Discussion of Dark Matter and Modified Gravity

To start the new year, I provide a link to a discussion I had with Simon White on Phil Halper’s YouTube channel:

In this post I’ll say little that we don’t talk about, but will add some background and mildly amusing anecdotes. I’ll also try addressing the one point of factual disagreement. For the most part, Simon & I entirely agree about the relevant facts; what we’re discussing is the interpretation of those facts. It was a perfectly civil conversation, and I hope it can provide an example for how it is possible to have a positive discussion about a controversial topic⁺ without personal animus.

First, I’ll comment on the title, in particular the “vs.” This is not really Simon vs. me. This is a discussion between two scientists who are trying to understand how the universe works (no small ask!). We’ve been asked to advocate for different viewpoints, so one might call it “Dark Matter vs. MOND.” I expect Simon and I could swap sides and have an equally interesting discussion. One needs to be able to do that in order to not simply be a partisan hack. It’s not like MOND is my theory – I falsified my own hypothesis long ago, and got dragged reluctantly into this business for honestly reporting that Milgrom got right what I got wrong.

For those who don’t know, Simon White is one of the preeminent scholars working on cosmological computer simulations, having done important work on galaxy formation and structure formation, the baryon fraction in clusters, and the structure of dark matter halos (Simon is the W in NFW halos). He was a Reader at the Institute of Astronomy at the University of Cambridge where we overlapped (it was my first postdoc) before he moved on to become the director of the Max Planck Institute for Astrophysics where he was mentor to many people now working in the field.

That’s a very short summary of a long and distinguished career; Simon has done lots of other things. I highlight these works because they came up at some point in our discussion. Davis, Efstathiou, Frenk, & White are the “gang of four” that was mentioned; around Cambridge I also occasionally heard them referred to as the Cold Dark Mafia. The baryon fraction of clusters was one of the key observations that led from SCDM to LCDM.

The subject of galaxy formation runs throughout our discussion. It is always a fraught issue how things form in astronomy. It is one thing to understand how stars evolve, once made; making them in the first place is another matter. Hard as that is to do in simulations, galaxy formation involves the extra element of dark matter in an expanding universe. Understanding how galaxies come to be is essential to predicting anything about what they are now, at least in the context of LCDM^*. Both Simon and I have worked on this subject our entire careers, in very much the same framework if from different perspectives – by which I mean he is a theorist who does some observational work while I’m an observer who does some theory, not LCDM vs. MOND.

When Simon moved to Max Planck, the center of galaxy formation work moved as well – it seemed like he took half of Cambridge astronomy with him. This included my then-office mate, Houjun Mo. At one point I refer to the paper Mo & I wrote on the clustering of low surface brightness galaxies and how I expected them to reside in late-forming dark matter halos^**. I often cite Mo, Mao, & White as a touchstone of galaxy formation theory in LCDM; they subsequently wrote an entire textbook about it. (I was already warning them then that I didn’t think their explanations of the Tully-Fisher relation were viable, at least not when combined with the effect we have subsequently named the diversity of rotation curve shapes.)

When I first began to worry that we were barking up the wrong tree with dark matter, I asked myself what could falsify it. It was hard to come up with good answers, and I worried it wasn’t falsifiable. So I started asking other people what would falsify cold dark matter. Most did not answer. They often had a shocked look like they’d never thought about it, and would rather not^***. It’s a bind: no one wants it to be false, but most everyone accepts that for it to qualify as physical science it should be falsifiable. So it was a question that always provoked a record-scratch moment in which most scientists simply freeze up.

Simon was one of the first to give a straight answer to this question without hesitation, circa 1999. At that point it was clear that dark matter halos formed central density cusps in simulations; so those “cusps had to exist” in the centers of galaxies. At that point, we believed that to mean all galaxies. The question was complicated by the large dynamical contribution of stars in high surface brightness galaxies, but low surface brightness galaxies were dark matter dominated down to small radii. So we thought these were the ideal place to test the cusp hypothesis.

We no longer believe that. After many attempts at evasion, cold dark matter failed this test; feedback was invoked, and the goalposts started to move. There is now a consensus among simulators that feedback in intermediate mass galaxies can alter the inner mass distribution of dark matter halos. Exactly how this happens depends on who you ask, but it is at least possible to explain the absence of the predicted cusps. This goes in the right direction to explain some data, but by itself does not suffice to address the thornier question of why the distribution of baryons is predictive of the kinematics even when the mass is dominated by dark matter. This is why the discussion focused on the lowest mass galaxies where there hasn’t been enough star formation to drive the feedback necessary to alter cusps. Some of these galaxies can be described as having cusps, but probably not all. Thinking only in those terms elides the fact that MOND has a better record of predictive success. I want to know why this happens; it must surely be telling us something important about how the universe works.

The one point of factual disagreement we encountered had to do with the mass profile of galaxies at large radii as traced by gravitational lensing. It is always necessary to agree on the facts before debating their interpretation, so we didn’t press this far. Afterwards, Simon sent a citation to what he was talking about: this paper by Wang et al. (2016). In particular, look at their Fig. 4:

Fig. 4 of Wang et al. (2016). The excess surface density inferred from gravitational lensing for galaxies in different mass bins (data points) compared to mock observations of the same quantity made from within a simulation (lines). Looks like excellent agreement.

This plot quantifies the mass distribution around isolated galaxies to very large scales. There is good agreement between the lensing observations and the mock observations made within a simulation. Indeed, one can see an initial downward bend corresponding to the outer part of an NFW halo (the “one-halo term”), then an inflection to different behavior due to the presence of surrounding dark matter halos (the “two-halo term”). This is what Simon was talking about when he said gravitational lensing was in good agreement with LCDM.

I was thinking of a different, closely related result. I had in mind the work of Brouwer et al. (2021), which I discussed previously. Very recently, Dr. Tobias Mistele has made a revised analysis of these data. That’s worthy its own post, so I’ll leave out the details, which can be found in this preprint. The bottom line is in Fig. 2, which shows the radial acceleration relation derived from gravitational lensing around isolated galaxies:

*The radial acceleration relation from weak gravitational lensing (colored points) extending existing kinematic data (grey points) to lower acceleration corresponding to very large radii (~ 1 Mpc)*. The dashed line is the prediction of MOND. Looks like excellent agreement.

This plot quantifies the radial acceleration due to the gravitational potential of isolated galaxies to very low accelerations. There is good agreement between the lensing observations and the extrapolation of the radial acceleration relation predicted by MOND. There are no features until extremely low acceleration where there may be a hint of the external field effect. This is what I was talking about when I said gravitational lensing was in good agreement with MOND, and that the data indicated a single halo with an r^-2 density profile that extends far out where we ought to see the r^-3 behavior of NFW.

The two plots above use the same method applied to the same kind of data. They should be consistent, yet they seem to tell a different story. This is the point of factual disagreement Simon and I had, so we let it be. No point in arguing about the interpretation when you can’t agree on the facts.

I do not know why these results differ, and I’m not going to attempt to solve it here. I suspect it has something to do with sample selection. Both studies rely on isolated galaxies, but how do we define that? How well do we achieve the goal of identifying isolated galaxies? No galaxy is an island; at some level, there is always a neighbor. But is it massive enough to perturb the lensing signal, or can we successfully define samples of galaxies that are effectively isolated, so that we’re only looking at the gravitational potential of that galaxy and not that of it plus some neighbors? Looks like there is some work left to do to sort this out.

Stepping back from that, we agreed on pretty much everything else. MOND as a fundamental theory remains incomplete. LCDM requires us to believe that 95% of the mass-energy content of the universe is something unknown and perhaps unknowable. Dark matter has become familiar as a term but remains a mystery so long as it goes undetected in the laboratory. Perhaps it exists and cannot be detected – this is a logical possibility – but that would be the least satisfactory result possible: we might as well resume counting angels on the head of a pin.

The community has been working on these issues for a long time. I have been working on this for a long time. It is a big problem. There is lots left to do.

⁺I get a lot of kill the messenger from people who are not capable of discussing controversial topics without personal animus. A lot – inevitably from people who know assume they know more about the subject than I do but actually know much less. It is really amazing how many scientists equate me as a person with MOND as a theory without bothering to do any fact-checking. This is logical fallacy 101.

