Progressive Approximations in Mass Modeling

I have said I wasn’t going to attempt to teach an entire graduate course on galaxy dynamics in this forum, and I’m not. But I can give some pointers for those who want to try it for themselves. It also provides some useful context for fans of Deur’s approach.

The go-to textbook for this topic is Galactic Dynamics by Binney & Tremaine. The first edition was published in 1987, conveniently when I switched to grad school in astronomy. It was already a deep and well-developed field at that time; this is a compendium of considerable scientific knowledge.

Fun story: a colleague in a joint physics & astronomy department once complained to me that she wanted to develop a course in galaxy dynamics, which is a staple of graduate programs in astronomy & astrophysics. However, there was a certain senior colleague who objected, saying that since it was astronomy, it couldn’t possibly be a rigorous course worthy of a full semester graduate course. This is a casual bias that astronomers often encounter when talking to physicists, many of whom have attitudes about the subject that were trapped in amber sometime in the Jurassic. I suggested that she walk into his office and drop a copy of Galactic Dynamics on his desk from on high, as (1) it would make a hefty impact, and (2) no one who so much as skims this book could persist in this toxic attitude.

She later reported that she had done this, and it had worked.

Galactic Dynamics is not a starter book. It is the textbook we use when teaching the graduate course that this is not. A useful how-to guide for the specific material I’ll discuss here is provided by Federico Lelli. In brief, to model the gravitational potential of an observed distribution of matter, we can make one of the following series of approximations:

*This is a slide I sometimes use to introduce mass modeling in science talks as a reminder for expert audiences.*

All science is an approximation at some level. The most crude approximation we can employ here is to imagine that all of the mass resides at a central point. In this limit, the potential is simply

V² = GM/R

where V is the orbital speed of a test particle on a circular orbit, G is Newton’s constant, M is the mass, and R is the distance from the point mass. Galaxies are not point masses, so this is a terrible approximation, as can be seen by the divergent V ~ R^-1/2 behavior as R → 0 (the dotted line above).

The next bad approximation one can make is a spherical cow: assume the mass is distributed in a sphere that is projected as the image we see on the sky. This at least incorporates the fact that the mass is not all concentrated at a point, so

V² = GM(R)/R

acknowledges that the mass M is spread out as a function of radius. This is a spherical cow. Since we cannot see dark matter, we almost always assume it to be a spherical cow.

For the luminous disk of a spiral galaxy, a common approximation is the so-called exponential disk:

Σ(R) = Σ₀ e^-R/R_d

where Σ₀ is the central surface density of stars and R_d is the scale length of the disk – the characteristic size over which the surface brightness declines exponentially. This can be integrated by parts to obtain an expression for the enclosed mass M(R) which I leave as an exercise for the eager reader. This provides a handy analytic formula, the rotation curve of which is illustrated above by the dashed line.

Spiral galaxies are fairly thin when seen edge-on, so the spherical cow is not a great approximation. In a classic paper, Freeman (1970) solved the Poisson equation for the case of a razor-thin exponential disk, where one meets modified Bessel functions of the first and second kind (denoted “ikik” above). These must be solved numerically, but one can make a tabulation for use with any choice of disk mass and scale length. Such a thin disk is illustrated by the grey line above for a choice of stellar mass and scale length appropriate to NGC 6946.

*The spiral galaxy NGC 6946, aka the fireworks galaxy.*

Spiral galaxies are not razor thin of course. We only see a projected image on the sky, so for a galaxy like NGC 6946, we may have a good measurement of its azimuthally averaged light (and presumable stellar mass) distribution Σ(R) but we have no idea how thick it is. Here, we have to make an educated guess based on observations of edge-on galaxies. A ballpark average is R:z = 8:1, but some galaxies are thicker and others thinner, so this becomes an approximation with an associated uncertainty. This uncertainty cannot be unambiguously eliminated; it is one of the known unknowns that comprise the inevitable systematic errors in astronomy. Fortunately, allowing for a finite thickness only takes the harsh edge off of the thin disk case, and the assumption one chooses makes little difference to the result (compare the lines labeled thick and thin above).

