Way is an outlier from scaling relations like Tully-Fisher. We can’t have it one way and not the other.

The traditional mass estimates that obtain ~10¹² M_☉ rely on dwarf satellite galaxies as tracers of the gravitational potential of the Milky Way. Maybe they’re not fair tracers? We have to make assumptions about their orbits to use them to infer a mass; perhaps these assumptions are wrong? It is conceivable that many of our satellites are on first infall rather than in well-established orbits. Indeed, the consensus is that our largest satellites, the Magellanic Clouds, are on first infall, and that they cause a substantial perturbation to the halo of the Milky Way. This was an absurd thought 15 years ago – the Magellanic clouds must have been here forever, and were far too small to do damage – but now this is standard lore.

There are tracers at large radii besides dwarf satellite galaxies. The figure above shows three: globular clusters (pink triangles) and two types of stars in the halo: blue horizontal branch stars (green squares) and K giants (red squares). These are well-known parts of the Milky Way that have been with us for many billions of years, so they’ve had plenty of time to become equilibrium tracers of the gravitational potential. They clearly indicate a larger enclosed mass than predicted by the Keplerian decline traced by the Gaia rotation curve, and are consistent with traditional satellite analyses. Perhaps these data are somehow misleading, but it is hard to see how.

Gaia is great, but has its limits. It is really optimized for nearby stars (within a few kpc). Outside of that, the statistics… leave something to be desired. Is it safe to push out beyond 20 kpc? I don’t know, but I did notice this panel from Fig. 8 of Wang et al.:

*Radial velocities of stars at different heights above the Galactic plane.*

The radial velocity is a minor component of disk motion, where azimuthal motion dominates. However, one does need to know it to solve the Jeans equation. Having it wrong will cause a perceptible systematic error. You notice the bifurcation in the data for R > 22 kpc? That, in technical terms, is Messed Up. I don’t know what goes awry there, but I’ve done this exercise enough times for the sight of this to scare the bejeepers out of me. No way I trust any of these data at R > 22 kpc, and I hope having seen this doesn’t give me nightmares tonight.

Perhaps the uncertainty caused by this is adequately reflected in the large error bars on the orange points above. Those with R > 22 kpc are nicely Keplerian, but also consistent with a lot of things, including the blue line that successfully predicts the halo stars and globular clusters. That’s not true for the data around R = 20 kpc where the error bars are much smaller: there the discrepancy with the blue line I take seriously. But that is a much more limited affair that might indicate the presence of a ring of mass – that’s what gives the bumps and wiggles at smaller radii – and certainly isn’t enough to imply the entire mass of the Milky Way has been enclosed.

But who knows? Perhaps fifteen years hence it will be the standard lore that all galaxies reside in dark matter halos that are only twice the mass of their luminous disks. At that mass ratio, all the galactic dark matter could be baryonic. I wouldn’t bet on it, but stranger things have happened before, and will happen again.

*A difference in interpretation is largely what the debate about dark matter and MOND boils down to. There is no doubt that there are acceleration discrepancies in extragalactic objects that require something beyond what you see is what you get with normal gravity. Whether we should blame what we can’t see or the assumption of normal gravity is open to interpretation. I would hope this is obvious, but this elementary point seems to be lost on many.

Wide Binary Results Favoring MOND

I think the time has come for another update on wide binaries. These were intensely debated at the conference in St. Andrews, with opposing camps saying they did or did not show MONDian behavior. Two papers by independent authors have recently been refereed and published: Chae (2023) in the Astrophysical Journal and Hernandez (2023) in Monthly Notices. These papers both find evidence for MONDian behavior in wide binaries.

If these new results are correct, they are the smoking gun for MOND. I’ve been trying to avoid that phrase, and think of how we would explain this with dark matter. I haven’t come up with any good ideas. This doesn’t preclude others from coming up with bad ideas, but the problem this result poses is profound.

The basic idea is that galaxies reside in dark matter halos. These are diffuse entities with a particular mass distribution that must contribute the right gravitational force to explain observations on galactic scales. On local scales, like the solar neighborhood, this leads to a very low space density of about 0.007 solar masses per cubic parsec, or 0.26 GeV/cm³. For comparison, the local density of stars and gas is about 0.11 solar masses per cubic parsec. Adding up all the dark matter in the solar system within the orbit of Pluto amounts to the equivalent mass of a one km-size asteroid. That doesn’t do anything noticeable to solar system dynamics, especially when it is spread out as expected rather than concentrated in an asteroid.

Wide binaries should encompass more dark matter than the solar system by virtue of their greater size, but the enclosed mass remains too tiny to affect the orbits of the stars. There could be the occasional lump of dark matter, but those should be few and far between: the conventional expectation for binary stars is purely Newtonian, with no hint of a mass discrepancy. In contrast, the expectation in MOND is that every system that experiences the low acceleration regime should show a discrepancy of predictable amplitude. I simply don’t see how to imitate that with any of the usual dark matter suspects.

Here is the results from Chae’s paper. There are many figures like this that explore all sorts of permutations on sample selection and other effects. The answer persistently comes up the same. There is a systematic deviation from Newtonian behavior that is consistent with MOND, and in particular with the nonlinear theory AQUAL proposed early on by Bekenstein & Milgrom.

*Part of Fig. 19 from Chae (2023)*. As one goes to lower acceleration, the data for wide binaries agrees well with the prediction of the Aquadratic Lagrangian theory of MOND (purple line in lower panel).

This figure subsumes many astronomical details, like the distribution of orbital eccentricities and the frequency of triple systems. Chae has simulated what to expect as a result of all these effects, with the results in the top panel distinguishing between the Newtonian expectation in blue and the data in red. At high accelerations, the red histogram is right on top of the blue histogram. These distributions are indistinguishable, as they should be in both theories. As one looks to lower accelerations, the red and blue histograms begin to part. They stand clearly apart in the lowest acceleration bin. This is as expected in MOND. In contrast, the histograms should never diverge in the Newtonian case, with or without dark matter.

A similar result has been obtained by Hernandez (2023), who emphasizes the importance of obtaining a clean sample for which one is sure that the binaries are genuinely bound and have radial velocities as well as proper motions. The data follow the Newtonian line until they don’t. The deviation is consistent with MOND.

*Part of Fig. A1 from Hernandez (2023). The MOND effect is apparent as the break of the red points from the purely Newtonian blue line.*

Again, there are many figures like this in the paper to explore all the possible permutations. These all paint the same picture: MOND. The published result Hernandez obtains is consistent with the result obtained by Chae, relieving a small tension that was present in the preprint stage.