^*The predictions of MOND are insensitive to the details of galaxy formation. Though of course an interesting question, we don’t need that in order to make predictions. All we need is the mass distribution that the kinematics respond to – we don’t need to know how it got that way. This is like the solar system, where it suffices to know Newton’s laws to compute orbits; we don’t need to know how the sun and planets formed. In contrast, one needs to know how a galaxy was assembled in LCDM to have any hope of predicting what its distribution of dark matter is and then using that to predict kinematics.

^**The ideas Mo & I discussed thirty years ago have reappeared in the literature under the designation “assembly bias.”

^***It was often accompanied by “why would you even ask that?” followed by a pained, constipated expression when they realized that every physical theory has to answer that question.

Holiday Concordance

Screw the Earth and its smoking habit. The end of 2023 approaches, so let’s talk about the whole universe, which is its own special kind of mess.

As I’ve related before, our current cosmology, LCDM, was established over the course of the 1990s through a steady drip, drip, drip of results in observational cosmology – what Peebles calls the classic cosmological tests. There were many contributory results; I’m not going to attempt to go through them all. Important among them were the age problem, the realization that the mass density was lower than expected, and that there was more structure on large scales⁺ than predicted. These established LCDM in the mid-1990s as the “concordance model” – the most probable flavor of FLRW universe. Here is the key figure from Ostriker & Steinhardt depicting the then-allowed region of the density parameter and Hubble constant:

The addition of the cosmological constant to the standard model – replacing SCDM with LCDM – was a brain-wrenching ordeal. Lambda had long been anathema, and there was a region in which an open universe was possible, even reasonable (stripes over shade in the figure above). Moreover, this strange new LCDM made the seemingly inconceivable prediction that not only was the universe expanding [itself the older mind-bender brought to us by Hubble (and Slipher and Lemaître)], the expansion rate should be accelerating. This sounded like crazy talk at the time, so it was greeted with great rejoicing when corroborated by observations of Type Ia supernovae.

A further prediction that could distinguish LCDM from then-viable open models was the geometry of the universe. Open models have a negative curvature (Ω_k < 0, in which initially parallel light beams diverge) while the geometry in LCDM should be uniquely flat (Ω_k = 0, in which initially parallel light beams remain parallel forever). Uniqueness is important, as it makes for a strong prediction, such as the location of the first peak of the acoustic power spectrum of the cosmic microwave background. In LCDM, this location was predicted to be ℓ ≈ 200 with little flexibility. For viable open models, it was more like ℓ ≈ 800 with a great deal of flexibility. The interpretation of the supernova data relied heavily on the assumption of a flat geometry, so I recall breathing a sigh of relief^* when ℓ ≈ 200 was clearly observed.

Where are we now? I decided to reconstruct the Ostriker & Steinhardt plot with modern data. Here it is, with the axes swapped for reasons unrelated to this post. Deal with it.

The concordance region (white space) in the mass density-expansion rate space where the allowed regions (colored bands) of many constraints intersect. Illustrated constraints include a direct measurement of the Hubble constant, the age of the universe, the cluster baryon fraction, and large scale structure. Also shown are the best-fit values from CMB fits labeled by their date of publication (WMAP in orange; Planck in yellow). These follow the green line of constant Ω_mH₀³; combinations of parameters along the line are tolerable but regions away from it are strongly excluded.

There is lots to be said here. First, note the scale. As the accuracy of data have improved, it has become possible to zoom in. My version of the figure is a wee postage stamp on that of Ostriker & Steinhardt. Nevertheless, the concordance region is in pretty much the same spot. Not exactly, of course; the biggest thing that has changed is that the age constraint is now completely incompatible with an open universe, so I haven’t bothered depicting it. Indeed, for the illustrated Hubble constant, the Hubble time (the age of a completely empty, “coasting” universe) is 13.4 Gyr. This is consistent with the illustrated age (13.80 ± 0.75 Gyr) only for Ω_m ≈ 0, which is far off the left edge of the plot.

Second, the CMB best-fit values follow a line of constant Ω_mH₀³. This is a deep trench in χ² space. The region outside this trench is strongly excluded – it’s kinda the grand canyon of cosmology. Even a little off, and you’re standing on the rim looking a long way down, knowing that a much better fit is only a short step away. Once you’re in the valley of χ², one must hunt along its bottom to find the true minimum. In the mid-`00s, a decade after Ostriker & Steinhardt, the best fit fell smack in the middle of the concordance region defined by completely independent data. It was this additional concordance that impressed me most, more than the detailed CMB fits themselves. This convinced the vast majority of scientists practicing in the field that it had to be LCDM and could only be LCDM and nothing but LCDM.

Since that time, the best-fit CMB value has wandered down the trench, away from the concordance region. These are the results that changed, not everything else. This temporal variation suggests a systematic in the interpretation of the CMB data rather than in the local distance scale.

I recall being at a conference (the Bright & Dark Universe in Naples in 2017) when the latest Planck results were announced. There was a palpable sense in the audience of having been whacked by a blunt object, like walking into a closed door you thought was open. We’d been doing precision cosmology for a long time and had settled on an answer informed by lots of independent lines of evidence, but they were telling us the One True answer was off over there. Not crazy far, but not consistent with the concordance we had come to expect. Worse, they had these crazy tiny error bars – not only were they getting an answer outside the concordance region, it was in tension with pretty much everything else. Not strong tension, but enough to make us all uncomfortable if not outright object. Indeed, there was a definite vibe that people were afraid to object. Not terrified, but nervous. Worried about being on the wrong side of the community. I get it. I know a lot about that.

People are remarkably talented at refashioning the past. Over the past five years, the Planck best-fit parameters have come to be synonymous with LCDM: all else is moot. Young scientists can be forgiven for not realizing it was ever otherwise, just as they might have been taught that cosmic acceleration was discovered by the supernova experiments totally out of the blue. These are convenient oversimplifications that elide so many pertinent events as to be tantamount to gaslighting. We refashion the past until there was never a serious controversy, then it seems strange that some of us think there still is. Sorry, not so fast, there definitely is: if you use the Planck value of the Hubble constant to estimate distances to local galaxies, you will get it wrong^%, along with all distance-dependent quantities.

I’m old enough to remember a time when there was a factor of two uncertainty in the Hubble constant (50 vs. 1000) and the age constraint was the most accurate one in this plot. Thanks to genuine progress, the Hubble constant is now the more precise. Consequently, of all the data one could plot above, this is the choice that matters most to where the concordance region falls. If I adopt our own estimate (H₀ = 75.1 ± 2.3 km/s/Mpc), then the concordance band gets wider and slides up a little but is basically the same as above. If instead I adopt the lowest highly accurate value, H₀ = 69.8 ± 0.8 km/s/Mpc, the window slides down, but not enough to be consistent with the Planck results. Indeed, it stays to the left of the CMB constraint, becoming inconsistent with the mass density as well as the expansion rate.

Dang it, now I want to make that plot. Processing… OK, here it is:

As above, but with a lower measurement of H₀. Only the range of statistical uncertainty is illustrated as a systematic uncertainty corresponds to a calibration error that slides H₀ up and down – i.e., the exact situation being illustrated relative to the figure above. These two plots illustrate the range of outcomes that are possible from slightly discordant direct modern measurements of the Hubble constant; it is hard to go lower. Doing so doesn’t really help as it would just shift the tension from H₀ to Ω_m.

Yes, as I expected: the allowed range slides down but remains to the left of the green line. It is less inconsistent with the Planck H₀, but that isn’t the only thing that matters. It is also inconsistent with the matter density. Indeed, it misses the CMB-allowed trench entirely. There is no allowed FLRW universe here.

These are only two parameters. Though arguably the most important, there are others, all of which matter to CMB fits. These are difficult to visualize simultaneously. We could, for starters, plot the baryon density as a third axis. If we did so, the concordance region would become a 3D object. It would also get squeezed, depending on what we think the baryon density actually is. Even restricting ourselves to the above-plotted constraints, there is some tension between the cluster baryon fraction and large scale structure constraint along the new third axis. I’m sure I could find in the literature more or less consistent values; this way the madness of cherry-picking lies.