The exponential disk formula Σ(R) is an azimuthal average over an image like that of NGC 6946. This approximation captures none of the spiral structure: it only tells us about the average rate at which the surface brightness falls off. It also imposes a smooth shape on that fall off that our eyes can see is not necessarily a great approximation. So the next level of approximation is to solve the Poisson equation numerically for the observed surface brightness profile, Σ(R), not just the exponential approximation thereto. This is the blue line in the bottom right graph above.

There are important differences between using the numerical solution for the observed light distribution and the exponential disk approximation. This has been known since the 1980s, but the analytic expression is so convenient that people need an occasional reminder not to trust it too much. Jerry Sellwood felt the need to provide this reminder in 1999:

Small apparent differences in the shape of the mass profile (left) correspond to pronounced differences in the rotation curve (right). I chose the example of NGC 6946 in part because the exponential approximation for it is pretty good. Nevertheless, the details matter, so the best practice is to build numerical mass models, as we did for SPARC.

Building numerical mass models is tractable for external galaxies, where we can see the entire light distribution. It is not possible for our own Milky Way, since we are located within it and cannot see it as a whole. Consequently, the vast majority of Milky Way models rely on the exponential approximation; so far as I’m aware, I’m the only one who has built a model that attempts to get beyond this.

Numerical mass models are still an approximation. We’re assuming that the gravitational potential is static and azimuthally symmetric. Taking the next step would require abandoning these assumptions to model the spiral arms. The Poisson equation can handle that, but it becomes dicey because the arms rotate with some pattern speed (generally unknown) and may grow or dissolve or reform on some unknown timescale. The potential at any given point is time variable even in equilibrium, so we need not just a numerical solution but a live numerical simulation to keep track of it. That can be done, but it has to be done on a case by case basis, and the answer will depend somewhat on additional assumptions that have to be introduced to run the simulation, like specifying a dark matter halo.

One can generalize further to consider the full 3D potential, e.g., to allow for asymmetry in the z-direction as well as in azimuth. One can further imagine non-equilibrium processes, such as an external perturbations. There is good evidence that the Milky Way suffers both of these effects, the passage of the Large Magellanic Cloud being one obvious and apparently large perturbation. So we are in the awkward position that the Gaia data now oblige us to consider the entire run of possible effects through non-equilibrium processes in a mass distribution that is not completely symmetric in any of the three spatial dimensions, but for the main mass component we are stuck with the inadequate approximation of an exponential disk.

Geometry appears to play a crucial role in the approach of Deur to the acceleration discrepancy problem. The essential claim is that the discrepancy correlates with flattening, with highly flattened systems like spirals evincing the classic discrepancy while spherical systems like E0 galaxies showing none. Big if true!

A useful plot appears on slide 44:

*Some measure of the discrepancy as a function of apparent ellipticity.*

This is the one example shown that goes into the plot of many determinations of the slope a on the following slide. It being the only one, it is the only thing I have to evaluate without chasing down every other case. Looking at this, I am not inclined to do so.

At first it looks persuasive: the best fit slope is clear. There is no reason why the discrepancy should depend on the projected ellipticity of a triaxial 3D blob of stars, so this must be telling us something important. I’d be on board with that if it were true, but I’ve seen too many non-correlations masquerading as correlations to believe this one. The fitted slope is strongly influenced by the one point at large ellipticity; absent that, a slope of zero works fine. Mostly what I see here is a lot of scatter, which is normal in extragalactic astronomy. Since there are only a few points at high and low ellipticity, we don’t know what would happen if we went out and got more data. But I bet that what would happen is that the high ellipticity points would wind up looking like those in the middle: a big blob of scatter, with no significant correlation.