Still outstanding is why Chae and Hernandez get a different answer from Pittordis & Sutherland (2023), who utilize many more binaries. This is a tradeoff that frequently arises in astronomical data analysis: numbers vs. quality. The risk with numbers is that the signal you’re searching for gets drowned out in a sea of noise. The risk in defining a high quality sample is that you unintentionally introduce a selection effect that causes a signal to appear where there isn’t one. It seems unlikely that this would result in MOND-like behavior – it could do any number of crazy things – but I don’t know enough about this specific subject to judge. Note that I’m willing to say when I’m out of my expertise; I expect it won’t be hard to find faux experts who don’t acknowledge the limitations of their qualifications and are perfectly happy to find flaws with studies they dislike but don’t understand.

What I hope to see in future is some convergence between the different groups, or at least for some understanding to emerge as to why their results differ. In the meantime, I expect most of the community will duck and cover.

Required dark matter properties

I was on vacation last week. As soon as I got back, the first thing I did was fall off my bike onto a tree stump, breaking my wrist. I’ll be okay, but I won’t be typing a lot. This post is being dictated to software; I hope I don’t have to do too much editing. I let the software generate the image above based on the prompt “dark matter properties illustrated” and I don’t think we should hold our breath for AI to help us out with this.

There were some good questions to the last post that I didn’t get to address. I went back and tried to answer some of them. Siriusactuary asked about the properties required for dark matter for galaxies vs. large scale structure. That’s a very deep question that requires a long answer with some historical perspective. Please bear with me as I attempt a quasi-coherent, off-the-cuff narrative that doesn’t invite a lot of editing, which it surely will.

I thought about this long and hard when I first encountered the problem. Which was almost thirty years ago now. So it is probably worth a short refresher.

We have been assuming all along, I think reasonably, that cosmological dark matter and galaxy dark matter are the same stuff, just different manifestations of the same problem. Perhaps they’re not, but there is a huge range of systems that show acceleration discrepancies, and it isn’t always trivial to split them into one camp or another. It seems common to talk about large and small scale problems, but I don’t think size is the right way to think about it. It’s more a difference between gravitationally bound systems that are in equilibrium and the dynamics of the expanding universe as an evolving entity that contains structures that develop within it.

The problem in bound systems is not just galaxy dynamics. It’s also clusters of galaxies. It’s also a star clusters that don’t show a discrepancy. The problem extends over a dynamic range of at least a billion in baryonic mass. It involves all sorts of dynamical questions where we do sometimes need to invoke dark matter or MOND or whatever. The evidence in bound systems is inevitably that when we apply the law of gravity as we know it to the stuff we can see, the visible baryons, then the dynamical mass doesn’t add up. We need something extra to explain the data.

The simple answer early on was that there was simply more mass there, i.e., dark matter. But that much is ambiguous. It could be that we infer the need for dark matter because the equations are inadequate and need to be generalized, i.e., something like MOND. But to start, at the beginning of the dark matter paradigm, there was no particular restriction on what the dark matter needed to be or what its properties needed to be. It could be baryonic, it could be non-baryonic. It could be black holes, brown dwarfs, all manner of things.

From a cosmological perspective, it became apparent in the early 1980s that we needed something extra – not just dark, but non-baryonic. By this time it was easy to believe because people like Vera Rubin and Albert Bosma had already established that we needed more than meets the eyes in galaxies. So dark matter was no longer a radical hypothesis, which it had been in 1970. The paradigm kinda snowballed – it had been around as a possibility since the 1930s, but it was only in the 1970s that it became firmly established dynamically. Even then it was like a factor of two and could be normal if hard to see baryons like brown dwarfs. By the early 1980s it was clear we needed more like a factor of ten, and it had to be something new: the cosmological constraint was that the gravitating mass density is greater than the baryon density allowed by big bang nucleuosynthesis. That means that there is a requirement on the nature of dark matter beyond there just being more mass.

The cosmic dark matter has to be something non-baryonic. That is to be say, it has to be some new kind of beast, presumably some kind of a particle that is not already in the standard model of particle physics. This was received with eagerness by particle physicists who felt that their standard model was complete and yet unsatisfactory and there should be something deeper and more to it. This was an indication in that direction. From a cosmological perspective, the key fact was that there was something more out there than met the eye. Gravitation gave a mass density then was higher than allowed in normal matter. Not only did you need dark matter, but you needed some kind of novel, new particle that’s not in the standard model of particle physics to be that dark matter.

The other cosmological imperative was to grow large scale structure. The initial condition that we see in the early universe is very smooth. That is the microwave background on the sky, with its very small temperature fluctuations, only one part in a hundred thousand. That’s the growth factor reached by redshift zero: structure has grown by a factor of a hundred thousand. Normal gravity will grow structure at a rate that is proportional to the rate at which the universe expands, which is basically a factor of a thousand since the microwave background was imprinted.

So we have another big discrepancy. We can only grow structure by a factor of a thousand, but we observe that it has grown by a factor of a hundred thousand. So we need something to goose the process. That something can be dark matter, provided that it does not interact with photons directly. It can be a form of particle that does not interact via the electromagnetic force. It can interacts through gravity and perhaps through the weak nuclear force, but not through the electromagnetic force.

Those are properties that are required of dark matter by cosmology. It has to be non-baryonic and not interact through electromagnetism. These properties are not necessary for galaxies. And that’s basically the picture that persists today. One additional constraint that we need from a cosmological perspective is that the dark matter needs to be slow-moving – dynamically cold so that structure can form. If you make it dynamically hot, like neutrinos that are born moving at very nearly the speed of light, those are not going to clump up and form structure even if they have a little mass.

So that was the origin of the cold dark matter paradigm. We needed some form of completely novel particle that had the right relic density – this is where the wimp miracle comes in. That worked fine for galaxies at the time. All you needed for galaxies early on was extra mass. It was cosmology that gave us these extra indications of what the dark matter needs to be.

We’ve learned a lot more about galaxies since then. I remember in the early nineties when I was still a staunch proponent of cold dark matter being approached at conferences by eminent dynamicists who confided in hushed tones so that the cosmologists wouldn’t hear that they thought the dark matter had to be baryonic, not non-baryonic.

I had come to this from the cosmological perspective that I had just described above. The total mass density had to be a lot bigger than the baryonic mass density. Therefore the dark matter had to be non-baryonic. To say otherwise was crazy talk, which is why they were speaking about it in hushed tones. But here were these very eminent people who were very quietly suggesting to me that their work on galaxies suggested that the dark matter had to be made a baryons not something non-baryonic. I asked why, and basically it boiled down to the fact that they could see clear connections between the dynamics and the baryons. It didn’t suffice just to have extra mass; the dark and luminous component seemed to know about each other*.