There are many other constraints that could be added here. I’ve tried to stay consistent with the spirit of the original plot without making it illegible by overburdening it with lots and lots of data that all say pretty much the same thing. Nor do I wish to engage in cherry-picking. There are so many results out there that I’m sure one could find some combination that slides the allowed box this way or that – but only a little.

Whenever I’ve taught cosmology, I’ve made it a class exercise^$ to investigate diagrams like this, with each student choosing an observational constraint to explore and champion. as a result, I’ve seen many variations on the above plots over the years, but since I first taught it in 1999 they’ve always been consistent with pretty much the same concordance region. It often happens that there is no concordance region; there are so many constraints that when you put them all together, nothing is left. We then debate which results to believe, or not, a process that has always been a part of the practice of cosmology.

We have painted ourselves into a corner. The usual interpretation is that we have painted ourselves into the correct corner: we live in this strange LCDM universe. It is also possible that there really is nothing left, the concordance window is closed, and we’ve falsified FLRW cosmology. That is a fate most fear to contemplate, and it seems less likely than mistakes in some discordant results, so we inevitably go down the path of cognitive dissonance, giving more credence to results that are consistent with our favorite set of LCDM parameters and less to those that do not. This is widely done without contemplating the possibility that the weird FLRW parameters we’ve ended up with are weird because they are just an approximation to some deeper theory.

So, as 2023 winds to an end, we [still] know pretty well what the parameters of cosmology are. While the tension between H₀ = 67 and 73 km/s/Mpc is real, it seems like small beans compared to the successful isolation of a narrow concordance window. Sure beats arguing between 50 and 100! Even deciding which concordance window is right seems like a small matter compared to the deeper issues raised by LCDM: what is the cold dark matter? Does it really exist, or is it just a mythical entity we’ve invented for the convenient calculation of cosmic quantities? What the heck do we even mean by Lambda? Does the whole picture hang together so well that it must be correct? Or can it be falsified? Has it already been? How do we decide?

I’m sure we’ll be arguing over these questions for a long time to come.

⁺Structure formation is often depicted as a great success of cosmology, but it was the failure of the previous standard model, SCDM, to predict enough structure on large scales that led to its demise and its replacement by LCDM, which now faces a similar problem. The observer’s experience has consistently been that there is more structure in place earlier and on larger scales than had been anticipated before its observation.

^*I believe in giving theories credit where credit is due. Putting on a cosmologist’s hat, the location of the first peak was a great success of LCDM. It was the amplitude of the second peak that came as a great surprise – unless you can take off the cosmology hat and don a MOND hat – then it was predicted. What is surprising from that perspective is the amplitude of the third peak, which makes more sense in LCDM. It seems impossible to some people that I can wear both hats without my head exploding, so they seem to simply assume I don’t think about it from their perspective when in reality it is the other way around.

^%As adjudicated by galaxies with distances known from direct measurements provided by Cepheids or the tip of the red giant branch or surface brightness fluctuations or geometric methods, etc., etc., etc.

^$This is a great exercise, but only works if CMB results are excluded. There has to be some narrative suspense: will the various disparate lines of evidence indeed line up? Since CMB fits constrain all parameters simultaneously, and brook no dissent, they suck the joy away from everything else in the sky and drain all interest in the debate.

Full speed in reverse!

People have been asking me about comments in a recent video by Sabine Hossenfelder. I have not watched it, but the quote I’m asked about is “the higher the uncertainty of the data, the better MOND seems to work” with the implication that this might mean that MOND is a systematic artifact of data interpretation. I believe, because they consulted me about it, that the origin of this claim emerged from recent work by Sabine’s student Maria Khelashvili on fitting the SPARC data.

Let me address the point about data interpretation first. Fitting the SPARC data had exactly nothing to do with attracting my attention to MOND. Detailed MOND fits to these data are not particularly important in the overall scheme of these things as I’ll discuss in excruciating detail below. Indeed, these data didn’t even exist until relatively recently.

It may, at this juncture in time, surprise some readers to learn that I was once a strong advocate for cold dark matter. I was, like many of its current advocates, rather derisive of alternatives, the most prominent at the time being baryonic dark matter. What attracted my attention to MOND was that it made a priori predictions that were corroborated, quite unexpectedly, in my data for low surface brightness galaxies. These results were surprising in terms of dark matter then and to this day remain difficult to understand. After a lot of struggle to save dark matter, I realized that the best we could hope to do with dark matter was to contrive a model that reproduced after the fact what MOND had predicted a priori. That can never be satisfactory.

So – I changed my mind. I admitted that I had been wrong to be so completely sure that the solution to the missing mass problem had to be some new form of non-baryonic dark matter. It was not easy to accept this possibility. It required lengthy and tremendous effort to admit that Milgrom had got right something that the rest of us had got wrong. But he had – his predictions came true, so what was I supposed to say? That he was wrong?

Perhaps I am wrong to take MOND seriously? I would love to be able to honestly say it is wrong so I can stop having this argument over and over. I’ve stipulated the conditions whereby I would change my mind to again believe that dark matter is indeed the better option. These conditions have not been met. Few dark matter advocates have answered the challenge to stipulate what could change their minds.

People seem to have become obsessed with making fits to data. That’s great, but it is not fundamental. Making a priori predictions is fundamental, and has nothing to do with fitting data. By construction, the prediction comes before the data. Perhaps this is one way to distinguish between incremental and revolutionary science. Fitting data is incremental science that seeks the best version of an accepted paradigm. Successful predictions are the hallmark of revolutionary science that make one take notice and say, hey, maybe something entirely different is going on.

One of the predictions of MOND is that the RAR should exist. It was not expected in dark matter. As a quick review of the history, here is the RAR as it was known in 2004 and now (as of 2016):

*The radial acceleration relation constructed from data available in 2004 and that from 2016.*

The big improvement provided by SPARC was a uniform estimate of the stellar mass surface density of galaxies based on Spitzer near-infrared data. These are what are used to construct the x-axis: g_bar is what Newton predicts for the observed mass distribution. SPARC was a vast improvement over the optical data we had previously, to the point that the intrinsic scatter is negligibly small: the observed scatter can be attributed to the various uncertainties and the expected scatter in stellar mass-to-light ratios. The latter never goes away, but did turn out to be at the low end of the range we expected. It could easily have looked worse, as it did in 2004, even if the underlying physical relation was perfect.

Negligibly small intrinsic scatter is the best one can hope to find. The issue now is the fit quality to individual galaxies (not just the group plot above). We already know MOND fits rotation curve data. The claim that appears in Dr. Hossenfelder’s video boils down to dark matter providing better fits. This would be important if it told us something about nature. It does not. All it teaches us about is the hazards of fitting data for which the errors are not well behaved.

While SPARC provides a robust estimate of g_bar, g_obs is based on a heterogeneous set of rotation curves drawn from a literature spanning decades. The error bars on these rotation curves have not been estimated in a uniform way, so we cannot blindly fit the data with our favorite software tool and expect that to teach us something about physical reality. I find myself having to say this to physicists over and over and over and over and over again: you cannot trust astronomical error bars to behave as Gaussian random variables the way one would like and expect in a controlled laboratory setting.

Astronomy is not conducted in a controlled laboratory. It is an observational science. We cannot put the entire universe in a box and control all the variables. We can hope to improve the data and approach this ideal, but right now we’re nowhere near it. These fitting analyses assume that we are.

Screw it. I really am sick of explaining this over and over, so I’m just going to cut & paste verbatim what I told Hossenfelder & Khelashvili by email when they asked. This is not the first time I’ve written an email like this, and I’m sure it won’t be the last.

Excruciating details: what I said to Hossenfelder & Khelashvili about the perils of rotation curve fitting on 22 September 2023 in response for their request for comments on the draft of the relevant paper:

First, the work of Desmond is a good place to look for an opinion independent of mine.