I’d kinda like to be wrong about this one, so I won’t even get into the theory side, which I find sorta compelling but ultimately unpersuasive. Why are gravitons confined to a disk? What happens way far out? Surely the flatness of the disk at tens of kpc is not dictating the flatness at 1000 kpc.

Surely.

Decision Trees & Philosophical Blunders

Given recent developments in the long-running hunt for dark matter and the difficulty interpreting what this means, it seems like a good juncture to re-up* this:

The history of science is a decision tree. Vertices appear where we must take one or another branching. Sometimes, we take the wrong road for the right reasons.

A good example is the geocentric vs. heliocentric cosmology. The ancient Greeks knew that in many ways it made more sense for the earth to revolve around the sun than vice-versa. Yet they were very clever. Ptolemy and others tested for the signature of the earth’s orbit in the seasonal wobbling in the positions of stars, or parallax. If the earth is moving around the sun, nearby stars should appear to move on the sky as the earth moves from one side of the sun to the other. Try blinking back and forth between your left and right eyes to see this effect, noting how nearby objects appear to move relative to distant ones.

Problem is, Ptolemy did not find the parallax. Quite reasonably, he inferred that the earth stayed put. We know now that this was the wrong branch to choose, but it persisted as the standard world view for many centuries. It turns out that even the nearest stars are so distant that their angular parallax is tiny (the angle of parallax is inversely proportional to distance). Precision sufficient for measuring the parallax was not achieved until the 19th century, by which time astronomers were already convinced it must happen.

Ptolemy was probably aware of this possibility, though it must have seemed quite unreasonable to conjecture at that time that the stars could be so very remote. The fact was that parallax was not observed. Either the earth did not move, or the stars were ridiculously distant. Which sounds more reasonable to you?

So, science took the wrong branch. Once this happened, sociology kicked in. Generation after generation of intelligent scholars confirmed the lack of parallax until the opposing branch seemed so unlikely that it became heretical to even discuss. It is very hard to reverse back up the decision tree and re-assess what seems to be such a firm conclusion. It took the Copernican revolution to return to that ancient decision branch and try the other one.

Cosmology today faces a similar need to take a few steps back on the decision tree. The problem now is the issue of the mass discrepancy, typically attributed to dark matter. When it first became apparent that things didn’t add up when one applied the usual Law of Gravity to the observed dynamics of galaxies, there was a choice. Either lots of matter is present which happens to be dark, or the Law of Gravity has to be amended. Which sounds more reasonable to you?

Having traveled down the road dictated by the Dark Matter decision branch, cosmologists find themselves trapped in a web of circular logic entirely analogous to the famous Ptolemaic epicycles. Not many of them realize it yet, much less admit that this is what is going on. But if you take a few steps back up the decision branch, you find a few attempts to alter the equations of gravity. Most of these failed almost immediately, encouraging cosmologists down the dark matter path just as Ptolemy wisely chose a geocentric cosmology. However, one of these theories is not only consistent with the data, it actually predicts many important new results. This theory is known as MOND (MOdified Newtonian Dynamics). It was introduced in 1983 by Moti Milgrom of the Weizmann Institute in Israel.

MOND accurately describes the effective force law in galaxies based only on the observed stars and gas. What this means is unclear, but it clearly means something! It is conceivable that dark and luminous matter somehow interact to mimic the behavior stipulated by MOND. This is not expected, and requires a lot of epicyclic thinking to arrange. The more straightforward interpretation is that MOND is correct, and we took the wrong branch of the decision tree back in the ’70s.

MOND has dire implications for much modern cosmological thought which has developed symbiotically with dark matter. As yet, no one has succeeded in writing down a theory which encompasses both MOND and General Relativity. This leaves open many questions in cosmology that were thought to be solved, such as the expansion history of the universe. There is nothing a scientist hates to do more than unlearn what was thought to be well established. It is this sociological phenomenon that makes it so difficult to climb back up the decision tree to the faulty branching.

