Author: tritonstation

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

Clusters of galaxies ruin everything

Clusters of galaxies ruin everything

A common refrain I hear is that MOND works well in galaxies, but not in clusters of galaxies. The oft-unspoken but absolutely intended implication is that we can therefore dismiss MOND and never speak of it again. That’s silly.

Even if MOND is wrong, that it works as well as it does is surely telling us something. I would like to know why that is. Perhaps it has something to do with the nature of dark matter, but we need to engage with it to make sense of it. We will never make progress if we ignore it.

Like the seventeenth century cleric Paul Gerhardt, I’m a stickler for intellectual honesty:

“When a man lies, he murders some part of the world.”

Paul Gerhardt

I would extend this to ignoring facts. One should not only be truthful, but also as complete as possible. It does not suffice to be truthful about things that support a particular position while eliding unpleasant or unpopular facts* that point in another direction. By ignoring the successes of MOND, we murder a part of the world.

Clusters of galaxies are problematic in different ways for different paradigms. Here I’ll recap three ways in which they point in different directions.

1. Cluster baryon fractions

An unpleasant fact for MOND is that it does not suffice to explain the mass discrepancy in clusters of galaxies. When we apply Milgrom’s formula to galaxies, it explains the discrepancy that is conventionally attributed to dark matter. When we apply MOND clusters, it comes up short. This has been known for a long time; here is a figure from the review Sanders & McGaugh (2002):

Figure 10 from Sanders & McGaugh (2002): (Left) the Newtonian dynamical mass of clusters of galaxies within an observed cutoff radius (rout) vs. the total observable mass in 93 X-ray-emitting clusters of galaxies (White et al. 1997). The solid line corresponds to Mdyn = Mobs (no discrepancy). (Right) the MOND dynamical mass within rout vs. the total observable mass for the same X-ray-emitting clusters. From Sanders (1999).

The Newtonian dynamical mass exceeds what is seen in baryons (left). There is a missing mass problem in clusters. The inference is that the difference is made up by dark matter – presumably the same non-baryonic cold dark matter that we need in cosmology.

When we apply MOND, the data do not fall on the line of equality as they should (right panel). There is still excess mass. MOND suffers a missing baryon problem in clusters.

The common line of reasoning is that MOND still needs dark matter in clusters, so why consider it further? The whole point of MOND is to do away with the need of dark matter, so it is terrible if we need both! Why not just have dark matter?

This attitude was reinforced by the discovery of the Bullet Cluster. You can “see” the dark matter.

An artistic rendition of data for the Bullet Cluster. Pink represents hot X-ray emitting gas, blue the mass concentration inferred through gravitational lensing, and the optical image shows many galaxies. There are two clumps of galaxies that collided and passed through one another, getting ahead of the gas which shocked on impact and lags behind as a result. The gas of the smaller “bullet” subcluster shows a distinctive shock wave.

Of course, we can’t really see the dark matter. What we see is that the mass required by gravitational lensing observations exceeds what we see in normal matter: this is the same discrepancy that Zwicky first noticed in the 1930s. The important thing about the Bullet Cluster is that the mass is associated with the location of the galaxies, not with the gas.

The baryons that we know about in clusters are mostly in the gas, which outweighs the stars by roughly an order of magnitude. So we might expect, in a modified gravity theory like MOND, that the lensing signal would peak up on the gas, not the stars. That would be true, if the gas we see were indeed the majority of the baryons. We already knew from the first plot above that this is not the case.

I use the term missing baryons above intentionally. If one already believes in dark matter, then it is perfectly reasonable to infer that the unseen mass in clusters is the non-baryonic cold dark matter. But there is nothing about the data for clusters that requires this. There is also no reason to expect every baryon to be detected. So the unseen mass in clusters could just be ordinary matter that does not happen to be in a form we can readily detect.

I do not like the missing baryon hypothesis for clusters in MOND. I struggle to imagine how we could hide the required amount of baryonic mass, which is comparable to or exceeds the gas mass. But we know from the first figure that such a component is indicated. Indeed, the Bullet Cluster falls at the top end of the plots above, being one of the most massive objects known. From that perspective, it is perfectly ordinary: it shows the same discrepancy every other cluster shows. So the discovery of the Bullet was neither here nor there to me; it was just another example of the same problem. Indeed, it would have been weird if it hadn’t shown the same discrepancy that every other cluster showed. That it does so in a nifty visual is, well, nifty, but so what? I’m more concerned that the entire population of clusters shows a discrepancy than that this one nifty case does so.

The one new thing that the Bullet Cluster did teach us is that whatever the missing mass is, it is collisionless. The gas shocked when it collided, and lags behind the galaxies. Whatever the unseen mass is, is passed through unscathed, just like the galaxies. Anything with mass separated by lots of space will do that: stars, galaxies, cold dark matter particles, hard-to-see baryonic objects like brown dwarfs or black holes, or even massive [potentially sterile] neutrinos. All of those are logical possibilities, though none of them make a heck of a lot of sense.

As much as I dislike the possibility of unseen baryons, it is important to keep the history of the subject in mind. When Zwicky discovered the need for dark matter in clusters, the discrepancy was huge: a factor of a thousand. Some of that was due to having the distance scale wrong, but most of it was due to seeing only stars. It wasn’t until 40 some years later that we started to recognize that there was intracluster gas, and that it outweighed the stars. So for a long time, the mass ratio of dark to luminous mass was around 70:1 (using a modern distance scale), and we didn’t worry much about the absurd size of this number; mostly we just cited it as evidence that there had to be something massive and non-baryonic out there.

Really there were two missing mass problems in clusters: a baryonic missing mass problem, and a dynamical missing mass problem. Most of the baryons turned out to be in the form of intracluster gas, not stars. So the 70:1 ratio changed to 7:1. That’s a big change! It brings the ratio down from a silly number to something that is temptingly close to the universal baryon fraction of cosmology. Consequently, it becomes reasonable to believe that clusters are fair samples of the universe. All the baryons have been detected, and the remaining discrepancy is entirely due to non-baryonic cold dark matter.

That’s a relatively recent realization. For decades, we didn’t recognize that most of the normal matter in clusters was in an as-yet unseen form. There had been two distinct missing mass problems. Could it happen again? Have we really detected all the baryons, or are there still more lurking there to be discovered? I think it unlikely, but fifty years ago I would also have thought it unlikely that there would have been more mass in intracluster gas than in stars in galaxies. I was ten years old then, but it is clear from the literature that no one else was seriously worried about this at the time. Heck, when I first read Milgrom’s original paper on clusters, I thought he was engaging in wishful thinking to invoke the X-ray gas as possibly containing a lot of the mass. Turns out he was right; it just isn’t quite enough.

All that said, I nevertheless think the residual missing baryon problem MOND suffers in clusters is a serious one. I do not see a reasonable solution. Unfortunately, as I’ve discussed before, LCDM suffers an analogous missing baryon problem in galaxies, so pick your poison.

It is reasonable to imagine in LCDM that some of the missing baryons on galaxy scales are present in the form of warm/hot circum-galactic gas. We’ve been looking for that for a while, and have had some success – at least for bright galaxies where the discrepancy is modest. But the problem gets progressively worse for lower mass galaxies, so it is a bold presumption that the check-sum will work out. There is no indication (beyond faith) that it will, and the fact that it gets progressively worse for lower masses is a direct consequence of the data for galaxies looking like MOND rather than LCDM.