Second, in my experience, the fit quality you find is what I’ve found before: DM halos with a constant density core consistently give the best fits in terms of chi^2, then MOND, then NFW. The success of cored DM halos happens because it is an extremely flexible fitting function: the core radius and core density can be traded off to fit any dog’s leg, and is highly degenerate with the stellar M*/L. NFW works less well because it has a less flexible shape. But both work because they have more parameters [than MOND].

Third, statistics will not save us here. I once hoped that the BIC would sort this out, but having gone down that road, I believe the BIC does not penalize models sufficiently for adding free parameters. You allude to this at the end of section 3.2. When you go from MOND (with fixed a₀ it has only one parameter, M*/L, to fit to account for everything) to a dark matter halo (which has at a minimum 3 parameters: M*/L plus two to describe the halo) then you gain an enormous amount of freedom – the volume of possible parameter space grows enormously. But the BIC just says if you had 20 degrees of freedom before, now you have 22. That does not remotely represent the amount of flexibility that represents: some free parameters are more equal than others. MOND fits and DM halo fits are not the same beast; we can’t compare them this way any more than we can compare apples and snails.

Worse, to do this right requires that the uncertainties be real random errors. They are not. SPARC provides homogeneous mass models based on near-IR observations of the stellar mass distribution. Those should be OK to the extent that near-IR light == stellar mass. That is a decent mapping, but not perfect. Consequently, we expect the occasional galaxy to misbehave. UGC 128 is a case where the MOND fit was great with optical data then became terrible with near-IR data. The absolute difference in the data are not great, but in terms of the formal chi^2 it is. So is that a failure of the model, or of the data to represent what we want it to represent?

This happens all the time in astronomy. Here, we want to know the circular velocity of a test particle in the gravitational potential predicted by the baryonic mass distribution. We never measure either of those quantities. What we measure is the (i) stellar light distribution and the (ii) Doppler velocities of gas. We assume we can map stellar light to stellar mass and Doppler velocity to orbital speed, but no mass model is perfect, nor is any patch of observed gas guaranteed to be on a purely circular orbit. These are known unknowns: uncertainties that we know are real but we cannot easily quantify. These assumptions that we have to make to do the analysis dominate over the random errors in many cases. We also assume that galaxies are in dynamical equilibrium, but 20% of spirals show gross side-to-side asymmetries, and at least 50% mild ones. So what is the circular motion in those cases? (F579-1 is a good example)

While SPARC is homogeneous in its photometry, it is extremely heterogeneous in its rotation curve measurements. We’re working on fixing that, but it’ll take a while. Consequently, as you note, some galaxies have little constraining power while others appear to have lots. That’s because many of the rotation curve velocity uncertainties are either grossly over or underestimated. To see this, plot the cumulative distribution of chi^2 for any of your models (or see the CDF published by Li et al 2018 for the RAR and Li et al 2020 for dark matter halos of many flavors. So many, I can’t recall how many CDF we published.) Anyway, for a good model, chi^2 is always close to one, so the CDF should go up sharply and reach one quickly – there shouldn’t be many cases with very low chi^2 or very high chi^2. Unfortunately, rotation curve data do not do this for any type of model. There are always way too many cases with chi^2 << 1 and also too many with chi^2 >> 1. One might conclude that all models are unacceptable – or that the error bars are Messed Up. I think the second option is the case. If so, then this sort of analysis will always have the power to mislead.

I insert Fig. 1 from Li et al. (2020) so you don’t have to go look it up. The CDF of a statistically good model would rise sharply, being an almost vertical line at chi^2 = 1. No model of any flavor does that. That’s in large part because the uncertainties on some rotation curves are too large, while those on others are too small. The greater flexibility of dark matter models make them incrementally better than MOND for the cases with error bars that are too small – hence the corollary statement that “the higher the uncertainty of the data, the better MOND seems to work.” *This happens because dark matter models are allowed to chase bogus outliers with tiny error bars in a way that MOND cannot. That doesn’t make dark matter better, it just makes it is easier to fool.*

A key thing to watch out for is the outsized effects of a few points with tiny error bars. Among galaxies with high chi^2, what often happens is that there is one point with a tiny error bar that does not agree with any of the rest of the data for any smoothly continuous rotation curve. Fitting programs penalize a model for missing this point by many sigma, so will do anything they can to make it better. So what happens is that if you let a₀ vary with a flat prior, it will got to some very silly values in order to buy a tiny improvement in chi^2. Formally, that’s a better fit, so you say OK, a₀ has to vary. But if you plot the fitted RCs with fixed and variable a₀, you will be hard pressed to see the difference. Chi^2 is different, sure, but both will have chi^2 >> 1, so a lousy fit either way, and we haven’t really gained anything meaningful from allowing for the greater fitting freedom. Really it is just that one point that is Wrong even though it has a tiny error bar – which you can see relative to the other points, never mind the model. Dark matter halos have more flexibility from the beginning, so this is less obvious for them even though the same thing happens.

So that’s another big point – what is the prior for a dark matter halo? [Your] Table 1 allows V₂₀₀ and C₂₀₀ to be pretty much anything. So yes, you will find a fit from that range. For Burkert halos, there is no prior, since these do not emerge from any theory – they’re just a flexible French curve. For NFW halos, there is a prior from cosmology – see McGaugh et al (2007) among a zillion other possible references, including Li et al (2020). In any[L]CDM cosmology, the parameters V200 and C200 correlate – they are not independent. So a reasonable prior would be a Gaussian in log(C200) at a given V200 as specified by some simulation (Macio et al; see Li et al 2020). Another prior is how V200 (or M200) relates to the observed baryonic mass (or stellar mass). This one is pretty dodgy. Originally, we expected a fixed ratio between baryonic and dark mass. So when I did this kind of analysis in the ’90s, I found NFW flunked hard compared to MOND. (I didn’t know about the BIC then.) Galaxy DM halos simply do not look like NFW halos that form in LCDM and host galaxies with a few percent of their mass in the luminous disk even though this was the standard model for many years (Mo, Mao, & White 1998). If we drop the assumption that luminous galaxies are always a fixed fraction of their dark matter halos, then better fits can be obtained. I suspect your uniform prior fits have halo masses all over the place; they probably don’t correlate well with the baryonic mass, nor are their C and V200 parameters likely to correlate as they are predicted to do. You could apply the expected mass-concentration and stellar mass-halo mass relations as priors, then NFW will come off worse in your analysis because you’ve restricted them to where they ought to live.

So, as you say – it all comes down to the prior.

Even applying a stellar mass-halo mass relation from abundance matching isn’t really independent information, though that’s the best you can hope to do. But I was saying 20+ years ago that fixed mass ratios wouldn’t work, but nobody then wanted to abandon that obvious assumption. Since then, they’ve been forced to do so. But there is no good physical reason for it (feedback is the deus ex machina of all problems in the field), what happened is that the data forced us to drop the obvious assumption. Data including kinematic data (McGaugh et al 2010). So adopting a modern stellar mass-halo mass relation will give you a stronger prior than a uniform prior, but that choice has already been informed by the kinematic data that you’re trying to fit. How do we properly penalize the model for cheating about its “prior” by peaking at past data?

So, as you say – it all comes down to the prior. I think it would be important here to better constrain the priors on the DM halo fits. Li et al (2020) discuss this. Even then we’re not done, because galaxy formation modifies the form of the halo function we’re fitting. They shouldn’t end up as NFW even if they start out that way – see Li et al 2022a & b. Those papers consider the inevitable effects of adiabatic compression, but not of feedback. If feedback really has the effects on DM halos that is frequently advertised, then neither NFW or Burkert are appropriate fitting functions – they’re not what LCDM+feedback predicts. Good luck extracting a legitimate prediction from simulations, though. So we’re stuck doing what you’re trying to do: adopt some functional form to represent the DM halo, and see what fits. What you’ve done here agrees with my experience: cored DM halos work best. But they don’t represent an LCDM prediction, or any other broader theory, so – so what?