The data for galaxies showed that the stuff we could see, the distribution of stars and gas, was clearly and intimately related to the total distribution of mass, including the dark matter. This led to a number of ideas, that do not sit well with the cold dark matter paradigm. One was HI scaling: basically, if you took the distribution of atomic gas, and scaled it up by a factor of roughly 10, then that was a decent predictor of what the dark matter was doing. Given that, one could imagine that maybe the dark matter was some form of unseen baryons that follow the same distribution as the atomic gas. There was even an elaborate paradigm built up around very cold molecular gas to do this. That seemed problematic for me, because if you have cold molecular gas, it should clump up and form stars, and then you see it. Even if you didn’t see it in it’s cold form you need a lot of it. Interestingly, you do not violate the BBN baryon density, just in galaxies. But you would on a cosmic scale, if that was the only form of dark matter. So then we we need multiple forms of dark matter, which violates parsimony.

Another important and frequent point is the concept of maximum disk. This came up last time in the case of NGC 1277, where the inner regions of that galaxy have its dynamics completely explained by the stars that you see. This is a very common occurrence in high surface brightness galaxies. In regions where the stars are dense, that’s all the mass that you need. It’s only when you get out to a much larger radius, where the accelerations become low, that you needed something extra, the dark matter effect.

It was pretty clear and widely accepted that the inner regions of many bright galaxies were star dominated. You did not need much dark matter in the center, only at the edges. So you had this picture of a pseudoisothermal halo with a low density central core. But by the mid-nineties, a lot of simulations all showed that cold dark matter halos should have cusps: they predicted there to be a lot of dark matter near the centers of galaxies.

This contradicted the picture that had been established. And so people got into big arguments as to whether or not high-surface brightness galaxies were indeed maximal. The people who actually worked on galaxies said Yes, we have established that they are maximal – we only need stars in the central regions; the dark matter only becomes necessary farther out. People who were coming at it from the cosmological perspective without having worked on individual galaxies saw the results of the simulations, saw that there’s always a little room to trade off between the stellar mass and the dark mass by adjusting the mass to light ratio of the stars, and said galaxies cannot be maximal.

I was perplexed by this contradiction. You had a strong line of evidence that galaxies were maximal and their centers. You had a completely different line of evidence, a top down cosmological view of galaxies that said galaxies should not and could not be maximal in nurse centers. Which of those interpretations you believe seemed to depend on which camp you came out of.

I came out of both camps. I was working on low surface brightness galaxies at the time and was hopeful that they would help to resolve the issue. Instead they made it worse, sticking us with a fine-tuning problem. I could not solve this fine-tuning problem. It caused me many headaches. It was only after I had suffered those headaches that I began to worry about the dark matter paradigm. And then by chance, I heard a talk by this guy Milgrom who, in a few lines on the board, derived as a prediction all of the things that I was finding problematic to interpret in terms of dark matter. Basically, a model with dark matter has to look like MOND to satisfy the data.

That’s just silly, isn’t it?

MOND made predictions. Those predictions came true. What am I supposed to report? That it had these predictions com true – therefore it’s wrong?

I had made my own prediction based on dark matter. It failed. Other people had different predictions based on dark matter. Those also did not come true. Milgrom was only the only one to correctly predict ahead of time what low surface brightness galaxies would do.

If we insist on dark matter, what this means is that we need, for each and every galaxy, the precise that looks like MOND. I wrote the equation for the required effects of dark matter in all generality in McGaugh (2004). The improvements in the data over the subsequent decade enable this to be abbreviated to

g_DM = g_bar/(e^{√(g_bar/a₀)} -1).

This is in McGaugh et al. (2016), which is a well known paper (being in the top percentile of citation rates). So this should be well known, but the implication seems not to be, so let’s talk it through. g_DM is the force per unit mass provided by the dark matter halo of a galaxy. This is related to the mass distribution of the dark matter – its radial density profile – through the Poisson equation. The dark matter distribution is entirely stipulated by the mass distribution of the baryons, represented here by g_bar. That’s the only variable on the right hand side, a₀ being Milgrom’s acceleration constant. So the distribution of what you see specifies the distribution of what you can’t.

This is not what we expect for dark matter. It’s not what naturally happens in any reasonable model, which is an NFW halo. That comes from dark matter-only simulations; it has literally nothing to do with g_bar. So there is a big chasm to bridge right from the start: theory and observation are speaking different languages. Many dark matter models don’t specify g_bar, let alone satisfy this constraint. Those that do only do so crudely – the baryons are hard to model. Still, dark matter is flexible; we have the freedom to make it work out to whatever distribution we need. But in the end, the best a dark matter model can hope to do is crudely mimic what MOND predicted in advance. If it doesn’t do that, it can be excluded. Even if it does do that, should we be impressed by the theory that only survives by mimicking its competitor?

The observed MONDian behavior makes no sense whatsoever in terms of the cosmological constraints in which the dark matter has to be non-baryonic and not interact directly with the baryons. The equation above implies that any dark matter must interact very closely with the baryons – a fact that is very much in the spirit of what earlier dynamicist had found, that the baryons and the dynamics are intimately connected. If you know the distribution of the baryons that you can see, you can predict what the distribution of the unseen stuff has to be.

And so that’s the property that galaxies require that is pretty much orthogonal to the cosmic requirements. There needs to be something about the nature of dark matter that always gives you MONDian behavior in galaxies. Being cold and non-interacting doesn’t do that. Instead, galaxy phenomenology suggests that there is a direct connection – some sort of direct interaction – between dark matter and baryons. That direct interaction is anathema to most ideas about dark matter, because if there’s a direct interaction between dark matter and baryons, it should be really easy to detect dark matter. They’re out there interacting all the time.

There have been a lot of half solutions. These include things like warm dark matter and self interacting dark matter and fuzzy dark matter. These are ideas that have been motivated by galaxy properties. But to my mind, they are the wrong properties. They are trying to create a central density core in the dark matter halo. That is at best a partial solution that ignores the detailed distribution that is written above. The inference of a core instead of a cusp in the dark matter profile is just a symptom. The underlying disease is that the data look like MOND.

MONDian phenomenology is a much higher standard to try to get a dark matter model to match than is a simple cored halo profile. We should be honest with ourselves that mimicking MOND is what we’re trying to achieve. Most workers do not acknowledge that, or even be aware that this is the underlying issue.

There are some ideas to try to build-in the required MONDian behavior while also satisfying the desires of cosmology. One is Blanchet’s dipole or dark matter. He imagined a polarizable dark medium that does react to the distribution of baryons so as to give the distribution of dark matter that gives MOND-like dynamics. Similarly, Khoury’s idea of superfluid dark matter does something related. It has a superfluid core in which you get MOND-like behavior. At larger scales it transitions to a non-superfluid mode, where it is just particle dark matter that reproduces the required behavior on cosmic scales.