Consequently, both paradigms suffer a residual missing baryon problem. One is seen as fatal while the other is barely seen.

2. Cluster collision speeds

A novel thing the Bullet Cluster provides is a way to estimate the speed at which its subclusters collided. You can see the shock front in the X-ray gas in the picture above. The morphology of this feature is sensitive to the speed and other details of the collision. In order to reproduce it, the two subclusters had to collide head-on, in the plane of the sky (practically all the motion is transverse), and fast. I mean, really fast: nominally 4700 km/s. That is more than the virial speed of either cluster, and more than you would expect from dropping one object onto the other. How likely is this to happen?

There is now an enormous literature on this subject, which I won’t attempt to review. It was recognized early on that the high apparent collision speed was unlikely in LCDM. The chances of observing the bullet cluster even once in an LCDM universe range from merely unlikely (~10%) to completely absurd (< 3 x 10-9). Answers this varied follow from what aspects of both observation and theory are considered, and the annoying fact that the distribution of collision speed probabilities plummets like a stone so that slightly different estimates of the “true” collision speed make a big difference to the inferred probability. What the “true” gravitationally induced collision speed is is somewhat uncertain because the hydrodynamics of the gas plays a role in shaping the shock morphology. There is a long debate about this which bores me; it boils down to it being easy to explain a few hundred extra km/s but hard to get up to the extra 1000 km/s that is needed.

At its simplest, we can imagine the two subclusters forming in the early universe, initially expanding apart along with the Hubble flow like everything else. At some point, their mutual attraction overcomes the expansion, and the two start to fall together. How fast can they get going in the time allotted?

The Bullet Cluster is one of the most massive systems in the universe, so there is lots of dark mass to accelerate the subclusters towards each other. The object is less massive in MOND, even spotting it some unseen baryons, but the long-range force is stronger. Which effect wins?

Gary Angus wrote a code to address this simple question both conventionally and in MOND. Turns out, the longer range force wins this race. MOND is good at making things go fast. While the collision speed of the Bullet Cluster is problematic for LCDM, it is rather natural in MOND. Here is a comparison:

A reasonable answer falls out of MOND with no fuss and no muss. There is room for some hydrodynamical+ high jinx, but it isn’t needed, and the amount that is reasonable makes an already reasonable result more reasonable, boosting the collision speed from the edge of the observed band to pretty much smack in the middle. This is the sort of thing that keeps me puzzled: much as I’d like to go with the flow and just accept that it has to be dark matter that’s correct, it seems like every time there is a big surprise in LCDM, MOND just does it. Why? This must be telling us something.

3. Cluster formation times

Structure is predicted to form earlier in MOND than in LCDM. This is true for both galaxies and clusters of galaxies. In his thesis, Jay Franck found lots of candidate clusters at redshifts higher than expected. Even groups of clusters:

Figure 7 from Franck & McGaugh (2016). A group of four protocluster candidates at z = 3.5 that are proximate in space. The left panel is the sky association of the candidates, while the right panel shows their galaxy distribution along the LOS. The ellipses/boxes show the search volume boundaries (Rsearch = 20 cMpc, Δz ± 20 cMpc). Three of these (CCPC-z34-005, CCPC-z34-006, CCPC-z35-003) exist in a chain along the LOS stretching ≤120 cMpc. This may become a supercluster-sized structure at z = 0.

The cluster candidates at high redshift that Jay found are more common in the real universe than seen with mock observations made using the same techniques within the Millennium simulation. Their velocity dispersions are also larger than comparable simulated objects. This implies that the amount of mass that has assembled is larger than expected at that time in LCDM, or that speeds are boosted by something like MOND, or nothing has settled into anything like equilibrium yet. The last option seems most likely to me, but that doesn’t reconcile matters with LCDM, as we don’t see the same effect in the simulation.

MOND also predicts the early emergence of the cosmic web, which would explain the early appearance of very extended structures like the “big ring.” While some of these very large scale structures are probably not real, there seem to be a lot of such things being noted for all of them to be an illusion. The knee-jerk denials of all such structures reminds me of the shock cosmologists expressed at seeing quasars at redshifts as high as 4 (even 4.9! how can it be so?) or clusters are redshift 2, or the original CfA stickman, which surprised the bejeepers out of everybody in 1987. So many times I’ve been told that a thing can’t be true because it violates theoretician’s preconceptions, only for them to prove to be true, ultimately to be something the theorists expected all along.

Well, which is it?

So, as the title says, clusters ruin everything. The residual missing baryon problem that MOND suffers in clusters is both pernicious and persistent. It isn’t the outright falsification that many people presume it to be, but is sure don’t sit right. On the other hand, both the collision speeds of clusters (there are more examples now than just the Bullet Cluster) and the early appearance of clusters at high redshift is considerably more natural in MOND than In LCDM. So the data for clusters cuts both ways. Taking the most obvious interpretation of the Bullet Cluster data, this one object falsifies both LCDM and MOND.

As always, the conclusion one draws depends on how one weighs the different lines of evidence. This is always an invitation to the bane of cognitive dissonance, accepting that which supports our pre-existing world view and rejecting the validity of evidence that calls it into question. That’s why we have the scientific method. It was application of the scientific method that caused me to change my mind: maybe I was wrong to be so sure of the existence of cold dark matter? Maybe I’m wrong now to take MOND seriously? That’s why I’ve set criteria by which I would change my mind. What are yours?

*In the discussion associated with a debate held at KITP in 2018, one particle physicist said “We should just stop talking about rotation curves.” Straight-up said it out loud! No notes, no irony, no recognition that the dark matter paradigm faces problems beyond rotation curves.

+There are now multiple examples of colliding cluster systems known. They’re a mess (Abell 520 is also called “the train wreck cluster“), so I won’t attempt to describe them all. In Angus & McGaugh (2008) we did note that MOND predicted that high collision speeds would be more frequent than in LCDM, and I have seen nothing to make me doubt that. Indeed, Xavier Hernandez pointed out to me that supersonic shocks like that of the Bullet Cluster are often observed, but basically never occur in cosmological simulations.

Quantifying the excess masses of high redshift galaxies

Quantifying the excess masses of high redshift galaxies

As predicted, JWST has been seeing big galaxies at high redshift. There are now many papers on the subject, ranging in tone from “this is a huge problem for LCDM” to “this is not a problem for LCDM at all” – a dichotomy that persists. So – which is it?

It will take some time to sort out. There are several important aspects to the problem, one of which is agreeing on what LCDM actually predicts. It is fairly robust at predicting the number density of dark matter halos as a function of mass. To convert that into something observable requires understanding how baryons find their way into dark matter halos at early times, how those baryons condense into regions dense enough to form stars, what kinds of stars form there (thus determining observables like luminosity and spectral shape), and what happens in the immediate aftermath of early star formation (does feedback shut off star formation quickly or does it persist or is there some distribution over all possibilities). This is what simulators attempt to do. It is hard work, and they are a long way from agreeing with each other. Many of them appear to be a long way from agreeing with themselves, as their answers continue to evolve – sometimes because of genuine progress in the simulations, but sometimes in response to unanticipated* observations.