Another detail to be wary of – the radial range over which the RC data constrain the DM halo fit is often rather limited compared to the size of the halo. To complicate matters further, the inner regions are often star-dominated, so there is not much of a handle on DM from where the data are best, at least beyond many galaxies preferring not to have a cusp since the stars already get the job done at small R. So, one ends up with V_DM(R) constrained from 3% to 10% of the virial radius, or something like that. V200 and C200 are defined at the notional virial radius, so there are many combinations of these parameters that might adequately fit the observed range while being quite different elsewhere. Even worse, NFW halos are pretty self-similar – there are combinations of (C200,V200) that are highly degenerate, so you can’t really tell the difference between them even with excellent data – the confidence contours look like bananas in C200-V200 space, with low C/high V often being as good as high C/low V. Even even even worse is that the observed V_DM(R) is often approximately a straight line. Any function looks like a straight line if you stretch it out enough. Consequently, the fits to LSB galaxies often tend to absurdly low C and high V200: NFW never looks like a straight line, but it does if you blow it up enough. So one ends up inferring that the halo masses of tiny galaxies are nearly as big as those of huge galaxies, or more so! My favorite example was NGC 3109, a tiny dwarf on the edge of the Local Group. A straight NFW fit suggests that the halo of this one little galaxy weighs more than the entire Local Group, M31 + MW + everything else combined. This is the sort of absurd result that comes from fitting the NFW halo form to a limited radial range of data.

I don’t know that this helps you much, but you see a few of the concerns.

How things go mostly right or badly wrong

People often ask me of how “perfect” MOND has to be. The short answer is that it agrees with galaxy data as “perfectly” as we can perceive – i.e., the scatter in the credible data is accounted for entirely by known errors and the expected scatter in stellar mass-to-light ratios. Sometimes it nevertheless looks to go badly wrong. That’s often because we need to know both the mass distribution and the kinematics perfectly. Here I’ll use the Milky Way as an example of how easily things can look bad when they aren’t.

First, an update. I had hoped to stop talking about the Milky Way after the recent series of posts. But it is in the news, and there is always more to say. A new realization of the rotation curve from the Gaia DR3 data has appeared, so let’s look at all the DR3 data together:

Gaia DR3 realizations of the Milky Way rotation curve. *The most recent* version of these data from Poder et al (2023) are shown as blue squares over the range 5 < R < 13 kpc. Other Gaia DR3 realizations include Ou et al. (2023, green circles), Wang et al. (2023, magenta downward pointing triangles), and Zhou et al. (2023, purple triangles).

The new Gaia realization does not go very far out, and has larger uncertainties. That doesn’t mean it is worse; it might simply be more conservative in estimating uncertainties, and not making a claim where the data don’t substantiate it. Neither does that mean the other realizations are wrong: these differences are what happens in different analyses. Indeed, all the independent realizations of the Gaia data are pretty consistent, despite the different stellar selection criteria and analysis techniques. This is especially true for R < 17 kpc where there are lots of stars informing the measurements. Even beyond that, I would say they are consistent at the level we’d expect for astronomy.

Zooming out to compare with other results:

The Milky Way rotation curve. The model line from McGaugh (2018) is shown with data from various sources. The abscissa switches from linear to logarithmic at 10 kpc to wedge it all in. The location of the Large Magellanic Cloud at 50 kpc is noted. Gaia DR3 data (Poder et al., Ou et al., Wang et al., and Zhou et al.) *are shown as in the plot above*. The small black squares are the Gaia DR2 realization of Eilers et al. (2019) reanalyzed to include the effect of bumps and wiggles by McGaugh (2019). Non-Gaia data include blue horizontal branch stars (light blue squares) and red giants (red squares) in the stellar halo (Bird et al. 2022), globular clusters (Watkins et al. 2019, pink triangles), VVV stars (Portail et al. 2017, dark grey squares at R < 2.2 kpc), and terminal velocities (McClure-Griffiths & Dickey 2007, 2016, light grey points from 3 < R < 8 kpc). These terminal velocities are the only data that inform the model line; everything else follows.

Overall, I would say the data paint a pretty consistent picture. The biggest tension amongst the data illustrated here is between the outermost Gaia points around R = 25 kpc and the corresponding results from halo stars. One is consistent with the model line and the other is not. We shouldn’t allow the model to inform our interpretation; the important point is that the independent data disagree with each other. This happens all the time in astronomy. Sometimes it boils down to different assumptions; sometimes it is a real discrepancy. Either way, one has to learn* to cope.

The sharp-eyed will also notice an apparent tension between the DR2 data (black squares) and DR3 around 6 and 7 kpc. This is not real – it is an artifact of different treatments of the term in the Jeans equation for the logarithmic derivative of the density profile of the tracer particles. That’s a choice made in the analysis. The data are entirely consistent when treated consistently.

Putting on an empiricist’s hat, I will say that the kink in the slope of the Gaia data around R = 18 kpc looks unnatural. That doesn’t happen in other galaxies. Rather than belabor the point further, I’ll simply say that this is how things mostly go right but also a little wrong. This is as good as we can hope for in [extra]galactic astronomy.

In contrast, it is easy to go very wrong. To give an example, here is a model of the Milky Way that was built to approximately match the rotation curve of Sofue (2020).

Fig. 1 from Dai et al. (2022). Note the logarithmic abscissa. Their caption: The rotation curve of the Milky Way. The data (solid dark circles with error bars) for r < 100kpc come from [22], while for r > 100kpc from [23]. The solid, dashed and doted lines describe the contribution from the bulge, stellar disk and dark matter halo respectively, within a ΛCDM model of the galaxy. The dashed-dot line is the total contribution of all three components.The parameters of each component are taken from [24]. For comparison, the Milky way rotation curve from Gaia DR2 is shown in color. The red dots are data from [34], the blue upward-pointing triangles are from [35], while the cyan downward-pointing triangles are from [36].

This realization of the rotation curve is very different from that seen above. Note that the rotation curve (black points) is very different from that of Gaia (red points) over the same radial range. These independent data are inconsistent; at least one of them is wrong. The data extend to very large radii, encompassing not only the LMC but also Andromeda (780 kpc away). I am already concerned about the effects of the LMC at 50 kpc; Andromeda is twice the baryonic mass of the Milky Way so anything beyond 260 kpc is more Andromeda’s territory than ours – depending on which side we’re talking about. The uncertainties are so big out there they provide no constraining power anyway.

In terms of MOND-required perfection, things fall apart for the Dai model already at very small radii. Dai et al. (2022) chose to fit their bulge component to the high amplitude terminal velocities of Sofue. That’s a reasonable thing to do, if we think the terminal velocities represent circular motion. Because of the non-circular motions that sustain the Galactic bar, they almost certainly do not – that’s why I restricted use of terminal velocities to larger radii. We also know something about the light distribution:

The inner 3 kpc of the Milky Way. The circles are the terminal velocities of Sofue (2020); the squares are the equivalent circular velocity of the potential reconstructed from the kinematics of stars in the VVV survey (Portail et al. 2017). The line is the bulge-bar model of McGaugh (2008) based on the light distribution reported by Binney et al (1997).

This is essentially the same graph as I showed before, but showing only the Newtonian bulge-bar component, and on a logarithmic abscissa for comparison with the plot of Dai et al. The two bulge models are very different. That of Dai et al. is more massive and more compact, as required to match the terminal velocities. There may be galaxies out there that look like this, but the Milky Way is not one of them.

Indeed, Newton’s prediction for the rotation curve of the bulge-bar component – the line labeled bulge/bar based on what the Milky Way looks like – is in good agreement with the effective circular speed curve obtained from stellar data. It is not consistent with the terminal velocities. We could increase the amplitude of the Newtonian prediction by increasing the mass-to-light ratio of the stars (I have adopted the value I expect for stellar populations), but the shape would still be wrong. This does not come as a surprise to most Galactic astronomers, because we know there is a bar in the center of the Milky Way and we know that bars induce non-circular motions, so we do not expect the terminal velocities to be a fair tracer of the rotation curve in this region. That’s why Portail et al. had to go to great lengths in their analysis to reconstruct the equivalent circular velocity, as did I just to build the bulge-bar model.

The thing about predicting rotation curves from the observed mass, as MOND does, is that you have to get both the kinematic data and the mass distribution right. The velocity predicted at any radius depends on the mass enclosed by that radius. So if we get the bulge badly wrong, everything spirals down the drain from there.