I don’t find any of these models completely satisfactory. It’s clearly a hard thing to do. You’re trying to mash up two very different sets of requirements. With these exceptions, the galaxy-motivated requirement that there is some physical aspect of dark matter that somehow knows about the distribution of baryons and organizes itself appropriately is not being used to inform the construction of dark matter models. The people who do that work seem to be very knowledgeable about cosmological constraints, but their knowledge of galaxy dynamics seems to begin and end with the statement that rotation curves are flat and therefore we need dark matter. That sufficed 40 years ago, but we’ve learned a lot since then. It’s not good enough just to have extra mass. That doesn’t cut it.

So in summary, we have two very different requirements on the dark matter. From a cosmological perspective, we need it to be dynamically cold. Something non baryonic that does not interact with photons or easily with baryons.

From a galactic perspective, we need something that knows intimately about what the baryons are doing. And when one does one thing, the other does a corresponding thing that always adds up to looking like MOND. If it doesn’t add up to looking like MOND, then it’s wrong.

So that’s where we’re at right now. These two requirements are both imperative – and contradictory.

* There is a knee-jerk response to say “mass tells light where to go” that sound wise but is actually stupid. This is a form of misdirection that gives the illusion of deep thought without the bother of actually engaging in it.

Is NGC 1277 a problem for MOND?

Alert reader Dan Baeckström recently asked about NGC 1277, as apparently some people have been making this out to be some sort of death knell for MOND.

My first reaction was NGC who? There are lots of galaxies in the New General Catalog (new in 1888, even then drawing heavily on earlier work by the Herschels). I’m well acquainted with many individual galaxies, and can recall many dozens by name, but I do not know every single thing in the NGC. So I looked it up.

*NGC 1277 in the Perseus cluster*. Photo credit: NASA, ESA, M. Beasley, & P. Kehusmaa

NGC 1277 is a lenticular galaxy. Early type. Lots of old stars. These types of galaxies tend to be baryon dominated in their centers. One might even describe them as having a dearth of dark matter. This is expected in MOND, as the stars are sufficiently concentrated that these objects are in the high acceleration regime near their centers. The modification only appears when the acceleration drops below a₀ = 1.2 x 10^-10 m/s/s; when accelerations are above this scale, everything is Newtonian – no modification, no need for dark matter.

So, is NGC 1277 special in some way? Why does this come up now?

There is a recent paper on NGC 1277 by Comerón et al. that seems to be the source of the claims of a death knell. The title is The massive relic galaxy NGC 1277 is dark matter deficient. That sounds normal for this type of galaxy, but I guess if you disliked MOND without understanding it, you might misinterpret that title to mean there was no mass discrepancy at all, hence a problem for MOND. I guess. I’m an expert on the subject; I don’t know where non-experts get their delusions.

The science paper by Comerón et al. is a nice analysis of reasonably high quality observations of the kinematics of this galaxy. Not seeing what the worry is. Here is their Fig. 19, which summarizes the enclosed mass distribution:

*Three-dimensional cumulative mass profiles of NGC 1277 (Fig. 19 of Comerón et al*.) Stars and the central black hole account for everything within the observed radius; dark matter (colored bands) is not yet needed.

The first thing I did was eyeball this plot and calculate the circular speed of a test particle at 10 kpc near the edge of the plot. Newton taught us that V² = GM/R, and the enclosed mass there looks to be just shy of 2 x 10¹¹ solar masses, so V = 290 km/s. That’s big, but also normal for a massive galaxy like this. The corresponding centripetal acceleration V²/R is about 2a₀. As expected, this galaxy is in the high acceleration regime, so MOND predicts Newtonian behavior. That means the stars suffice to explain the dynamics; no need for dark matter over this range of radii.

The second thing I did was check to see what Comerón et al. said about it themselves. They specifically address the issue, saying

One might be tempted to use the fact that NGC 1277 lacks detectable dark matter to speculate about the (in)existence of Milgromian dynamics (also known as MOND; Milgrom 1983) or other alternatives to the ΛCDM paradigm. Given a centrally concentrated baryonic mass of M_⋆ ≈ 1.6 × 10¹¹ M_⊙ and an acceleration constant a₀ = 1.24 × 10⁻¹⁰ m s⁻² (McGaugh 2011), a radius R = 13 kpc should be explored to be able to probe the fully Milgromian regime. This is about twice the radius that we cover and therefore our data do not permit studying the Milgromian regime
Comerón et al. (2023)

which is what I just said. These observations do not probe the MOND regime, and do not test theory. So, in order to think this work poses a problem for MOND, you have to (i) not understand MOND and (ii) not bother to read the paper.

I wish I could say this was unusual. Unfortunately, it is only a bit sub-par for the course. A lot of people seem to hate MOND. I sympathize with that; I was really angry the first time it came up in my data. But I got over it: anger is not conducive to a rational assessment of the evidence. A lot of people seem to let their knee-jerk dislike of the idea completely override their sense of objectivity. All too often, they don’t even bother to do minimal fact checking.

As Romanowsky et al. pointed out, the dearth of dark matter near the centers of early type galaxies is something of a problem for the dark matter paradigm. As always, this depends on what dark matter actually predicts. The most obvious expectation is that galaxies form in cuspy dark matter halos with a high concentration of dark matter towards the center. The infall of baryons acts to further concentrate the central dark matter. So the nominal expectation is that there should be plenty of dark matter near the centers of galaxies rather than none at all. That’s not what we see here, so nominally NGC 1277 presents more of a challenge for the dark matter paradigm than it does for MOND. It makes no sense to call foul on one theory without bothering to check if the other fares better. But we seem to be well past sense and well into hypocrisy.

Checking in on Troubles with Dark Matter

It is common to come across statements like “There is overwhelming astrophysical and cosmological evidence that most of the matter in our Universe is dark matter.” This is a gross oversimplification. The astronomical data that indicates the existence of acceleration discrepancies also test the ideas we come up with to explain them. I never considered MOND until I was persuaded by the data that there were serious problems with its interpretation in terms of dark matter.

The community seems to react to problems with the dark matter interpretation in one of several ways. Physicists often seem to simply ignore them, presuming that any problems are mere astronomical details that aren’t relevant to fundamental physics. Among more serious scientists, there is a tendency to bicker over solutions, settle on something (satisfactory or not), then forget that there was ever a problem.

Benoit Famaey and I wrote a long review for Living Reviews in Relativity about a decade ago. In it, we listed some of the problems that afflicted LCDM. It is instructive to review what those were, and examine what progress has been made. The following is based on section 4 of the review. I will skip over the discussion of coincidences, which remain an issue, to focus on specific astronomical problems.

Unobserved predictions

A problem for LCDM, and indeed, any theory, is when it makes predictions that are not confirmed. Here are a list of challenges stemming from observational reality deviating from the expectations or LCDM that we identified in our review, together with an assessment of whether they remain a concern.
The bulk flow challenge
Peculiar velocities of galaxy clusters are predicted to be on the order of 200 km/s in the ΛCDM model: as massive, recently formed objects, they should be nearly at rest with respect to the frame of the cosmic microwave background. Instead, they are observed to have bulk flows of order 1000 km/s.