Observationally, we can hope to measure at least two distinct things: the masses of individual galaxies, and their number density – how many galaxies of a particular mass exist in a specified volume. I have mostly been worried about the first issue, as it appears that individual galaxies got too big too fast. In the hierarchical galaxy formation picture of LCDM, the massive galaxies of today were assembled from many smaller protogalaxies over an extended period of time, so big galaxies don’t emerge until comparatively late: it takes about seven billion years for a typical bright galaxy to assemble half its stellar mass. (The same hierarchical process is accelerated in MOND so galaxies can already be massive at z ≈ 10.) That there are examples of individual galaxies that are already massive in the early universe is a big issue.

How common should massive galaxies be? There are always early adopters: objects that grew faster than average for their mass. We’ll always see the brightest things first, so is what we’re seeing with JWST typical? Or is it just the bright tip of an iceberg that is perfectly reasonable in LCDM? This is what the luminosity function helps quantify: just how many galaxies of each mass are there? If we can quantify that, then we can quantify how many we should be able to see with a given survey of specified depth and sky coverage.

Astronomers have been measuring the galaxy luminosity function for a long time. Doing so at high redshift has always been an ambition, so JWST is hardly the first telescope to contribute to the subject. It is the newest and best, opening a regime where we had hoped to see protogalactic fragments directly. Instead, the first thing we see are galaxies bigger than we expected (in LCDM). This has been building for some time, so let’s take a step back to provide some context.

Steinhardt et al. (2016) pointed out what they call “the impossibly early galaxy problem.” They quantified this by comparing the observed luminosity function in various redshift bins to that predicted by LCDM. We’ve discussed their Fig. 1 before, so let’s look now at their Fig. 4:

Figure 4 from Steinhardt et al. (2016). Colors correspond to redshift, with z = 4, 5, 6, 7, 8, 9, 10 being represented by blue, green, yellow, orange, red, pink, and black: there are fewer objects at high redshift where they’ve had less time to form. (a) Expected halo mass to monochromatic UV luminosity ratio, along with the required evolution to reconcile observation with theory, and (b) resulting corrected halo-mass functions derived as in Figure 1 with Mhalo/LUV evolving due to a stellar population starting at low metallicity at z = 12 and aging along the star-forming main sequence, as described in Section 4.1.1. Such a model would be reasonable given observational constraints, but cannot produce agreement between measured UV luminosity functions and simulated halo-mass functions.

In a perfect model, the points (data) would match the lines (theory) of the same color (redshift). This is not the case – observed galaxies are persistently brighter than predicted. Making that prediction is subject to all the conversions from dark matter mass to stellar mass to observed luminosity we mentioned above, so they also show what they expect and what it would take to match the data. These are the different lines in the top panel. There is a lot of discussion of this in their paper that boils down to these lines are different, and we cannot plausibly make them the same.

The word “plausibly” is doing a lot of work in that last sentence. Just because one set of authors finds something to be impossible (despite their best efforts) doesn’t mean anyone else accepts that. We usually don’t, even when we should**.

It occurs to me that not every reader may appreciate how redshift corresponds to cosmic time. So here is a graph for vanilla LCDM parameters:

The age-redshift relation for the vanilla LCDM cosmology. Everything at z > 3 is in the early universe, i.e., the first two billion years after the Big Bang. Everything at z > 10 is in the very early universe, the first half billion years when there has not yet been time to form big galaxies hierarchically.

Things don’t change much if we adopt slightly different cosmologies: this aspect of LCDM is well established. We used to think it would take a least a couple of billion years to form a big galaxy, so anything at z > 3 is surprising from that perspective. That’s not wrong, as there is an inverse relation between age and redshift, with increasing redshifts crammed into an ever smaller window of time. So while z = 5 and 10 sound very different, there is only about 700 Myr between them. That sounds like a long time to you and me, but the sun will only complete 3 orbits around the Galaxy in that time. This is why it is hard to imagine an object as large as the Milky Way starting from the near-homogeneity of the very early universe then having time to expand, decouple, recollapse, and form into something coherent so “quickly.” There is a much larger distance for material to travel than the current circumference of the solar circle, and not much time in which to do it. If we want to get it done by z = 10, there is less than 500 Myr available – about two orbits of the sun. We just can’t get there fast enough.

We’ve quickly become jaded to the absurdly high redshifts revealed by JWST, but there’s not much difference in cosmic time between these seemingly ever higher redshifts. Very early epochs were already being probed before JSWT; JWST just brings them into excruciating focus. To provide some historical perspective about what “high redshift” means, here is a quote from Schramm (1992). The full text is behind a paywall, so I’ll just quote a relevant paragraph:

Pushing the opposite direction from the “zone of mystery” epoch [the dark ages] between the background radiation and the existence of objects at high redshift is the discovery of objects at higher and higher redshift. The higher the redshift of objects found, the harder it is to have the slow growth of Figure 5 [SCDM] explain their existence. Some high redshift objects can be dismissed as statistical fluctuations if the bulk of objects still formed late. In the last year, the number of quasars with redshifts > 4 has gone to 30, with one having a redshift as large as 4.9… While such constraints are not yet a serious problem for linear growth models, eventually they might be.

David Schramm, 1992

Here we have a cosmologist already concerned 30 years ago that objects exist at z > 4. Crazy, that! Back then, the standard model was SCDM; one of the reasons to switch to LCDM was to address exactly this problem. That only buys us a couple of billion years, so now we’re smack up against the same problem all over again, just shifted to higher redshift. Some people are even invoking statistical fluctuations: same as it ever was.

Consequently, a critical question is how common these massive galaxies are. Sure, massive galaxies exist before we expected them. But are they just statistical fluctuations? This is a question we can address with the luminosity function.

Here is the situation just before JWST was launched. Yung et al. (2019) made a good faith effort to establish a prior: they made predictions for what JWST would see. This is how science is supposed to work. In the figure below, I compare that to what was known (Stefanon et al. 2021) from the Spitzer Space Telescope, in many ways the predecessor to JSWT:

Figure 4 from McGaugh (2024). The number density Φ of galaxies as a function of their stellar mass 𝑀∗, color coded by redshift with 𝑧=6, 7, 8, 9, 10 in dark blue, light blue, green, orange, and red, respectively. The left panel shows predicted stellar mass functions [lines] with the corresponding data [circles]. The right panel shows the ratio of the observed-to-predicted density of galaxies. There is a clear excess of massive galaxies at high redshifts.

If you just look at the mass functions in the left panel, things look pretty good. This is one of the dangers of the logarithmic plots necessary to illustrate the large dynamic range of astronomical data: large differences may look small in log-log space. So I also plot the ratio of densities at right. There one can see a clear excess in the number density of high mass galaxies. There are nearly an order of magnitude more 1010 M☉ galaxies than expected at z ≈ 8!

For technical reasons I don’t care to delve into, it is difficult to get the volume estimate right when constructing the luminosity function. So I can imagine there might be some systematic effects to scale the ratio up or down. That wouldn’t do anything to explain the bump at high masses, and it is rather harder to get the shape wrong, especially at the bright end. The faint end of the luminosity function is the hard part!