Once one returns and takes the correct branch, the way forward is not necessarily obvious. The host of questions which had been assigned seemingly reasonable explanations along the faulty branch must be addressed anew. And there will always be those incapable of surrendering the old world view irrespective of the evidence.

In my opinion, the new successes of MOND can not occur by accident. They are a strong sign that we are barking up the wrong tree with dark matter. A grander theory encompassing both MOND and General Relativity must exist, even if no one has as yet been clever enough to figure it out (few have tried).

These all combine to make life as a cosmologist interesting. Sometimes it is exciting. Often it is frustrating. Most of the time, “interesting” takes on the meaning implied by the old Chinese curse:

MAY YOU LIVE IN INTERESTING TIMES

Like it or not, we do.

*I wrote this in 2000. I leave it to the reader to decide how much progress has been made since then.

Why’d it have to be MOND?

I want to take another step back in perspective from the last post to say a few words about what the radial acceleration relation (RAR) means and what it doesn’t mean. Here it is again:

The Radial Acceleration Relation over many decades. The grey region is forbidden – there cannot be less acceleration than caused by the observed baryons. The entire region above the diagonal line (yellow) is accessible to dark matter models as the sum of baryons and however much dark matter the model prescribes. MOND is the blue line.

This information was not available when the dark matter paradigm was developed. We observed excess motion, like flat rotation curves, and inferred the existence of extra mass. That was perfectly reasonable given the information available at the time. It is not now: we need to reassess as we learn more.

There is a clear organization to the data at both high and low acceleration. No objective observer with a well-developed physical intuition would look at this and think “dark matter.” The observed behavior does not follow from one force law plus some arbitrary amount of invisible mass. That could do literally anything in the yellow region above, and beyond the bounds of the plot, both upwards and to the left. Indeed, there is no obvious reason why the data don’t fall all over the place. One of the lingering, niggling concerns is the 5:1 ratio of dark matter:baryons – why is it in the same ballpark, when it could be pretty much anything? Why should the data organize in terms of acceleration? There is no reason for dark matter to do this.

Plausible dark matter models have been predicted to do a variety of things – things other than what we observe. The problem for dark matter is that real objects only occupy a tiny line through the vast region available to them in the plot above. This is a fine-tuning problem: why do the data reside only where they do when they could be all over the place? I recognized this as a problem for dark matter before I became aware^$ of MOND. That it turns out that the data follow the line uniquely predicted^* by MOND is just chef’s kiss: there is a fine-tuning problem for dark matter because MOND is the effective force law.

The argument against dark matter is that the data could reside anywhere in the yellow region above, but don’t. The argument against MOND is that a small portion of the data fall a little off the blue line. Arguing that such objects, be they clusters of galaxies or particular individual galaxies, falsify MOND while ignoring the fine-tuning problem faced by dark matter is a case of refusing to see the forest for a few outlying trees.^%

So to return to the question posed in the title of this post, I don’t know why it had to be MOND. That’s just what we observe. Pretending dark matter does the same thing is a false presumption.

^$I’d heard of MOND only vaguely, and, like most other scientists in the field, had paid it no mind until it reared its ugly head in my own data.

*I talk about MOND here because I believe in giving credit where credit is due. MOND predicted this; no other theory did so. Dark matter theories did not predict this. My dark matter-based galaxy formation theory did not predict this. Other dark matter-based galaxy formation theories (including simulations) continue to fail to explain this. Other hypotheses of modified gravity also did not predict what is observed. Who⁺ ordered this?

*Modified Dynamics. Very dangerous. You go first.*

Many people in the field hate MOND, often with an irrational intensity that has the texture of religion. It’s not as if I woke up one morning and decided to like MOND – sometimes I wish I had never heard of it – but disliking a theory doesn’t make it wrong, and ignoring it doesn’t make it go away. MOND and only MOND predicted the observed RAR a priori. So far, MOND and only MOND provides a satisfactory explanation of thereof. We might not like it, but there it is in the data. We’re not going to progress until we get over our fear of MOND and cope with it. Imagining that it will somehow fall out of simulations with just the right baryonic feedback prescription is a form of magical thinking, not science.