This appears to remain a problem, and is related to the high collision speeds of objects like the bullet cluster, which basically shouldn’t exist.

The high-z clusters challenge
Structure formation is reputed to be one of the greatest strengths of LCDM, but the observers’ experience has consistently been to find more structure in place earlier than expected. This goes back at least to the 1987 CfA redshift survey stick man figure, which may seem normal now but surprised the bejeepers out of us at the time. It also includes clusters of galaxies, which appear at higher redshift than they should. At the time, we pointed out XMMU J2235.3-2557 with a mass of of ∼ 4 × 10¹⁴ M_⊙ at z = 1.4 as being very surprising.

More recently we have El Gordo, so this remains a problem.

The Local Void challenge
Peebles has been pointing out for a long time that voids are more empty than they should be, and do not contain the population of galaxies expected in LCDM. They’re too normal, too big, and gee it would help if structure formed faster. In our review, we pointed out that the “Local Void” hosts only 3 galaxies, which is much less than the expected ∼ 20 for a typical similar void in ΛCDM.

I am not seeing much in the literature in the way of updates, so I guess this one has been forgotten and remains a problem.

The missing satellites challenge
LCDM predicts that there are many subhalos in every galactic halo, and one would naturally expect each of these to host a dwarf satellite galaxy. While galaxies like the Milky Way do have dwarf satellites, they number in the dozens when there should be thousands of subhalos. This is manifestly not the case.

The trick with this test is mapping the predicted number of halos to the corresponding galaxies that inhabit them. If there is a nonlinear relation between mass and light, then there can be fewer (or more) dwarf galaxies than halos. People seem to have decided that this problem has been solved.

It is not clear to me how the solutions map to the (contemporaneous with our review) Too Big To Fail problem in which the most massive predicted subhaloes are incompatible with hosting any of the known Milky Way satellites. It isn’t a simple nonlinearity in mass-to-light; some biggish subhalos simply don’t host galaxies, apparently, while many smaller ones do. That doesn’t make sense in terms of the many mass-dependent mechanisms that are invoked to suppress dwarf galaxy formation. Nevertheless, we are assured that it all works out.

The satellites phase-space correlation challenge
This is also known as the planes of satellites problem. At the time of our review, it had recently been recognized that the satellite galaxies of the Milky Way are observed to correlate in phase-space, lying in a seemingly rotation-supported disk. This is pretty much the opposite of what one expects in LCDM, in which subhalos are on randomly oriented, radial orbits.

The problem has gotten worse with more planes now being known around Andromeda and Centaurus A and other galaxies. There have been a steady stream of papers asserting that this is not a problem, but the “solution” seems to be to declare planes to be “common” if their incidence in simulations is a few percent. That is, they seem to agree with the observers who point out that this is a problem, and simply declare it not to be a problem.

The cusp-core challenge
The cusp-core problem is that cold dark matter halos are predicted to have cuspy central regions in which the density of dark matter rises continuously towards their centers, while fitting a dark matter mass distribution to observed galaxies prefers cored halos with a rougly constant density within some finite radius. This has a long history. Observers traditionally used the pseudoisothermal halo profile (with a constant density core) to fit rotation curve data. This was the standard model for a decade before CDM simulations predicted the presence of a central cusp. The pseudoisothermal halo continues to provide a better description of the data. The initial reaction of the theoretical community was to blame the data for not conforming to their predictions: they came up with a series of lame excuses (beam smearing, slit misplacement) for why the data were wrong. Serial improvements in the quality of data showed that these ideas were wrong, and effort switched from reality denial to model modification.

People generally seem to think this problem is solved through the use of baryon feedback to erase the cusps from galaxy halos. I do not find these explanations satisfactory, as they require a just-so fine-tuning to get things right. More generally, this is just one aspect of the challenge presented by galaxy kinematic data. This is what happens if you insist on fitting dark matter halos to data the looks like what MOND predicts. Lots of people seem to think that explaining the cusp-cpore problem solves everything, but this is just one piece of a more general problem, which is not restricted to the central regions. Ultimately, the question remains why MOND works at all in a universe run by dark matter.

I mention all this because it is the prototypical example of why one should take the claims of theorists to have solved a problem with a huge grain of salt. Here, the problem has been redefined into something more limited, then the limited problem has been solved in a seemingly-plausible yet unconvincing way, victory is declared, and the original, more difficult problem (MOND works when it should not) is forgotten or considered to be solved by extension.

The angular momentum challenge
During galaxy formation, the baryons sink to the centers of their dark matter halos. A persistent idea is that they spin up as they do so (like a figure skater pulling her arms in), ultimately establishing a rotationally supported equilibrium in which the galaxy disk is around ten or twenty times smaller than the dark matter halo that birthed it, depending on the initial spin of the halo. This is a seductively simple picture that still has many adherents despite never having really worked. In live simulations, in which baryonic and dark matter particles interact, there is a net transfer of angular momentum from the baryonic disk to the dark halo. This results in simulated disks being much too small.

This problem is solved by invoking just-so feedback again. Whether the feedback one needs to solve this problem is consistent with the feedback one needs to solve the cusp-core problem is unclear, in large part because different groups have different implementations of feedback that all do different things. At most one of them can be right. Given familiarity with the approximations involved, a more likely number is Zero.

The pure disk challenge
Structure forms hierarchically in CDM: small galaxies merge into larger ones. This process is hostile to the existence of dynamically cold, rotating disks, preferring instead to construct dynamically hot, spheroidal galaxies. All the merging destroys disks. Yet spiral galaxies are ubiquitous, and many late type galaxies have no central bulge component at all. At some point it was recognized that the existence of quiescent disks didn’t make a whole lot of sense in LCDM. To form such things, one needs to let gas dissipate and settle into a plane without getting torqued and bombarded by lots of lumps falling onto it from random directions. Indeed, it proved difficult to form large, bulgeless, thin disk galaxies in simulations.

The solution seems to be just-so feedback again, though I don’t see how that can preclude the dynamical chaos caused by merging dark matter halos regardless of what the baryons do.

The stability challenge
One of the early indications of the need for spiral galaxies to be embedded in dark matter halos was the stability of disks. Thin, dynamically cold spiral disks are everywhere around us, yet Newton can’t hold them together by himself: simulated spirals self destruct on a short timescale (a few orbits). A dark matter halo precludes this from happening by counterbalancing the self-gravity of the disk. This is a somewhat fine-tuned situation: too little halo, and a disk goes unstable; too much and disk self-gravity is suppressed – and spiral arms and bars along with it.