The Spitzer data already probes the early universe, before JWST reported results. As those have come in, it has started to be possible to construct luminosity functions at very high redshift. Here are some measurements from Harikane et al. (2023), Finkelstein et al. (2023), and Robertson et al. (2023) together with revised predictions from Yung et al. (2024).

Figure 5 from McGaugh (2024). The number density of galaxies as a function of their rest-frame ultraviolet absolute magnitude observed by JWST, a proxy for stellar mass at high redshift. The left panel shows predicted luminosity functions [lines], color coded by redshift: blue, green, orange, red for 𝑧=9, 11, 12, 14, respectively. Data in the corresponding redshift bins are shown as squares, circles, and triangles. The right panel shows the ratio of the observed-to-predicted density of galaxies. The observed luminosity function barely evolves, in contrast to the prediction of substantial evolution as the first dark matter halos assemble. There is a large excess of bright galaxies at the highest redshifts observed.

Again, we see that there is an excess of bright galaxies at the highest redshifts.

As we look to progressively higher redshift, the light we observe shifts from familiar optical bands to the ultraviolet. This was a huge part of the motivation to build JWST: it is optimized for the infrared, so we can observed the redshifted optical light as our eyes would see it. Astronomers always push to the edge of what a telescope can do, so we start to run into this problem again at the highest redshifts. The mapping of ultraviolet light to stellar mass is one of the harder tasks in stellar population work, much less mapping that to a dark matter halo mass. So one promising conventional idea is “the up-scattering in UV luminosity of small, abundant halos due to stochastic, high efficiency star formation during the initial phases of galaxy formation (unregulated star formation)” discussed$ by Finkelstein et al. (2023). I like this because, yeah, we expect lots of little halos, star formation is messy and star formation during the first phases of galaxy formation should be especially messy, so it is easy to imagine little halos stochastically lighting up in the UV. But can this be enough?

It remains to be seen if the observations can be explained by this or any of the usual tweaks to star formation. It seems like a big gap to overcome. I mean, just look at the left panel of the final figure above. The observed UV luminosity function is barely evolving while the prediction of LCDM is dropping like a rock. Indeed, the mass functions get jagged, which may be an indication that there are so few dark matter halos in the simulation volume at the redshift in question that they do not suffice to define a smooth mass function. Indeed, Harikane et al. estimate a luminosity density of ∼7 × 10−6 mag.−1 Mpc−3 at 𝑧≈16. This point is omitted from the figure above because the corresponding prediction is NAN (not a number): there just isn’t anything big enough in the simulation to do be so bright that early.

There is good reason to be skeptical of the data at 𝑧≈16. There is also good reason to be skeptical of the simulations. These have yet to converge, and even the predictions of the same group continue to evolve. Yung et al. (2019) did the right thing to establish a prior before JWST’s launch, but they haven’t stuck by it. The density of rare, massive galaxies has gone up by a factor of 2 to 2.5 in Yung et al. (2024). They attribute this to the use of higher resolution simulations, which may very well be correct: in order to track the formation of the earliest structures, you have to resolve them. But it doesn’t exactly inspire confidence that we actually know what LCDM predicts, and it feels like the same sort of moving of the goalposts that I’ve witnessed over and over and over and over and over again.

It always seems to come down to special pleading:

Please don’t falsify LCDM! I ran out of computer time. I had a disk crash. I didn’t have a grant for supercomputer time. My simulation data didn’t come back from the processing center. A senior colleague insisted on a rewrite. Someone stole my laptop. There was an earthquake, a terrible flood, locusts! It wasn’t my fault! I swear to God!

And the community loves LCDM, so we fall for it every time.

Oh, LCDM. LCDM, honey.

*There is always a danger in turning knobs to fit the data, and there are plenty of knobs to turn. So what LCDM predicts is a very serious matter – a theory is only as good as its prior, and we should be skeptical if theorists keep adjusting what that is in response to observations they failed to predict. This is true even in the absence of the existential threat of MOND which implies that the entire field of cosmological simulations is betrayed by its most fundamental assumptions, reducing it to “garbage in, garbage out.”

**When I first found that MOND had predicted our observations of low surface brightness galaxies where dark matter had not, despite my best efforts to make it work out, Ortwin Gerhard asked me if he “had to believe it.” My instant reaction was “this is astronomy, we don’t have to believe anything.” More seriously, this question applies on many levels: do we believe the data? do we believe the interpretation? is this the only possible conclusion? At the time, I had already tried very hard to fix it, and had failed. Still, I was willing to imagine there might be some way out, and maybe someone could figure out something I had not. Since that time, lots of other people have tried and also failed. This has not kept some of them from claiming that they have succeeded, but they never seem to address the underlying problem, and most of these models are mere variations on things I tried and dismissed as obviously unworkable.

Now, as then, what we are obliged to believe is the data, to the limits of their accuracy. The data have improved substantially, and at this point it is clear that the radial acceleration relation exists+ and has remarkably small intrinsic scatter. What we can always argue about is the interpretation: sure, it looks exactly like MOND, and MOND was the only theory that predicted it in advance, and we haven’t been able to come up with a reasonable explanation in terms of dark matter, but perhaps one can be found in some dark matter model that does not yet exist.

+Of course, there will always be some people behind the times and in a state of denial, as this subject seems to defeat rationalism in the hearts and minds of particle physicists in the same way Darwin still enrages some of the more religiously inclined.

$I directly quote Finkelstein’s coauthor Mauro Giavalisco from an email exchange.

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

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 (Ωk < 0, in which initially parallel light beams diverge) while the geometry in LCDM should be uniquely flat (Ωk = 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 ΩmH03; 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 Ωm ≈ 0, which is far off the left edge of the plot.

Second, the CMB best-fit values follow a line of constant ΩmH03. 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 Ωm.

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.

Global climate basics

Global climate basics

Last time, I expressed extreme disappointment that fossil fuel executives had any role in leading the climate meeting COP28. This is a classic example of putting the the fox in charge of the hen house. The issue is easily summed up:

It’s difficult to get a man to understand something when his salary depends on not understanding it.

Upton Sinclair

Setting aside economic self-interest and other human foibles, it is clear from the comments that the science is not as clear to everyone as it is to me. That’s fair; I’ve followed this subject for half a lifetime, and it is closely related to my own field.

Stars are fusion reactors surrounded by big balls of gas; understanding how they work was a major triumph of 20th century astrophysics. We understand these things. Planetary atmospheres are also balls of gas; there is some rich physics there but the problem is in many ways simpler when they aren’t acting as the container for a giant fusion reactor. We understand these things. The atmospheres of Venus and Mars come up when teaching Astronomy 101, these planets represent opposite extremes of climate change run amok. From that perspective, Earth is a nice problem to have. We understand these things.

It is easy to get distracted by irrelevant details. No climate model is ever perfect, but that doesn’t mean we don’t understand what’s going on. The issue is basic physics, which has been understood for well over a century. Not only is the physics incredibly clear; so too is the need to take collective action to ameliorate the effects of climate change. The latter has itself been clear since 1990+ at least.

The temperature of a planet is the balance between heating by the sun during the day and re-radiation of that heat at night. The effectiveness of both depend on the properties of the planet. What is the albedo? That is, how much of the incident radiation is reflected into space without heating? Once heated, how efficiently can the heated surface cool by radiating energy to space?