⁺Milgrom. Milgrom ordered this.

^%I expect many cosmologists would argue the same in reverse for the cosmic microwave background (CMB) and other cosmological constraints. I have some sympathy for this. The fit to the power spectrum of the CMB seems too good to be an accident, and it points to the same parameters as other constraints. Well, mostly – the Hubble tension might be a clue that things could unravel, as if they haven’t already. The situation is not symmetric – where MOND predicted what we observe a priori with a minimum of assumptions, LCDM is an amalgam of one free parameter after another after another: dark matter and dark energy are, after all, auxiliary hypotheses we invented to save FLRW cosmology. When they don’t suffice, we invent more. Feedback is single word that represents a whole Pandora’s box of extra degrees of freedom, and we can invent crazier things as needed. The results is a Frankenstein’s monster of a cosmology that we all agree is the same entity, but when we examine it closely the pieces don’t fit, and one cosmologist’s LCDM is not really the same as that of the next. They just seem to agree because they use the same words to mean somewhat different things. Simply agreeing that there has to be non-baryonic dark matter has not helped us conjure up detections of the dark matter particles in the laboratory, or given us the clairvoyance to explain^# what MOND predicted a prioi. So rather than agree that dark matter must exist because cosmology works so well, I think the appearance of working well is a chimera of many moving parts. Rather, cosmology, as we currently understand it, works if and only if non-baryonic dark matter exists in the right amount. That requires a laboratory detection to confirm.

^#I have a disturbing lack of faith that a satisfactory explanation can be found.

The Radial Acceleration Relation starting from high accelerations

In the previous post, we discussed how lensing data extend the Radial Acceleration Relation (RAR) seen in galaxy kinematics to very low accelerations. Let’s zoom out now, and look at things at higher accelerations and from a historical perspective.

This all started with Kepler’s Laws of Planetary Motion, which are explained by Newton’s Universal Gravitation – the inverse square law g_bar = GM/r² is exactly what is needed to explain the observed centripetal acceleration, g_obs = V²/r. It also explains the surface gravity of the Earth. Indeed, it was the famous falling apple that is reputed to have given Newton the epiphany that it was the same force that made the apple fall to the ground that made the Moon circle the Earth that made the planets revolve around the sun.

The inverse square law holds over more than six decades of observed acceleration in the solar system, from the one gee we feel here on the surface of the Earth to the outskirts patrolled by Neptune.

Planetary motion in the radial acceleration plane. The dotted line is Newton’s inverse square law of universal gravity.*

The inverse square force law is what it takes to make the planetary data line up. A different force law would give a line with a different slope in this plot. No force law at all would give chaos, with planets all over the place in this plot, if, say, the solar system were run by a series of deferents and epicycles as envisioned for Ptolemaic cosmologies. In such a system, there is no reason to expect the organization seen above. It would require considerable contrivance to make it so.

Newtonian gravity and General Relativity are exquisitely well-tested in the solar system. There are also some very precise tests at higher accelerations that GR passes with flying colors. The story to lower accelerations is another matter. The most remote solar system probes we’ve launched are the Voyger and Pioneer missions. These probe down to ~10^-6 m/s/s; below that is uncharted territory.

The RAR extended from high solar system accelerations to much low accelerations typical of galaxies – not the change in scale. Some early rotation curves (of NGC 55, NGC 801, NGC 2403, NGC 2841, & UGC 2885) are shown as lines. These probed an entirely new regime of acceleration. The departure of these lines from the dotted line are the flat rotation curves indicating the acceleration discrepancy/need for dark matter. This discrepancy was clear by the end of the 1970s, but the amplitude of the discrepancy then was modest.