I recognized this as a potential test early on. Dark matter halos tend to over-stabilize low surface density disks against the formation of bars and spirals. You need a lot of dark matter to explain the rotation curve, but not too much so as to allow for spiral structure. These tensions can be contradictory, and the tension I anticipated long ago has been realized in subsequent analyses.

The low surface brightness spiral F568-1 (left) and its rotation curve (right). The heavy line indicates the stellar disk mass required to sustain the observed spiral arms; the light line shows what is reasonable for a normal stellar population for which the galaxy consistent with the BTFR and RAR. We can’t have it both ways; this is the predicted contradiction to invoking dark matter to explain both disk stability and kinematics.

I’m not aware of this problem being addressed in the context of cold dark matter models, much less solved. The problem is very much present in modern hydrodynamical simulations, as illustrated by this figure from the enormous review by Banik & Zhao:

The pattern speeds of bars as observed and simulated. Real bars are fast (R = 1) while simulated bars are slow (R > 2) due to the excessive dynamical friction from cuspy dark matter halos. (Fig. 21 from Banik & Zhao 2022).

The missing baryons challenge
The cosmic fraction of baryons – the ratio of normal matter to dark matter – is well known (16 ± 1%). One might reasonably expect individual CDM halos to be in in possession of this universal baryon fraction: the sum of the stars and gas in a galaxy should be 16% of the total, mostly dark mass. However, most objects fall well short of this mark, with the only exception being the most massive clusters of galaxies. So where are all the baryons?

The answer seems to be that we don’t have to answer that. Initially, the poroblem was overcooling: low mass galaxies should turn more of their baryons into stars than is observed. Feedback was invoked to prevent that, and it seems to be widely accepted that feedback from those stars that do form heat much of the surrounding gas so it remains mixed in with the halo in some conveniently unobservable form, or that the feedback is so vigorous that it expells the excess baryons entirely. That the observed baryon fraction declines with declining mass is attributed to the lesser potential wells of smaller galaxies not being able to hang on to their baryons as well – they are more readily expelled. That sounds reasonable at a hand-waving level, but getting it right quantitatively presents a fine-tuning problem: the observed baryon fraction correlates strongly with mass with practically no scatter. One would expect feedback to be rather stochastic and result in a lot of scatter, but if it did it would propagate straight into the Tully-Fisher relation, which has practically no scatter. This fine-tuning problem is addressed by ignoring it.

The more things change

So those are the things that concerned us a decade ago. Looking back on them, there has been some progress on some items and less on others. Being generous, I would say there has at least been progress on the missing satellite problem, cusp-core, angular momentum, and pure disks. There has been no perceptible progress on the other problems, some of which (high-z clusters, disk stability) have gotten worse.

This is all written in the context of dark matter, with only passing reference to MOND. How does MOND fare for these same issues? MOND is good at making things move fast; it naturally predicts the scale of the bulk flows. It also predicted early structure formation, and is good at sweeping the voids clean. It has nothing to say about missing satellites. There are no subhalos that might be populated with dwarfs in MOND, so the question doesn’t arise. It might provide an explanation for the planes of satellites, but I am underwhelmed by this idea (or any others that I’ve heard for this particular problem). MOND is the underlying cause of the cusp-core problem, which arises entirely from trying to fit dark matter halos to galaxies that obey MOND. MOND suffers no angular momentum problem; what you see is what you get. It is noteworthy that angular momentum is not an additonal free parameter as there is no dark component with an unspecified quantity of it; it is specified entirely by the observed distribution of baryons and their motions. Similarly, making pure disks is not a problem for MOND. One can have hierarchical structure formation, but it is not required to the degree that it wipes out nascent disks in the way it did in LCDM simulations before steps were taken to make them stop doing that. Disk stability in MOND stems from the longer range of the force law rather than piling on dark matter; it is comparable for high surface brightness galaxies in both theories, but readily distinguishable for low surface brightness galaxies. This test clearly prefers MOND. Finally, the missing baryon problem doesn’t really pertain in MOND. Objects just have the baryons they have; only in rich clusters of galaxies is there a residual missing baryon problem (albeit a serious one!)

At a conservative count, that is four distinct items that have nothing to do with rotation curves where MOND performs better than LCDM. But go ahead, tell me again how MOND only explains rotation curves and nothing else.

This was basically just section 4.2 of the review. Section 4.3 was about unexpected observations – observations that were surprising in the context of LCDM. I think this post is been long enough, so I won’t go there except to say that these unexpected things were either predicted a priori by MOND, or follow so naturally from it that they could have been if the question had been posed. So it’s not just that MOND explains some things better than dark matter, it’s that it correctly predicted in advance things that were not predicted by dark matter, and that are often not well-explained by it.

The situation remains incommensurate.

The MOND at 40 conference

I’m back from the meeting in St. Andrews, and am mostly recovered from the jet lag and the hiking (it was hot and sunny, we did not pack for that!) and the driving on single-track roads like Mr. Toad. The A835 north from Ullapool provides some spectacular mountain views, but the A837 through Rosehall is more perilous carnival attraction than well-planned means of conveyance.

As expected, the most contentious issue was that of wide binaries. The divide was stark: there were two talks finding nary a hint of MONDian signal, just old Newton, and two talks claiming a clear MONDian signal. Nothing was resolved in the sense of one side convincing the other it was right, but there was progress in terms of [mostly] amicable discussion, with some sensible suggestions for how to proceed. One suggestion was that a neutral party should provide all the groups with several sets of mock data, one Newtonian, one MONDian, and one something else, to see if they all recovered the right answers. That’s a good test in principle, but it is a hassle to do in practice, as it is highly nontrivial to produce realistic mock Gaia data, so no one was leaping at the opportunity to stick their hand in this particular bear trap.

Xavier Hernandez made the excellent point that one should check that one’s method recovers Newtonian behavior for close binaries before making any claims to require/exclude such behavior for wide binaries. Neither MOND nor dark matter predicts any deviation from Newtonian behavior where stars are orbiting each other well in excess of a₀, of which there are copious examples, so they provide a touchstone on which all should agree. He also convinced me that it was a Good Idea to have radial velocities as well as proper motions. This limits the sample size, but it helps immensely to insure that sample binaries are indeed bound pairs of binary stars. Doing this, he finds MOND-like behavior.

Previously, I linked to a talk by Indranil Banik, who found Newtonian behavior. This led to an exchange with Kyu-Hyun Chae, who has now posted an update to his own analysis in which he finds MONDian behavior. It is a clear signal, and if correct, could be the smoking gun for MOND. It wouldn’t be the first one; that honor probably goes to NGC 1560, and there have been plenty of other smoking guns since then. The trick seems to be finding something than cannot be explained with dark matter, and this could play that role since dark matter shouldn’t be relevant to binary stars. But dark matter is pretty much the ultimate Rube Goldberg machine of science, so we’ll see explanation people come up with, should they need to do so.