If a planet has no atmosphere, it is a straightforward calculation to find the balance point. If the Earth had no atmosphere, the average temperature would be much colder than it is, about -18 C. Thankfully, we have an atmosphere. There is a natural greenhouse effect – nothing to do with human activity – that makes the actual average temperature more like +15 C. I, for one, am grateful for this. It also means that changing the composition of the atmosphere will change the balance point.

The bulk of Earth’s atmosphere is nitrogen and oxygen. These gases are transparent to the incoming optical radiation from the sun that heats the surface. They are also transparent to the outgoing infrared radiation that cools the surface. Despite composing the bulk of the atmosphere, they play basically zero role in the greenhouse effect. As far as climate goes, having only these gases in the atmosphere returns the same answer as the zero atmosphere case.

The natural greenhouse effect is entirely due to trace gases like water vapor and carbon dioxide. I note this because one reasonable-sounding falsehood that gets repeated a lot is that CO2 is a trace gas, so it can’t possibly make a difference. That’s like saying adding a small dash of poison to a beverage isn’t dangerous. Or that it makes no difference to draw a shade over a window. The shade may be much thinner than the glass of the window, but unlike the transparent glass, the shade is opaque. That’s the property greenhouse gases provide, even in trace quantities: they are opaque to the infrared radiation that is trying to cool the surface by escaping to space.

If we looked down on the Earth with eyes that saw in the infrared part of the spectrum where greenhouse gases trap heat, we’d wouldn’t see the surface of the planet. Instead, we’d see a hazy ball: the effective altitude in the atmosphere from which infrared radiation can escape to space. This isn’t a solid surface any more than the edge of a cloud is – to you and me. To the photons seeking escape, it is an effective barrier. Some don’t make it out.

The greenhouse gases are like a fog bank that has to be traversed before the heat carried by the infrared radiation can escape into space. If we add greenhouse gases to the atmosphere, it makes the fog bank thicker, effectively trapping more heat. At a basic level, the issue is that simple. The science is entirely settled; no one seriously* debates this. It has been known for over a century.

Article published in 1912 in the Braidwood Dispatch and Mining Journal, via the National Library of Australia

The leading greenhouse gas in Earth’s atmosphere is water vapor. You don’t need a fancy scientific instrument to detect this effect, just your own senses. High humidity leads to hot, sultry nights while low humidity allows rapid cooling. To feel this, visit a humid place like New Orleans and an arid one like the desert of the US west. These places feel very different at night even when their daytime temperatures are similar. The humid place cannot cool effectively because of the greenhouse effect provided by water vapor, and nighttime temperatures can remain unpleasantly high. In the dry desert, the temperature drops like a rock as soon as the sun sets, and it can get rather chilly even if it was baking hot all day long. I’ve personally experienced both conditions many times; the difference is stark and obvious.

The amount of water vapor the atmosphere can hold is a function of temperature, but on bulk it is always less than half a percent. That trace gas is nevertheless 100% of what you care about in the morning weather forecast, as it leads to rain, snow, sleet, hail, cloud cover, and all the other weather phenomena that makes life near the triple point of water interesting. Indeed, clouds increase the albedo of the planet, reflecting some of the incoming solar radiation, so water in the atmosphere prevents some heating as well as helping to retain warmth once heated. This is pretty much in balance, as the limit on how much water vapor the atmosphere can hold means that equilibrium is achieved on a short time scale: too much humidity, and it rains. The sources and sinks of H2O in the atmosphere balance out on short timescales readily perceptible to humans. It’s what we call weather.

The next most important greenhouse gas is CO2. That too has a natural level with sources and sinks. The issue that induces human-caused climate change is the extra CO2 we put in the atmosphere by burning coal, oil, etc. for the energy it provides. This does not balance out on a short timescale, so there is a cumulative effect on the climate, as anticipated in 1912.

Producing energy is a good thing; no one here is advocating that we stop doing this add return to living like cavemen. Heck, even cavemen had an environmental impact: they burned enough wood to blacken many a cave roof. Human activity has always left a mark; the problem today is that there are 8+ billion of us doing a lot more than making campfires. That adds up to a measurable change in the composition of the atmosphere.

The natural pre-industrial level of CO2 was about 277 parts per million (ppm). Here is a graph of the CO2 content of the atmosphere over the past few centuries, extending to back before the onset of the industrial revolution when our collective experiment in atmospheric physics got going. We know how much carbon we’ve burned (that’s economic activity with profits and receipts, we know this number quite well) and we can measure how much CO2 is in the atmosphere directly. They ramp up together.

Mass of CO2 in the atmosphere (in gigatonnes) since 1700. Modern measurements (blue line) come from the Mauna Loa observatory courtesy of the NOAA Global Monitoring Laboratory; older measurements (black line) come from the Law Dome Antarctic ice core data. The red line is the cumulative CO2 added to the atmosphere by human activity. I’ve added the pre-industrial value to this in the upper (thin) red line to show how it compares with the measured CO2 content.

There is lots that can be said about this plot. Just some basic points: the amount of CO2 in the atmosphere has gone up as we have burned coal and oil to generate energy. We have measurably changed the composition of the atmosphere we all breathe. The current CO2 content of the atmosphere is 424 ppm, which is much larger than the pre-industrial level of 277 ppm. That by itself ought to give one pause: we are conducting an uncontrolled experiment in atmospheric physics on a global scale. That seems like a bad idea, even if we didn’t understand heat propagation in the atmosphere, which we do.

Not only has the amount of CO2 in the atmosphere increased as we’ve burned things, it is accumulating. There are natural sinks, which is why the extra amount of CO2 in the atmosphere is less than what we’ve added: not all of it sticks around. Much of it has been absorbed by the ocean, which is acidifying as a result. But lots of CO2 persists in the atmosphere: the timescale for it to “rain out” is much longer than for water. It will take many decades and probably centuries to restore anything resembling equilibrium. We aren’t just adding CO2 to the atmosphere, we’re making a long-term investment in having it there. Future generations will have to contend with the consequences of what we’ve already done.

What have we already done? I’ve outlined the basic physics; let’s now check the predictions of one of the earliest forecasts. This is from a 1982 report generated by Exxon scientists:

Forecast atmospheric CO2 content of the atmosphere (upper line; left axis) and the corresponding temperature change (lower line; right axis). I’ve added current values for both CO2 (green) and the temperature anomaly (orange). Looks like they pretty much nailed it. Note also that the null hypothesis of no climate change, i.e., a constant temperature with increasing CO2 content, is strongly rejected.

The study made over forty years ago accurately forecast where we are today. These predictions have repeatedly been corroborated. Some models may miss minor details here and there, but the basic picture is crystal clear. Anyone who tells you otherwise has some fossil fuels to sell.

Enough has been written on this subject; I won’t suggest solutions nor delve into likely impacts. But there is absolutely no doubt that climate change is real and that we caused it. None. That this simple, plain fact is not obvious to everyone at this point is a credit to the power of disinformation and propaganda. The best course forward from here is debatable. Pretending like it isn’t a problem is straight-up reality denial.

+It has become a trope of wingnut politics in the U.S. that scientists only say climate change is real so they can get research grants. That’s ridiculous on many levels. One reason that such grants exist is that right wing politicians asked for more research. This was a delaying tactic employed in the early 1990s by then-president and oil magnate George H. W. Bush.