Dai et al. (2022) compare their model to the acceleration residuals predicted by MOND for their mass model. If all is well, the data should scatter around the constant line at zero in this graph:

Fig. 4 from Dai et al. (2022). Their caption: [The radial acceleration relation] recast as a comparison between the total acceleration, a, and the MOND prediction, a_M , as a function of the acceleration due to baryons a_B. The solid horizontal line is a = a_M. The circles and squares with error bars represent the Milky Way and M31 data, while the gray dots are from the EAGLE simulation of ΛCDM in [1]. For a_B > 10⁻¹⁰m/s² any difference between a and a_M is unclear. However, once a_B drops well below 10⁻¹¹m/s2, the discrepancy emerges. The short-dashed line is the ΛCDM fitting curve of the MW. The dash-dot line is the ΛCDM fitting curve of M31. The mass range** of galaxies in EAGLE’s data is chosen to be between 5 × 10¹⁰M_⊙ to 5 × 10¹¹M_⊙. For comparison, the Milky way rotation curve from GAIA data release II is shown in color. The red dots are data from [34], the blue triangles are from [35], while the cyan down triangles are from [36]. While the EAGLE simulation does not match the data perfectly, these plots indicate that it is much easier to accommodate a systematic downward trend with the ΛCDM model than with MOND.

Things are not well.

The interpretation that is offered (right in the figure caption) is that MOND is wrong and the LCDM-based EAGLE simulation does a better if not perfect job of explaining things. We already know that’s not right. The alternate interpretation is that this is not a valid representation of the prediction of MOND, because their mass model does not follow from the observed distribution of light. They get neither the baryonic mass distribution and its predicted acceleration a_b nor the total acceleration a right in the plot above.

In terms of dark matter, the model of Dai et al. may appear viable. In terms of MOND, it is way off, not just a little off. The residuals are only zero, as they should be, for a narrow range of accelerations, 2 to 3 x 10^-10 m/s/s. That’s more Newton than MOND, and appears to correspond to the limited range in radii over which their model matches the rotation curve data in their Fig. 1 (roughly 4 to 6 kpc). It doesn’t really fit the data elsewhere, and the restrictions on a MOND fit are considerably more stringent than on the sort of dark matter model they construct: there’s no reason to expect their model to behave like MOND in the first place.

And, hoo boy, does it ever not behave like MOND. Look at how far those red points – the Gaia DR2 data – deviate from zero in their Fig. 4. Those are the exact same data that agree well with the model line I show above – the data that were correctly predicted in advance. This model is a reasonable representation of the radial force predicted by MOND, with the blue line in my plot being equivalent to the zero line in theirs.

This is how things can go badly wrong. To properly apply MOND, we need to measure both the kinematics and baryonic mass distribution correctly. If we screw either up, as is easy to do in astronomy, then the result will look very wrong, even if it shouldn’t. Combine this with the eagerness many people have to dismiss MOND outright, and you wind up with lots of articles claiming that MOND is wrong – even when that’s not really the story the data tell. Happens over and over again, so the field remains stagnant.

*This is a large part of the cultural difference between physics and astronomy. Physicists are spoiled by laboratory experiments done in controlled conditions in which one can measure to the sixth place of decimals. In contrast, astronomy is an observational rather than experimental science. We can’t put the universe in a box and control all the systematics – measuring most quantities to 1% is a tall order. Consequently, astronomers are used to being wrong. While I wouldn’t say that astronomers cope with it gracefully, they’re well aware that it happens, that is has happened a lot historically, and will continue to happen in the future. It is a risk we all take in trying to understand a universe so much vaster than ourselves. This makes astronomers rather more tolerant of surprising results – results where the first response is “that can’t be right!” but also informed by the experience that “we’ve been wrong before!” Physicists coming to the field generally lack this experience and take the error bars way too seriously. I notice this attitude is creeping into the younger generation of astronomers; people who’ve received their data from distant observatories and performed CPU-intensive MCMC error analyses, so want to believe them, but often lack the experience of dozens of nights spent at the observatory sweating a thousand ill-controlled but consequential details, like walking out to a beautiful sunrise decorated by wisps of cirrus clouds. When did those arrive?!?

**The data that define the radial acceleration relation come from galaxies spanning six decades in stellar mass, so this one decade range from the simulations is tiny – it is literally comparing a factor of ten to a a factor of a million. What happens outside the illustrated mass range? Are lower masses even resolved?

A Response to Recent Developments Concerning the Gravitational Potential of the Milky Way

In the series of recent posts I’ve made about the Milky Way, I missed an important reply made in the comments by Francois Hammer, one of the eminent scientists doing the work. I was on to writing the next post when he wrote it, and simply didn’t see it until yesterday. Dr. Hammer has some important things to say that are both illustrative of the specific topic and also of how science should work. I wanted to highlight his concerns with their own post, so, with his permission, I cut & paste his comments below, making this, in effect, a guest post by Francois Hammer.

There are two aspects we’d like to mention, as they may help to clarify part of the debate:
1- When saying “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?”, one may wonder whether the significance of Gaia data has been really understood.
In the Eilers et al. 2019 DR2 rotation curve, you may see points with small error bar up to 21-22 kpc. Gaia DR3 provides proper motion (systematics) uncertainties that are 2 times smaller than from Gaia DR2, so it can easily goes to 25 kpc or more.
The gain in quality for parallaxes is indeed smaller (30% gain). However, our results cannot be affected by distance estimates, since the large number of stars with parallax estimates in Wang et al. (2023) is giving the same rotation curve than that from (a lower number of) RGB stars with spectrophotometric distances (Ou et al. 2023), i.e., following Eilers et al. 2019. And both show a Keplerian decline, which was already noticeable with DR2 results from Eilers et al 2019. The latter authors said in their conclusions: “We do see a mild but significant deviation from the straightly declining circular velocity curve at R≈19–21 kpc of Δv≈15 km s−1.” Our work using Gaia DR3 is nothing else than having a factor 2 better in accounting for systematics, and then being able to resolve what looks like a Keplerian decrease of the rotation curve.
We may also mention here that one of us participated to an unprecedented study of the kinematics LMC (Gaia Collaboration 2021, Luri’s paper), which is at 50 kpc. Unless one proves everything that people has done about the LMC and MW is wrong, and that the data are too uncertain to conclude anything about what happens at R=17-25 kpc, the above clarifications about Gaia accuracy are truly necessary for people reading your blog.
2- The argument that the result “violates a gazillion well-established constraints.” has to be taken with some caution, since otherwise, no one can do any progress in the field. In fact, the problem with many probes (so-called “satellites”) in the MW halo, is the fact that one cannot guarantee whether or not their orbits are at equilibrium with the MW potential. This is the reverse for the MW disk, for which stars are rotating in the disk, and, e.g., at 25 kpc, they have likely experience 7-8 orbits since the last merger (Gaia-Sausage-Enceladus), about 9 billion years go. In other words, the mass provided by a system mostly at equilibrium, likely supersedes masses provided by systems that equilibrium conditions are not secured. An interesting example of this is given by globular clusters (GCs). If taken as an ensemble of 156 GCs (from Baumgardt catalog), just by removing Pyxis and Terzan 8, the MW mass inside 50 kpc passes from 5.5 to 2.1 10^11 Msun. This is likely because these two GCs may have come quite recently, meaning that their initial kinetic energy is still contributing to their total energy. A similar mass overestimate could happen if one accounts the LMC or Leo I as MW satellites at equilibrium with the MW potential. So we agree that near 25 kpc the disk of the MW may show signs of less-equilibrium, or sign of slightly less circular orbits due to different phenomenas discussed in the blog. However, why taking into account objects for which there is no proof they are at equilibrium as being the true measurements?
In our work, we have considerably focused in understanding and expanding the whole contribution of systematics, which may comes from Gaia data, but also from assumptions about stellar profile (i.e., deviations from exponential profiles), from the Sun distance and proper motion and so on. You may find a description in Ou et al.’s Figure 5 and Jiao et al.’s Figure 4, both showing that systematics cannot gives much more than 10% error on circular velocity estimates. This is an area where we are considered by the Local Group community as being quite conservative, and following Gaia specialists with who we have worked to deliver the EDR3 catalog of dwarf galaxy motions (Li, Hammer, Babusiaux et al 2021) up to about 150 kpc. Jiao et al. paper main contribution is the fair accounting of systematics, which analysis shows error bars that are much larger than those from other sources of errors especially in MW outskirts (see Fig. 2).
Francois Hammer, 24 September 2023

The image at top is Fig. 2 from Jiao et al. illustrating their assessment of the rotation curve and its systematic uncertainties.