At present, the facts of the matter are still in dispute, so that’s the first thing to get straight.

Thanks to everyone I met at the conference who told me how useful this blog is. That’s good to know. Communication is inefficient at best, counterproductive at worst, and most often practically nonexistent. So it is good to hear that this does some small good.

Commentary on Wide Binaries

Wide Binary Weirdness

Last time, I commented on the developing situation with binary stars as a test of MOND. I neglected to enable comments for that post, so have done so now.

Indranil Banik has shared his perspective on wide binaries in a talk on the subject that is available on Youtube, included below.

Indranil and his collaborators are not seeing a MOND effect in wide binaries. Others have, as I discussed in the previous post. After the video posted above, Indranil comments on the work of Kyu-Hyun Chae:

Regarding the article by Chae (https://arxiv.org/abs/2305.04613), equation 7 of MNRAS 506, 2269–2295 (2021) shows that the relative velocity is limited such that the v_tilde parameter (ratio of relative velocity within the sky plane to the Newtonian circular velocity at the projected separation) is at most 1 for 5 M_Sun binaries and in general is sqrt(5 M_Sun/M) for a binary of total mass M. This means v_tilde only goes up to 2 for M = 1.25 M_Sun, but more generally it goes up to a higher value at lower mass. Since the main signal in MOND is a broader v_tilde distribution at lower acceleration and a lower mass reduces the acceleration, this can lead to an artificial signal whereby lower mass systems have a larger rms v_tilde. Now a simple rms statistic is not exactly what Chae did, but this does highlight the kind of problem that can arise. Indeed, the v_tilde distribution prepared by Chae for the article in its figure 25 does show a rather sharp decline in the v_tilde distribution – there is not much of an extended tail, even less than in the model! This is obviously not due to measurement errors and contaminating effects like chance alignments, which would broaden the tail further. Rather, it is due to the upper limit to v_tilde imposed from the sample selection. This just means the underlying sample used is not well suited to the wide binary test, since it was quite clear a priori that the main signal for MOND would be in the region of v_tilde = 1-1.5 or so. One possibility is to try and restrict the analysis to a narrower range of binary total mass to try and alleviate the above concern, in which case the upper limit to v_tilde would be perhaps above 2 for the full sample used. There is however another issue in that lower accelerations generally correspond to higher separations and thus lower orbital velocities, so the fractional uncertainty in the velocity is likely to be larger. Thus, the v_tilde distribution is likely to be broader at low accelerations. This can be counteracted by having low errors across the board, but then the key quantity is the uncertainty on v_tilde. This aspect is not handled very rigorously – it is assumed that if the proper motions are accurate to better than 1%, then v_tilde will be sufficiently well known. But if the tangential velocity is about 20 km/s, a 1% error means an error of 200 m/s on the velocity of each star, so the relative velocity has an uncertainty of about 280 m/s. This is quite large compared to typical wide binary relative velocities, which are generally a few hundred m/s. Without doing a more detailed analysis, perhaps one thing to do would be to change this 1% requirement to 0.5% or 1.5% and see what happens. I am therefore not convinced that the MOND signal claimed by Chae is genuine.
I. Banik

Kyu-Hyun Chae responded to that, but apparently many people are not able to see his response on Youtube. I cannot. So I asked him about it, and he shares it here:

Since Indranil sent this concern to me in person, I’m replying here. No cut on v_tilde is used in my analysis because it is a gravity test. I did not use equation 7 of El-Badry et al. (MNRAS 506, 2269–2295 (2021)) to cut out high v_tilde data, so there are some (though relatively small number of) data points above equation (7). I removed chance alignment cases by requiring R < 0.01 (El-Badry et al. convincingly show that R can be used to effectively remove chance alignment cases). This is the main reason why there is no high velocity tail. I have already considered varying proper motion (PM) relative errors: there are three cases PM rel error < 0.01 (nominal case), <0.003 (smaller case), and <0.2 (larger case). The conclusion on gravity anomaly (MOND signal) is the same in all three cases although the fitted f_multi (multiplicity fraction) varies. We can have more discussion in the st Andrews June meeting. I’m sure it will take some time but you will be convinced that my results are correct.
K.-H. Chae

He also shares this figure:

This is how the science sausage is made. As yet, there is no consensus.

Wide Binary Weirdness

My last post about the Milky Way was intended to be a brief introduction to our home galaxy in order to motivate the topic of binary stars. There’s too much interesting to say about the Milky Way as a galaxy, so I never got past that. Even now I feel the urge to say more, like with this extended rotation curve that I included in my contribution to the proceedings of IAU 379.

The RAR-based model rotation curve of the Milky Way extrapolated to large radii (note the switch to a logarithmic scale at 20 kpc!) for comparison to the halo stars of Bird et al (2022) and the globular clusters of Watkins et al (2019). The location of the solar system is noted by the red circle.

But instead I want to talk about data for binary stars from the Gaia mission. Gaia has been mapping the positions and proper motions of stars in the local neighborhood with unprecedented accuracy. These can be used to measure distances via trigonometric parallax, and speeds along the sky. The latter once seemed impossible to obtain in numbers with much precision; thanks to Gaia such data now outnumber radial (line of sight) velocities of comparable accuracy from spectra. That is a mind-boggling statement to anyone who has worked in the field; for all of my career (and that of any living astronomer), radial velocities have vastly outnumbered comparably well-measured proper motions. Gaia has flipped that forever-reality upside down in a few short years. It’s third data release was in June of 2022; this provides enough information to identify binary stars, and we’ve had enough time to start (and I do mean start) sorting through the data.

OK, why are binary stars interesting to the missing mass (really the acceleration discrepancy) problem? In principle, they allow us to distinguish between dark matter and modified gravity theories like MOND. If galactic mass discrepancies are caused by a diffuse distribution of dark matter, gravity is normal, and binary stars should orbit each other as Newton predicts, no matter their separation: the dark matter is too diffuse to have an impact on such comparatively tiny scales. If instead the force law changes at some critical scale, then the orbital speeds of widely separated binary pairs that exceed this scale should get a boost relative to the Newtonian case.

The test is easy to visualize for a single binary system. Imagine two stars orbiting one another. When they’re close, they orbits as Newton predicts. This is, after all, how we got Newtonian gravity – as an explanation for Kepler’s Laws or planetary motion. Ours is a lonely star, not a binary, but that makes no difference to gravity: Jupiter (or any other planet) is an adequate stand-in. Newton’s universal law of gravity (with tiny tweaks from Einstein) is valid as far out in the solar system as we’ve been able to probe. For scale, Pluto is about 40 AU out (where Earth, by definition, is 1 AU from the sun).