Fresh off the success of regulatory repair to the ozone hole problem in the late 1980s, it was reasonable to hope that we could start tackling the threat of climate change. This was a much bigger problem encompassing a broader range of human activity, but the basic science is far simpler than the atmospheric chemistry that threatened ozone. Industries that didn’t want to be regulated whined about that as usual, but no one seriously questioned the science. After the usual wailing and gnashing of teeth, appropriate regulatory action was taken, and it worked.

When it came to doing the same thing with the oil industry when an oil baron was president, well, harrumph harrumph, more research was needed. The first Bush was a Republican, but he wasn’t a backwards science-denying goon, so he offered to fund more research. It was an obvious delaying tactic, but the argument in favor of it was to make the case more convincing. So the science community was like, sure, the basic answer is already clear, but there are things that we could understand better, so we’ll do more research if that helps you to also understand the problem. But it hasn’t helped people who don’t want to understand to do so, and never will, because the problem is with them, not with the science. So now, thirty years on from Bush I, the same political party that demanded more research be done now routinely attacks scientists for doing the research they asked scientists to do.

Sorry, not sorry: just because you don’t like the answer science gives doesn’t make it wrong. It is well past time for climate denying snowflakes to stop having emotional meltdowns and grow up already.

*Sometimes it is asserted that the opacity of CO2 is already saturated, so adding more doesn’t matter. Yes on one, no on two. Even at saturation we can still make the fog bank thicker by adding more CO2 – just ask Venus. Indeed, we’re dang lucky that the CO2 bands are already saturated; if not for that, the response of the climate to adding as much CO2 as we have would be much stronger. If these features were not saturated the response would be linear instead of incremental, so the temperature would have already increased by about an extra 7 C, not the mere 1 C we’ve so far** accomplished.

**Just how much we’ve added depends on how you define “before.” Modern studies often seem to adopt the average temperature measured between 1980 and 2000, presumably because the data with which to do so are very good. This gives an increase since then around 1.1 C, which is a remarkable amount of growth in just a few decades: we’ve tipped the climate system out of anything resembling equilibrium hard and fast. Of course, the impact of human activity was already palpable before 1980, so the total change since the industrial revolution is closer to 1.5 C. We’re not quite to that arbitrary threshold yet, but I see no way to avoid blowing past it. Talk of doing so is predicated on giving us a half degree mulligan by defining “average” during a period that is not average. So if you think portrayals of the problem are exaggerated, it is actually already worse than generally depicted.

Cop28 president not even trying to hide his obvious bias

In 1986, I was a grad student at Princeton, working in the atomic physics lab of Will Happer. It was at a department colloquium that I first heard a science talk that raised serious concerns about our use of fossil fuels potentially impacting the climate. This was not received well.

People asked all sorts of questions, with much of the discussion revolving around feedback effects. Perhaps warmer weather from CO2 will result in higher humidity, making more clouds*, and reflecting more sunlight into space. It does not. What about ice cover? This is actually a positive feedback – as the globe warms, ice coverage is replaced by darker surfaces, leading to more absorption of the incident solar radiation. And so on.

I thought the speaker did a creditable job of answering the concerns raised, repeatedly making the point that most feedback effects would make things worse, not better. It was entirely new to me at the time; I didn’t have any context to judge the relative merits of the discussion. Prof. Happer is one of those remarkable people who seems to know a lot about everything. So, as I related before, I asked him. His immediate and harsh retort was

“We can’t turn off the wheels of industry, and go back to living like cavemen.”

I relate this story again because the same language comes up today in a story I saw in the Guardian:

The president of Cop28, Sultan Al Jaber, has claimed there is “no science” indicating that a phase-out of fossil fuels is needed** to restrict global heating to 1.5C, the Guardian and the Centre for Climate Reporting can reveal.

Al Jaber also said a phase-out of fossil fuels would not allow sustainable development “unless you want to take the world back into caves”.

COP23 article, 3 December 2023

This is exactly the same solution aversion that Happer displayed, using exactly the same language. It doesn’t address the actual question. It leaps ahead to the worst conceivable consequence, doesn’t like it, and so reverts to reality denial: We don’t want that to happen, so the evidence must be wrong!

We humans excel at reality-denial. It is not helpful. Rather than starting to deal with the problem of climate change thirty years ago – the science was already crystal clear by then – we’ve dug ourselves a much deeper hole. That’s not to say we should abandon all hope and revert to living in caves, but we do need to take serious and rapid steps to reform the ways in which we generate power. It is doing more of the same that risks sending us back into caves.

The quoted reaction to this assertion in the story is predictably tepid. The quote in the Guardian is “The comments were `incredibly concerning’ and `verging on climate denial’, scientists said.” As a scientist who is not directly involved with dealing with these people, let me be more blunt:

ARE YOU FUCKING KIDDING ME?

Al Jaber’s attitude isn’t verging on climate denial, it is the archetype of climate denial. Literally the same thing that climate deniers said in the 1980s. It expresses an attitude that was clearly wrong and dangerously backwards by the early 1990s. And this guy is the president of COP28? I say again

ARE YOU FUCKING KIDDING ME?

And who is this guy? The Guardian reports “Al Jaber is also the chief executive of the United Arab Emirates’ state oil company, Adnoc, which many observers see as a serious conflict of interest.” A conflict of interest? Really? Do you think? Again I say

ARE YOU FUCKING KIDDING ME?

This is obviously a conflict of interest, of the worst sort. His personal wealth, and the sovereign wealth of his nation, is entirely based on the production and sale of fossil fuels. Mitigating climate change means reducing our consumption of fossil fuels, which is a direct threat to the economic interests he represents. Talk about putting the fox in charge of the hen house.

I am impressed by how the moneyed interests have managed to slither their way into positions of consequence on discussions in which they have an obvious conflict. I guess money always finds a way in. But can we please stop being so polite that we fail to call out obvious bullshit wherever it crops up? It seems to be spreading at the rate of made-up conspiracy nonsense on that site formerly known as Twitter. We should stop putting up with it already.

*Ironically, SO2 pollution from ocean-going vessels does have this effect, and as these emissions have been cleaned up, we can see the effect in global temperatures. This is not to advocate for SO2 pollution! though injecting aerosols like SO2 into the stratosphere is one of geoengineering approaches that gets discussed. Before going down that path, the obvious first step is to stop pouring petrol on the fire by continuing to add CO2 to the atmosphere.

**We’re already committed to 1.5C. I see no conceivable way that we can curb emissions fast enough to avoid that. So I guess this statement is true, from a certain point of view – that of a liar. A less misleading statement would be that a phase-out of fossil fuels is necessary to prevent things from getting much worse than forecast for the 1.5C threshold.

A post in which some value judgements are made about the situation with wide binaries

A post in which some value judgements are made about the situation with wide binaries

I have tried very hard to remain objective and even handed, but I find that I weary of the wide binary debate. I don’t know what the right answer will turn out to be. But I do have opinions.

For starters, it is a big Galaxy. There is just too much to know. When I wrote about the Milky Way earlier this year, the idea was to set up an expectation value for wide binaries in the solar neighborhood. That devolved into at least eight other posts on the Milky Way itself, because our Galaxy is too damn interesting, and has its own controversies. So it occurs to me that I never really got on with the regularly scheduled program.