Galaxies (and extragalactic data in general) probe an acceleration range that is unprecedented from the perspective of solar system tests. General Relativity has passed so many precise tests that the usual presumption is that is applies at all scales. But it is an assumption that it applies to scales where it hasn’t been tested. Galaxies and cosmology pose such a test. That we need to invoke dark matter to save the phenomenon would be interpreted as a failure if we had set out to test the theory rather than assume it applied.

It was clear from flat rotation curves that something extra was needed. However, when we invented the dark matter paradigm, it was not clear that the data were organized in terms of acceleration. As the data continued to improve, it became clear that the vast majority of galaxies adhered to a single, apparently universal⁺ radial acceleration relation. What had been a hint of systematic behavior in early data became clean and clear. The data did not exhibit the scatter that as was expected from a sum of a baryonic disk and a non-baryonic dark matter halo – there is no reason that these two distinct components should sum to the single effective force law that is observed.

The RAR with modern data for both early (red triangles) and late (cyan circles) morphological types. The blue line is the prediction of MOND: there is a transition at an acceleration scale to a force law that is universal but no longer inverse-square.

The observed force-law happened to already have a name: MOND. If it had been something else, then we could have claimed to discover something new. But instead we were obliged to admit that the unexpected thing we had found had in fact been predicted by Milgrom.

This predictive power now extends to much lower accelerations. Again, only MOND got this prediction right in advance.

*The RAR as above, extended by weak gravitational lensing observations. These follow the prediction of MOND as far as they are credible.*

The data could have done many different things here. It could have continued along the dotted line, in which case we’d have need for no dark matter or modified gravity. It could have scattered all over the place – this is the natural expectation of dark matter theories, as there is no reason to expect the gravitational potential of the dominant dark matter halo to be dictated by the distribution of baryons. One expects that not to happen. Yet the data evince the exceptional degree of organization seen above.

It requires considerable contrivance to explain the RAR with dark matter. No viable explanation yet exists, despite many unconvincing claims to this effect. I have worked more on trying to explain this in terms of dark matter than I have on MOND, and all I can tell you is what doesn’t work. Every explanation I’ve seen so far is a special case of a model I had previously considered and rejected as obviously unworkable. At this point, I don’t see how dark matter can ever plausibly do what the data require.

I worry that dark matter has become an epicycle theory. We’re sure it is right, so whatever we observe, no matter how awkward or unexpected, must be what it does. But what if it is wrong, and it does not exist? How do we ever disabuse ourselves of the notion that there is invisible mass once we’ve convinced ourselves that there has to be?

Of course, MOND has its own problems. Clusters of galaxies are systems^$ for which it persistently fails to explain the amplitude of the observed acceleration discrepancy. So let’s add those to the plot as well:

*As above, with clusters of galaxies added (x: Sanders 2003; +: Li et al. 2023).*

So: do clusters violate the RAR, or follow it? I’d say yes and yes – the offset, thought modest in amplitude in this depiction, is statistically significant. But there is also a similar scaling with acceleration, only the amplitude is off. The former makes no sense in MOND; the latter makes no sense in terms of dark matter which did not predict a RAR at all.

Clusters are the strongest evidence against MOND. Just being evidence against MOND doesn’t automatically make it evidence in favor of dark matter. I often pose myself the question: which theory requires me to disbelieve the least amount of data? When I first came to the problem, I was shocked to find that the answer was clearly MOND. Since then, it has gone back and forth, but rather than a clear answer emerging, what has happened is more a divergence of different lines of evidence: that which favors the standard cosmology is incommensurate with that which favors MOND. This leads to considerable cognitive dissonance.

One way to cope with cognitive dissonance is to engage with a problem from different perspectives. If I put on a MOND hat, I worry about the offset seen above for clusters. If I put on a dark matter hat, I worry about the same kind of offset for every system that is not a rich cluster of galaxies. Most critics of MOND seem unconcerned about this problem for dark matter, so how much should a critic of dark matter worry about it in MOND?