Let’s start with a pair of stars orbiting at a distance that is comfortably in the Newtonian regime, say with a separation of 40 AU. If we know the mass of the stars, we can calculate what their orbital speed will be. Now imagine gradually separating the stars so they are farther and farther apart. For any new separation s, we can predict what the new orbital speed will be. According to Newton, this will decline in a Keplerian fashion, v ~ 1/√s. This will continue indefinitely if Newton remains forever the law of the land. If instead the force law changes at some critical scale s_c, then we would expect to see a change when the separation exceeds that scale. Same binary pair, same mass, but relatively faster speed – a faster speed that on galaxy scales leads to the inference of dark matter. In essence, we want to check if binary stars also have flat rotation curves if we look far enough out.

We have long known that simply changing the force law at some length scale s_c does not work. In MOND, the critical scale is an acceleration, a₀. This will map to a different s_c for binary stars of different masses. For the sun, the critical acceleration scale is reached at s_c ≈ 7000 AU ≈ 0.034 parsecs (pc), about a tenth of a light-year. That’s a lot bigger than the solar system (40 AU) but rather smaller than the distance to the next star (1.3 pc = 4.25 light-years). So it is conceivable that there are wide binaries in the solar neighborhood for which this test can be made – pairs of stars with separations large enough to probe the MOND regime without being so far apart that they inevitably get broken up by random interactions with unrelated stars.

Gaia is great for identifying binaries, and space is big. There are thousands of wide binaries within 200 pc of the sun where Gaia can obtain excellent measurements. That’s not a big piece of the galaxy – it is a patch roughly the size of the red circle in the rotation curve plot above – but it is still a heck of a lot of stars. A signal should emerge, and a number of papers have now appeared that attempt this exercise. And ooooo-buddy, am I confused. Frequent readers will have noticed that it has been a long time between posts. There are lots of reasons for this, but a big one is that every time I think I understand what is going on here, another paper appears with a different result.

OK, first, what do we expect? Conventionally, binaries should show Keplerian behavior whatever their separation. Dark matter is not dense enough locally to have any perceptible impact. In MOND, one might expect an effect analogous to the flattening of rotation curves, hence higher velocities than predicted by Newton. And that’s correct, but it isn’t quite that simple.

In MOND, there is the External Field Effect (EFE) in which the acceleration from distant sources can matter to the motion of a local system. This violates the strong but not the weak Equivalence Principle. In MOND, all accelerative tugs matter, whereas conventionally only local effects matter.

This is important here, as we live in a relatively high acceleration neighborhood that is close to a₀. The acceleration the sun feels towards the Galactic center is about 1.8 a₀. This applies to all the stars in the solar neighborhood, so even if one finds a binary pair that is widely separated enough for the force of one star on another to be less than a₀, they both feel the 1.8 a₀ of the greater Galaxy. A lot of math intervenes, with the net effect being that the predicted boost over Newton is less than it would have been in the absence of this effect. There is still a boost, but its predicted amplitude is less than one might naively hope.

*The location of the solar system along the radial acceleration relation is roughly (g_bar, g_obs) = (1.2, 1.8)* a₀*. At this acceleration, the effects of MOND are just beginning to appear, and the external field of the Galaxy can affect local binary stars.*

One of the first papers to address this is Hernandez et al (2022). They found a boost in speed that looks like MOND but is not MOND. Rather, it is consistent with the larger speed that is predicted by MOND in the absence of the EFE. This implies that the radial acceleration relation depicted above is absolute, and somehow more fundamental than MOND. This would require a new theory that is very similar to MOND but lacks the EFE, which seems necessary in other situations. Weird.

A thorough study has independently been made by Pittordis & Sutherland (2023). I heard a talk by them over Zoom that motivated the previous post to set the stage for this one. They identify a huge sample of over 73,000 wide binaries within 300 pc of the sun. Contrary to Hernandez et al., they find no boost at all. The motions of binaries appear to remain perfectly Keplerian. There is no hint of MOND-like effects. Different.

OK, so that is pretty strong evidence against MOND, as Indranil Banik was describing to me at the IAU meeting in Potsdam, which is why I knew to tune in for the talk by Pittordis. But before I could write this post, yet another paper appeared. This preprint by Kyu-Hyun Chae splits the difference. It finds a clear excess over the Newtonian expectation that is formally highly significant. It is also about right for what is expected in MOND with the EFE, in particular with the AQUAL flavor of MOND developed by Bekenstein & Milgrom (1984).

So we have one estimate that is MOND-like but too much for MOND, one estimate that is straight-laced Newton, and one estimate that is so MOND that it can start to discern flavors of MOND.

I really don’t know what to make of all this. The test is clearly a lot more complicated than I made it sound. One does not get to play God with a single binary pair; one instead has to infer from populations of binaries of different mass stars whether a statistical excess in orbital velocity occurs at wide separations. This is challenging for lots of reasons.

For example, we need to know the mass of each star in each binary. This can be gauged by the mass-luminosity relation – how bright a main sequence star is depends on its mass – but this must be calibrated by binary stars. OK, so, it should be safe to use close binaries that are nowhere near the MOND limit, but it can still be challenging to get this right for completely mundane, traditional astronomical reasons. It remains challenging to confidently infer the properties of impossibly distant physical objects that we can never hope to visit, much less subject to laboratory scrutiny.

Another complication is the orientation and eccentricity of orbits. The plane of the orbit of each binary pair will be inclined to our line of sight so that the velocity we measure is only a portion of the full velocity. We do not have any way to know what the inclination of any one wide binary is; it is hard enough to identify them and get a relative velocity on the plane of the sky. So we have to resort to statistical estimates. The same goes for the eccentricities of the orbits: not all orbits are circles; indeed, most are not. The orbital speed depends on where an object is along its elliptical orbit, as Kepler taught us. So yet again we must make some statistical inference about the distribution of eccentricities. These kinds of estimates are both doable and subject to going badly wrong.

The net effect is that we wind up looking at distributions of relative velocities, and trying to perceive whether there is an excess high-velocity tail over and above the Newtonian expectation. This is far enough from my expertise that I do not feel qualified to judge between the works cited above. It takes time to sort these things out, and hopefully we can all come to agreement on what it is that we’re seeing. Right now, we’re not all seeing eye-to-eye.

There is a whole session devoted to this topic at the upcoming meeting on MOND. The primary protagonists will be there, so hopefully some progress can be made. At least it should be entertaining.

Triton Station

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

Author: tritonstation

Recent Developments Concerning the Gravitational Potential of the Milky Way. II. A Closer Look at the Data

Recent Developments Concerning the Gravitational Potential of the Milky Way. I.

The total mass of the Milky Way

Is the Milky Way’s rotation curve declining?

Wide Binary Results Favoring MOND

Required dark matter properties

Is NGC 1277 a problem for MOND?

Checking in on Troubles with Dark Matter

Unobserved predictions

The more things change

The MOND at 40 conference

Commentary on Wide Binaries

Wide Binary Weirdness

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