In my assessment, the radial acceleration at the solar circle is 2.2 x 10-10 m/s/s, which in terms of the MOND acceleration scale is 1.8 a0. We live on the Newtonian side of the transition to the MOND regime. The ideal place to test MOND with wide binaries would be the deep MOND regime, well below a0. That is in a part of the Galaxy that is far, far away, and not currently accessible to us. What is accessible are wide binaries in the solar neighborhood (within 250 pc, about 1% of the Galaxy’s radius) as mapped by Gaia. Locally, the MOND effect is modest, but nonzero. We’re close enough to the transition for there to be a small, detectable effect.

Local binaries that are widely separated enough for their internal acceleration to drop below a0 find themselves in the regime dominated by the field of the rest of the Galaxy and subject to the so-called External Field Effect (EFE). This situation is illustrated in the lower right panel below.

Mass estimators in different regimes of acceleration. The top row illustrates pure Newtonian (left) and MOND (right) regimes. The bottom row illustrates the case of small systems embedded in larger systems. A low acceleration system embedded in a Newtonian external field is Newtonian (left) while a very low acceleration system embedded in a merely low acceleration system is quasi-Newtonian (right). Wide binaries fall in the last category.

Intriguingly, orbits in the EFE regime remain Keplerian. The rotation curves of nearby binaries, if you could map them, are not expected to be flat in MOND. They should, however, experience enhanced speeds, with a boost to the effective value of Newton’s constant: G → γG. The value of γ depends on the sum of internal and external acceleration as well as the shape of the interpolation function when near a0. That’s one reason to prefer to do this experiment in the deep MOND regime, where the shape of the interpolation function doesn’t matter. But that’s not where we live. For Galactic data and viable possibilities* for the interpolation function, a reasonable expectation value is γ = 1.4 ± 0.1. This is what the wide binary papers attempt to measure. So, what do they find?

Hernandez et al. find γ = 1.0±0.1 for 466 close binaries with 2D separations less than 0.01 pc (about 2000 AU) and γ = 1.5±0.2 for 108 wide binaries with 2D separations greater than 0.01 pc. A purely Newtonian result (γ = 1) is recovered in the high acceleration regime of relatively close binaries where this is expected to be the case. For wider binaries, one finds a boost value consistent with the prediction of MOND and differing from Newton with modest significance (2.6σ).

Chae reports+ γ = 1.49(+0.21/-0.19) for 2,463 “pure” binaries in the low acceleration regime, consistent with his earlier result γ = 1.43±0.06 for 26,615 wide binaries. The larger numbers make the formal error smaller, hence a formally more significant departure from Newton. Many of these binaries are impure in the sense of being triples with one member being itself a close binary as discussed previously, an effect that has to be modeled in large samples. The point of the smaller samples is to select true binaries so that this modeling is unnecessary. For his smaller pure binary sample, Chae finds a smooth transition from γ ≈ 1 at high acceleration (10-8 m/s/s ≈ 100a0) through γ ≈ 1.11 around 7a0 to γ ≈ 1.49 at local Galactic saturation (1.8 a0).

Banik et al. use a slightly different language. Translating, they find γ = 1 at high confidence (16σ)$ from 8,611 wide binaries with separations from 2,000 to 30,000 AU. Newtonian behavior persists at all scales and accelerations; they find no significant deviations from γ = 1 anywhere. Note that despite going out very far, to 30,000 AU, they do not reach especially low accelerations because the EFE of the Galaxy is effectively constant in the solar neighborhood. There is no getting away from the Galaxy’s 1.8 a0. They also do not reach particularly high, purely Newtonian accelerations: 2,000 AU is in the transition regime where MOND effects are perceptible.

Here is Fig. 11 from Banik et al., the key figure Dr. Banik was advocating in his comments to the previous post:

Fig. 11 of Banik et al. shows the median dimensionless characteristic velocity as a function of dimensionless binary separation for several bins of the data (solid lines). The predicted MOND effect increases with separation until saturating in the Galactic field (dashed lines). This is not seen in most of the data, with only a hint in the highest velocity bin that represents only a few percent of the data.

A flat line in this plot indicates no boost in velocity with diminishing acceleration, so one can clearly see the source of the claim that Newton works better than MOND. There is little indication that the velocity increases at wider separations. Effectively, γ ≈ 1 pretty much everywhere.

Both Chae and Hernandez have pointed out that the lack of a constraint on the high acceleration Newtonian regime is problematic. Orbits are Keplerian in the quasi-Newtonian regime, so the behavior looks Newtonian. Lacking an anchor in the high acceleration regime, it is conceivable that the analysis of Banik is detecting the predicted MOND quasi-Newtonian behavior and defining it to be purely Newtonian. It’s just a modest offset in qualitatively similar behavior. In this context, it is worth noting that Chae and Hernandez independently measure γ ≈ 1 at high acceleration as well as γ ≈ 1.5 at low acceleration: they [claim to] detect the difference between these regimes in a way Banik does not probe.

Now let’s look at the plot that gave me the heebie-jeebies with data on it, Banik et al.’s Fig. 12:

The left two panels of Fig. 12 from Banik et al. showing the probability of observing a particular dimensionless velocity in two bins of radial separation: 2,000 to 3,000 AU (top) and 5,000 to 12,000 AU (bottom). The histograms are the Gaia data. The black lines show the Newtonian prediction while the blue lines show that of MOND. These predictions depend on many things besides the underlying theory, sampling over many astrophysical complications like the distribution of stellar masses, orbital orientations to the line of sight, orbital phase, orbital eccentricity, the close binary fraction, and probably other things that I don’t instantly recall.

One can see the basis of the concern. At high acceleration, the prediction of Newton and MOND are identical. The top bin is the closest we get to that, yet there is a clear difference in the predictions. This bin is in the transition region; there is no bin at sufficiently high acceleration for the predictions to align and provide the self-calibration that both Chae and Hernandez independently exploit.

Looking at the data in the top panel, it clearly agrees better with the Newtonian prediction. I can believe that; what concerns me is the lack grounding at still higher acceleration where the black and blue lines should coincide. I do not have a sufficiently clear understanding of all the machinations (and their inevitable foibles) that go into the predicted lines to trust that this constitutes a definitive test.

Looking at the data in the bottom panel, it clearly agrees better with the prediction of MOND. The histogram of the data follows the blue line of MOND more closely than the Newtonian black line. This is so obvious that I wondered if the colors were wrong – maybe there had been some inadvertent switcheroo in the line color in the plotting code. Apparently not, as this point is addressed in the text of Banik et al.: [in the bottom panel,] “MOND performs somewhat better in a handful of pixels around the peak region…” Yes. Yes it does. That’s… a really weird way of putting it. They go on to say “…though given the uncertainties, the Newtonian model is not that far off.” One could just as well say that about MOND in the top panel. Just looking at this figure, one might conclude that they have detected MOND at large separations.

One of the things that gives me the heebie-jeebies about this figure is that there isn’t much difference in the location of the peaks of the distributions. Some, yes, but not much: Newton and MOND apparently predict very nearly the same typical velocity. Yet that is what Fig. 11 traces: the typical (median) normalized velocity. That only tells a tiny bit of the story that is in Fig. 12, and does not appear to be a particularly sensitive indicator of the effect we’re testing for.