Recent Developments Concerning the Gravitational Potential of the Milky Way. III. A Closer Look at the RAR Model

I am primarily an extragalactic astronomer – someone who studies galaxies outside our own. Our home Galaxy is a subject in its own right. Naturally, I became curious how the Milky Way appeared in the light of the systematic behaviors we have learned from external galaxies. I first wrote a paper about it in 2008; in the process I realized that I could use the RAR to infer the distribution of stellar mass from the terminal velocities observed in interstellar gas. That’s not necessary in external galaxies, where we can measure the light distribution, but we don’t get a view of the whole Galaxy from our location within it. Still, it wasn’t my field, so it wasn’t until 2015/16 that I did the exercise in detail. Shortly after that, the folks who study the supermassive black hole at the center of the Galaxy provided a very precise constraint on the distance there. That was the one big systematic uncertainty in my own work up to that point, but I had guessed well enough, so it didn’t make a big change. Still, I updated the model to the new distance in 2018, and provided its details on my model page so anyone could use it. Then Gaia data started to pour in, which was overwhelming, but I found I really didn’t need to do any updating: the second data release indicated a declining rotation curve at exactly the rate the model predicted: -1.7 km/s/kpc. So far so good.

I call it the RAR model because it only involves the radial force. All I did was assume that the Milky Way was a typical spiral galaxy that followed the RAR, and ask what the mass distribution of the stars needed to be to match the observed terminal velocities. This is a purely empirical exercise that should work regardless of the underlying cause of the RAR, be it MOND or something else. Of course, MOND is the only theory that explicitly predicted the RAR ahead of time, but we’ve gone to great lengths to establish that the RAR is present empirically whether we know about MOND or not. If we accept that the cause of the RAR is MOND, which is the natural interpretation, then MOND over-predicts the vertical motions by a bit. That may be an important clue, either into how MOND works (it doesn’t necessarily follow the most naive assumption) or how something else might cause the observed MONDian phenomenology, or it could just be another systematic uncertainty of the sort that always plagues astronomy. Here I will focus on the RAR model, highlighting specific radial ranges where the details of the RAR model provide insight that can’t be obtained in other ways.

The RAR Milky Way model was fit to the terminal velocity data (in grey) over the radial range 3 < R < 8 kpc. Everything outside of that range is a prediction. It is not a prediction limited to that skinny blue line, as I have to extrapolate the mass distribution of the Milky Way to arbitrarily large radii. If there is a gradient in the mass-to-light ratio, or even if I guess a little wrong in the extrapolation, it’ll go off at some point. It shouldn’t be far off, as V(R) is mostly fixed by the enclosed mass. Mostly. If there is something else out there, it’ll be higher (like the cyan line including an estimate of the coronal gas in the plot that goes out to 130 kpc). If there is a bit less than the extrapolation, it’ll be lower.

The RAR model Milky Way (blue line) together with the terminal velocities to which it was fit (light grey points), VVV data in the inner 2.2 kpc (dark grey squares), and the Zhou et al. (2023) realization of the Gaia DR3 data. Also shown are the number of stars per bin from Gaia (right axis).

From 8 to 19 kpc, the Gaia data as realized by Zhao et al. fall bang on the model. They evince exactly the slowly declining rotation curve that was predicted. That’s pretty good for an extrapolation from R < 8 kpc. I’m not aware of any other model that did this well in advance of the observation. Indeed, I can’t think of a way to even make a prediction with a dark matter model. I’ve tried this – a lot – and it is as easy to come up with a model whose rotation curve is rising as one that is falling. There’s nothing in the dark matter paradigm that is predictive at this level of detail.

Beyond R > 19 kpc, the match of the model and Zhou et al. realization of the data is not perfect. It is still pretty damn good by astronomical standards, and better than the Keplerian dotted line. Cosmologists would be wetting themselves with excitement if they could come this close to predicting anything. Heck, they’re known to do that even when they’re obviously wrong*.

If the difference between the outermost data and the blue line is correct, then all it means is that we have to tweak the model to have a bit less mass than assumed in the extrapolation. I call it a tweak because it would be exactly that: a small change to an assumption I was obliged to make in order to do the calculation. I could have assumed something else, and almost did: there is discussion in the literature that the disk of the Milky Way is truncated at 20 kpc. I considered using a mass model with such a feature, but one can’t make it a sharp edge as that introduces numerical artifacts when solving the Poisson equation numerically, as this procedure depends on derivatives that blow up when they encounter sharp features. Presumably the physical truncation isn’t unphysically sharp anyway, rather being a transition to a steeper exponential decline as we sometimes see in other galaxies. However, despite indications of such an effect, there wasn’t enough data to constrain it in a way useful for my model. So rather than introduce a bunch of extra, unconstrained freedom into the model, I made a straight extrapolation from what I had all the way to infinity in the full knowledge that this had to be wrong at some level. Perhaps we’ve found that level.

That said, I’m happy with the agreement of the data with the model as is. The data become very sparse where there is even a hint of disagreement. Where there are thousands of stars per bin in the well-fit portion of the rotation curve, there are only tens per bin outside 20 kpc. When the numbers get that small, one has to start to worry that there are not enough independent samples of phase space. A sizeable fraction of those tens of stars could be part of the same stellar stream, which would bias the results to that particular unrepresentative orbit. I don’t know if that’s the case, which is the point: it is just one of the many potential systematic uncertainties that are not represented in the formal error bars. Missing those last five points by two sigma is as likely to be an indication that the error bars have been underestimated as it is to be an indication that the model is inadequate. Trying to account for this sort of thing is why the error bars of Jiao et al. are so much bigger than the formal uncertainties in the three realization papers.

That’s the outer regions. The place where the RAR model disagrees the most with the Gaia data is from 5 < R < 8 kpc, which is in the range where it was fit! So what’s going on there?

Again, the data disagree with the data. The stellar data from Gaia disagree with the terminal velocity data from interstellar gas at high significance. The RAR model was fit to the latter, so it must per force disagree with the former. It is tempting to dismiss one or the other as wrong, but do they really disagree?

Adapted from Fig. 4 of McGaugh (2019). Grey points are the first and fourth quadrant terminal velocity data to which the model (blue line) was matched. The red squares are the stellar rotation curve estimated with Gaia DR2 (DR3 is indistinguishable). The black squares are the stellar rotation curve after adjustment to be consistent with a mass profile that includes spiral arms. This adjustment for self-consistency remedies the apparent discrepancy between gas and stellar data.

In order to build the model depicted above, I chose to split the difference between the first and fourth quadrant terminal velocity data. I fit them separately in McGaugh (2016) where I made the additional point that the apparent difference between the two quadrants is what we expect from an m=2 mode – i.e., a galaxy with spiral arms. That means these velocities are not exactly circular as commonly assumed, and as I must per force assume to build the model. So I split the difference above in the full knowledge that this is not the exact circular velocity curve of the Galaxy, it’s just the best I can do at present. This is another example of the systematic uncertainties we encounter: the difference between the first and fourth quadrant is real and is telling us that the galaxy is not azimuthally symmetric – as anyone can tell by looking at any spiral galaxy, but is a detail we’d like to ignore so we can talk about disk+dark matter halo models in the convenient limit of axisymmetry.

Though not perfect – no model is – the RAR model Milky Way is a lot better than models that ignore spiral structure entirely, which is basically all of them. The standard procedure assumes an exponential disk and some form of dark matter halo. Allowance is usually made for a central bulge component, but it is relatively rare to bother to include the interstellar gas, much less consider deviations from a pure exponential disk. Having adopted the approximation of an exponential disk, one inevitably get a smooth rotation curve like the dashed line below:

Fig. 1 from McGaugh (2019). Red points are the binned fourth quadrant molecular hydrogen terminal velocities to which the model (blue line) has been fit. The dotted lines shows the corresponding Newtonian rotation curve of the baryons. The dashed line is the model of Bovy & Rix (2013) built assuming an exponential disk. The inset shows residuals of the models from the data. The exponential model does not and cannot fit these data.