*For the hyper-pedantic: the eccentricity of each orbit causes the exact location of each planet in the first plot to oscillate up and down along the dotted line. The extent of this oscillation is smaller than the size of each symbol with the exception of Mercury, which has a relatively high eccentricity (but nowhere near enough to reach Venus).

⁺There are a few exceptions, of course – there are always exceptions in astronomy. The issue is whether these are physically meaningful, or the result of systematic uncertainties or non-equilibrium processes. The claimed discrepancies range from dubious to unconvincing to obviously wrong.

^$I’ve heard some people criticize MOND because the centroid of the lensing signal does not peak around the gas in the Bullet cluster. This assumes that the gas represents the majority of the baryons. We know the is not the case, and that there is some missing mass in clusters. Whatever it is, it is clearly more centrally concentrated than the gas, so we don’t expect the lensing signal to peak where the gas is. All the Bullet cluster teaches us is that whatever this stuff is, it is collisionless. So this particular complaint is a logical fallacy of the a red herring and/or straw man variety born of not understanding MOND well enough to criticize it accurately. Why bother to do that when you come to the problem already sure that MOND is wrong? I understand this line of thought extraordinarily well, because that’s the attitude I started with, and I’ve seen it repeated by many colleagues. The difference is that I bothered to educate myself.

A personal note – I will be on vacation next week, so won’t be quick to respond to comments.

The Radial Acceleration Relation to very low accelerations

Flat rotation curves and the Baryonic Tully-Fisher relation (BTFR) both follow from the Radial Acceleration Relation (RAR). In Mistele et al. (2024b) we emphasize the exciting aspects of the former; these follow from the RAR in the Mistele et al. (2024a). It is worth understanding the connection.

First, the basic result:

**Figure 2** from Mistele et al. (2024a). The RAR from weak lensing data (yellow diamonds) is shown together with the binned kinematic RAR from Lelli et al. (2017, gray circles). The solid line is Newtonian gravity without dark matter (g_obs = g_bar). The shaded region at g_bar < 10⁻¹³ m/s² indicates where the isolation criterion may be less reliable according to the estimate by Brouwer et al. (2021). Our results suggest that late type galaxies (LTGs) may be sufficiently isolated down to g_bar ≈ 10⁻¹⁴ m/s². We shade this region where LTGs may still be reliable in a lighter color.

The RAR of weak lensing extends the RAR from kinematics to much lower accelerations. How low we can trust we’ll come back to, but certainly to g_bar ≈ 10⁻¹³ m/s² and probably to g_bar ≈ 10⁻¹⁴ m/s². For the mass of the typical galaxy in the KiDS sample, this corresponds to a radius of 300 kpc and 1.1 Mpc, respectively. Hence our claim that the effective gravitational potentials of isolated galaxies are consistent with rotation curves that remain flat indefinitely far out: a million light years at least, and perhaps a million parsecs.

Note that the kinematic and lensing data overlap at log(g_bar) = -11.5. These independent methods give the same result. Moreover, this region corresponds to the regions in galaxies where atomic gas rather than stars dominates the baryonic mass budget, which minimizes the systematic uncertainty due to stellar population mass estimates. The lensing results still depend on these, but they agree with the gas-dominated portion of the RAR, and merge smoothly into the star-dominated portion of the kinematic data when the same stellar pop models are used for both. To wit: the agreement is really good.

A flat rotation curve projects into the log(g_obs)-log(g_bar) plane as a line with slope 1/2. The data adhere closely to this slope, so I knew as soon as I saw the lensing RAR that the implied rotation curves remained flat indefinitely. How far, in radius, depends on galaxy mass, since for a point mass (a good approximation at radii beyond 100 kpc), g_bar = GM_bar/R². We can split the lensing data into different mass bins, for which the RAR looks like

**Figure 5** from Mistele et al. (2024a). The RAR implied by weak lensing for four baryonic mass bins. The dashed line has the slope a flat rotation curve has when projected into the acceleration plane. That different masses follow the same RAR implies the Baryonic Tully-Fisher relation.