Returning to the matter of statistics, the attentive reader might have noted that I have not said much about the number of binaries included in each analysis. These range from a few hundred to many thousands to tens of thousands. More is better, right?

In this case, I think not. There is always a tension between data quality and quantity. Quantity helps with the statistics, but only so long as there is a signal to be dug out. At some point, it becomes a matter of garbage in, garbage out. In this respect, I am inclined to agree with Ernest Rutherford:

If your experiment requires statistics, you ought to have done a better experiment.

Ernest Rutherford$

I suspect that we’re squeezing the stone of statistics too hard here. When we do this, we get the appearance of a signal when really we’re just grinding metal. I am reminded that any time we do a big experiment like this (dark matter searches are a great example), the first thing we learn about are all the false signals we didn’t anticipate. That happens no matter how well we construct the experiment, and I give Banik et al. credit for planning this out ahead of time. That doesn’t guarantee that everything comes out right on the first attempt. There is just so much junk that the universe can and does throw at us that it is easy to imagine that the samples with large numbers of binaries bring with them too much junk (e.g., false binaries). If the fraction of junk is high, then it will look like junk at all scales – there will be no trend with increasing separation even if there is a signal buried in junk.

Consequently, I am at present inclined to trust more the super-clean sample of Hernandez – the high quality binaries where there is a chance that we’re actually measuring what we want to measure. There are only a few hundred such binaries, so the statistical confidence is modest (2.6σ). I worry that in setting the highly restrictive standards necessary to select the best binaries that we might unintentionally omit objects that could change the answer. But at least there is some confidence that these are real binaries that stands above the accumulation of all the gratuitously enormous amount of junk the universe has to throw our way.

I hope the principal scientists can come to agreement about what the data show and not just wind up having the same argument over and over forever more. That’s what usually happens. I ask all parties to remember that it is important to retain the ability to change one’s mind. All of them have demonstrated the ability to do this previously, and somebody will need to do it again.

*In principle, one might also hope to distinguish between specific theories of MOND. Examples of modified gravity like AQUAL and QUMOND give slightly different predictions. To be able to do this seems… optimistic at this point.

In modified inertia theories, the interpolation function is a chimera that depends on each orbital trajectory, so its effective realization may differ between the nearly circular motions of rotation curves and eccentric wide binaries. If so, the interpolation function defined by external galaxies may not be relevant to the problem. Ultimately, we need a theory that automatically results in the MONDian phenomenology in galaxies. Whatever it is wide binaries are doing should help inform this theory development as well as test alternatives and hopefully exclude some of them.

+The work of Chae has gone through some revisions in response to a referee, but the basic findings are unchanged. Having read an earlier version, I appreciate the clarity provided by the additions: this is a case where the refereeing process was beneficial. (I was not the referee of this or any of these papers. Editors keep me busy enough as it is, thank you very much.)

$Outside of long-established observations like the value of Gauss’s constant, there is no such thing as 16σ confidence in astronomy. The tails of real probability distributions are never as tiny as a pure Gaussian. This is a remarkably naive assertion.

Full speed in reverse!

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: gbar 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 gbar, gobs 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. 

Wide binary debate heats up again

Wide binary debate heats up again

One of the most interesting and contentious results concerning MOND this year has been the dynamics of wide binaries. When last I wrote on this topic, way back at the end of August, Chae (2023) and Hernandez (2023) both had new papers finding evidence for MONDian behavior in wide binaries. Since that time, they each have written additional papers on the subject. These independent efforts both report strong evidence for MONDian behavior in wide binaries, so for all of October it seemed like Game Over for conventional* dark matter.

I refrained from writing a post then because I was still waiting to see if there would be a contradictory paper. Now there is. And boy, is it contradictory! Where Hernandez et al. find 2.6σ evidence for non-Newtonian behavior and Chae finds ~5σ evidence for non-Newtonian behavior, both consistent with MOND, Banik et al. find purely Newtonian behavior and claim to exclude MOND at 19σ. That’s pretty high confidence!

Well, which is it, young feller? You got proof of non-Newtonian dynamics, or you want to insist that’s impossible?

After the latest results appeared, a red-hot debate [re]ignited on e-mail, largely along the lines of what was discussed at the conference in St. Andrews. Banik et al say that they can reproduce the MOND-like signal of Chae, but that it goes away when the data quality restriction is applied to physical velocity uncertainties (arguing that this is what you want to know) rather than to raw observational uncertainties. Chae and Hernandez counter that the method Banik et al. apply is not grounded in the Newtonian regime where everyone agrees on what should happen, so they could be calibrating the signal away. This is one thing that I had the impression that everyone had agreed to work on in St. Andrews, but it doesn’t appear that we’re there yet.

Banik et al. do a carefully planned Bayesian analysis. This approach in principle allows one to separate many effects simultaneously, one of which is close binaries (CB**). I look at the impact that close binaries have on the analysis, and it gives me the heebie-jeebies:

One panel from Fig. 10 of Banik et al.

This figure illustrates the probability of measuring a characteristic velocity in MOND for the noted range of projected sky separation. If it is just wide binaries (WB), you get the blue line. If there are some close binaries, the expected distribution changes dramatically. This change is rather larger than the signal expected from the nominal difference in gravity. You can in principle fit for everything simultaneously, but extracting the right small signal when there is a big competing signal can be tricky. Bayesian analyses can help, but they are also a double-sided sledge-hammer: a powerful tool with which to pound the data, but also a tool that can bounce back and smack you in the face. Having done such analyses, and been smacked around a few times (and having seen others get smacked around), looking at this plot really does give me the heebie-jeebies. There are lots of ways in which this can go wrong – or even just overstate the confidence of a correct result.

Everyone uses Bayesian methods these days.***

I expect people are expecting me to comment on this hot mess. Some have already asked me to do so. I really don’t want to. I’ve already said more than I should.

There are very earnest, respectable people doing this work; I don’t think anyone is being intentionally misleading. Somebody must be wrong, but it isn’t my job to sort out who. Moreover, these are long and involved analyses; it will take me time to read all the papers and make sense of them. Maybe once I do, I’ll have something more cogent to say.

I make no promises.

*By conventional dark matter, I mean new particles that only communicate with baryons via gravity.

**CB: In principle, some of the wide binaries detected by Gaia will also be close binaries, in the sense that one of the two widely separated stars is itself not a single star but an unrecognized close binary. We know this happen in nature: the nearest star system, αCentauri, is an example. The main A&B components compose a close binary with Proxima Centauri being widely separated. Modeling how often this happens in the Gaia data gives me the willies.

***To paraphrase Churchill: Many forms of statistics have been tried, and will be tried in this science of sin and woe. No one pretends+ that Bayes is perfect or all-wise. Indeed it has been said that Bayes is the worst form of statistics except for all those other forms that have been tried from time to time.

+Lots of people pretend that Bayes is perfect and all-wise.

How things go mostly right or badly wrong

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, aM , as a function of the acceleration due to baryons aB. The solid horizontal line is a = aM. 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 aB > 10−10m/s2 any difference between a and aM is unclear. However, once aB 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 ab 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?