The common assumption of exponential disk precludes the possibility of fitting the bumps and wiggles observed in the terminal velocities. These occur because of deviations from a pure exponential profile caused by features like spiral arms. By making this assumption, the variations in mass due to spiral arms is artificially smoothed over. They are not there by assumption, and there is no way to recover them in a dark matter fit that doesn’t know about the RAR.

Depending on what one is trying to accomplish, an exponential model may suffice. The Bovy & Rix model shown above is perfectly reasonable for what they were trying to do, which involved the vertical motions of stars, not the bumps and wiggles in the rotation curve. I would say that the result they obtain is in reasonable agreement with the rotation curve, given what they were doing and in full knowledge that we can’t expect to hit every error bar of every datum of every sort. But for the benefit of the chi-square enthusiasts who are concerned about missing a few data points at large radii, the reduced chi-squared of the Bovy & Rix model is 14.35 while that of the RAR model is 0.6. A good fit is around 1, so the RAR model is a good fit while the smooth exponential is terrible – as one can see by eye in the residual inset: the smooth exponential model gets the overall amplitude about right, but hits none of the data. That’s the starting point for every dark matter model that assumes an exponential disk; even if they do a marginally better job of fitting the alleged Keplerian downturn, they’re still a lot worse if we consider the terminal velocity data, the details of which are usually ignored.

If instead we pay attention the details of the terminal velocity data, we discover that the broad features seen there in are pretty much what we expect for the kinematic signatures of photometrically known spiral arms. That is, the mass density variations inferred by fitting the RAR correspond to spiral arms that are independently known from star counts. We’ve discussed this before.

Spiral structure in the Milky Way (left) as traced by HII regions and Giant Molecular Clouds (GMCs). These correspond to bumps in the surface density profile inferred from kinematics with the RAR (right).

If we accept that the bumps and wiggles in the terminal velocities are tracers of bumps and wiggles in the stellar mass profiles, as seen in external galaxies, then we can return to examining the apparent discrepancy between them and the stellar rotation curve from Gaia. The latter follow from an application of the Jeans equation, which helps us sort out the circular motion from the mildly eccentric orbits of many stars. It includes a term that depends on the gradient of the density profile of the stars that trace the gravitational potential. If we assume an exponential disk, then that term is easily calculated. It is slowly and smoothly varying, and has little impact on the outcome. One can explore variations of the assumed scale length of the disk, and these likewise have little impact, leading us to infer that we don’t need to worry about it. The trouble with this inference is that it is predicated on the assumption of a smooth exponential disk. We are implicitly assuming that there are no bumps and wiggles.

The bumps and wiggles are explicitly part of the RAR model. Consequently, the gradient term in the Jeans equation has a modest but important impact on the result. Applying it to the Gaia data, I get the black points:

*The red squares are the Gaia DR2 data. The black squares are the same data after including in the Jeans equation the effect of variations in the tracer gradient. This term dominates the uncertainties.*

The velocities of the Gaia data in the range illustrated all go up. This systematic effect reconciles the apparent discrepancy between the stellar and gas rotation curves. The red points are highly discrepant from the gray points, but the black points are not. All it took was to drop the assumption of a smooth exponential profile and calculate the density gradient numerically from the data. This difference has a more pronounced impact on rotation curve fits than any of the differences between the various realizations of the Gaia DR3 data – hence my cavalier attitude towards their error bars. Those are not the important uncertainties.

Indeed, I caution that we still don’t know what the effective circular velocity of the potential is. I’ve made my best guess by splitting the difference between the first and fourth quadrant terminal velocity data, but I’ve surely not got it perfectly right. One might view the difference between the quadrants as the level at which the perfect quantity is practically unknowable. I don’t think it is quite that bad, but I hope I have at least given the reader some flavor for some of the hidden systematic uncertainties that we struggle with in astronomy.

It gets worse! At small radii, there is good reason to be wary of the extent to which terminal velocities represent circular motion. Our Galaxy hosts a strong bar, as artistically depicted here:

*Artist’s rendition of the Milky Way. Image credit: NASA/JPL-Caltech*.

Bars are a rich topic in their own right. They are supported by non-circular orbits that maintain their pattern. Consequently, one does not expect gas in the region where the bar is to be on circular orbits. It is not entirely clear how long the bar in our Galaxy is, but it is at least 3 kpc – which is why I have not attempted to fit data interior to that. I do, however, have to account for the mass in that region. So I built a model based on the observed light distribution. It’s a nifty bit of math to work out the equivalent circular velocity corresponding to a triaxial bar structure, so having done it once I’ve not been keen to do it again. This fixes the shape of the rotation curve in the inner region, though the amplitude may shift up and down with the mass-to-light ratio of the stars, which dominate the gravitational potential at small radii. This deserves its own close up:

Colored points are terminal velocities from Marasco et al. (2017), from both molecular (red) and atomic (green) gas. Light gray circles are from Sofue (2020). These are plotted assuming they represent circular motions, which they do not. Dark grey squares are the equivalent circular velocity inferred from stars in the VVV survey. The black line is the Newtonian mass model for the central bar and disk, and the blue line is the corresponding RAR model as seen above.

Here is another place where the terminal velocities disagree with the stellar data. This time, it is because the terminal velocities do not trace circular motion. If we assume they do, then we get what is depicted above, and for many years, that was thought to be the Galactic rotation curve, complete with a pronounced classical bulge. Many decades later, we know the center of the Galaxy is not dominated by a bulge but rather a bar, with concominant non-circular motions – motions that have been observed in the stars and carefully used to reconstruct the equivalent circular velocity curve by Portail et al. (2017). This is exactly what we need to compare to the RAR model.

Note that 2008, when the bar model was constructed, predates 2017 (or the 2016 appearance of the preprint). While it would have been fair to tweak the model as the data improved, this did not prove necessary. The RAR model effectively predicted the inner rotation curve a priori. That’s a considerably more impressive feat than getting the outer slope right, but the model manages both sans effort.

No dark matter model can make an equivalent boast. Indeed, it is not obvious how to do this at all; usually people just make a crude assumption with some convenient approximation like the Hernquist potential and call it a day without bothering to fit the inner data. The obvious prediction for a dark matter model overshoots the inner rotation curve, as there is no room for the cusp predicted in cold dark matter halos – stars dominate the central potential. One can of course invoke feedback to fix this, but it is a post hoc kludge rather than a prediction, and one that isn’t supposed to apply in galaxies as massive as the Milky Way. Unless it needs to, of course.

So, lets’s see – the RAR model Milky Way reconciles the tension between stellar and interstellar velocity data, indicates density bumps that are in the right location to correspond to actual spiral arms, matches the effective circular velocity curve determined for stars in the Galactic bar, correctly predicted the slope of the rotation curve outside the solar circle out to at least 19 kpc, and is consistent with the bulk of the data at much larger radii. That’s a pretty successful model. Some realizations of the Gaia DR3 data are a bit lower than predicted, but others are not. Hopefully our knowledge of the outer rotation curve will continue to improve. Maybe the day will come when the data have improved to the point where the model needs to be tweaked a little bit, but it is not this day.

*To give one example, the BICEP II experiment infamously claimed in March of 2014 to have detected the Inflationary signal of primordial gravitational waves in their polarization data. They held a huge press conference to announce the result in clear anticipation of earning a Nobel prize. They did this before releasing the science paper, much less hearing back from a referee. When they did release the science paper, it was immediately obvious on inspection that they had incorrectly estimated the dust foreground. Their signal was just that – excess foreground emission. I could see that in a quick glance at the relevant figure as soon as the paper was made available. Literally – I picked it up, scanned through it, saw the relevant figure, and could immediately spot where they had gone wrong. And yet this huge group of scientists all signed their name to the submitted paper and hyped it as the cosmic “discovery of the century”. Pfft.

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.

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.

Triton Station

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

Category: Data Interpretation