Most dark matter models that I’ve seen or constructed myself predict a mass-dependent shift in the RAR, if they predict a RAR at all (many do not). We see no such shift. But the math is such that the flat rotation speed implied by the slope 1/2 RAR varies with mass in such a way that they only fall on the same RAR, as observed, if there is a Baryonic Tully-Fisher relation with slope 4. So I knew from examination of the above figure that the BTFR was sure to follow, but that’s because I’ve been working on these things for a long time. It isn’t necessarily obvious to everyone else, so it was worth explicitly showing.

Our result differs from the original of Brouwer et al. in two subtle but important ways. The first is that we use stellar population models that are the same as we use for the kinematic data. This self-consistency is important to the continuity of the data. We (especially Jim Schombert) took a deep dive into this, and the models used by Brouwer et al. are consistent with ours for late type (spiral) galaxies (LTGs). However, ours are somewhat heavier^ for early type galaxies (ETGs). That’s part of the reason that they find an offset in the RAR between morphological types and we do not.

Another important difference is the strictness of the isolation criterion. We are trying to ascertain the average gravitational potential of isolated galaxies, those with no big neighbors to compound the lensing signal. Brouwer et al. required that there be no galaxies more than a tenth of the luminosity of the primary within 3 Mpc. That seems reasonable, but we explored lots of variations on both aspects of that limit. It seems to be fine for LTGs, but insufficient for ETGs. That in itself is not surprising, as ETGs are known to be more strongly clustered than LTGs, so it is harder to find isolated examples.

To illustrate this, we show the deviation of the data from the kinematic RAR fit as a function of the isolation criterion:

**Figure 4** from Mistele et al. (2024a). Top: the difference between the radial accelerations inferred from weak lensing and the RAR fitting function, measured in sigmas, as a function of how isolated the lenses are, quantified by Ri_sol. We separately show the result for ETGs (red) and LTGs (blue) as well as for small (triangles with dashed lines) and large accelerations (diamonds with solid lines). LTGs are mostly unaffected by making the isolation criterion stricter. In contrast, ETGs do depend on R_isol, but tend towards with increasing R_isol. Middle and bottom: the accelerations behind these sigma values for R_isol = 3 Mpc/h₇₀ and R_isol = 4 Mpc/h₇₀.

The top panel shows that LTGs do not deviate from the RAR as we vary the radius of isolation. In contrast, ETGs deviate a lot for small R_isol. This is what Brouwer et al. found, and it would be a problem for MOND if LTGs and ETGs genuinely formed different sequences: it would be as if they were both obeying their own version of a similar but distinct MOND-like force law rather than a single universal force law.

That said, the ETGs converge towards the same RAR as the LTGs as we make the isolation criterion more strict. The distinction between ETGs and LTGs that appears to be clear for the R_isol = 3 Mpc/h₇₀ used by Brouwer et al. (middle panel) goes away when R_isol = 4 Mpc/h₇₀ (bottom panel). The random errors grow because fewer galaxies⁺ meet the stricter criterion, but this seems a price well worth paying to be rid of the systematic variation seen in the top panel. This also dictates how far out we can trust the data, which show no clear deviation from the RAR until below the limit g_bar = 10⁻¹⁴ m/s².

Regardless of the underlying theory, the data paint a consistent picture. This can be summarized by three empirical laws of galactic rotation:

Rotation curves become approximately* flat at large radii and remain so indefinitely.
The amplitude of the flat rotation speed scales with the baryonic mass as M_bar ~ V_f⁴ (the BTFR).
The observed centripetal acceleration follows from that predicted by the baryons (the RAR).

These are the galactic analogs of Kepler’s Laws for planetary motion. There is no theory in these statements; their just a description of what the data do. That’s useful, as they provide an empirical touchstone that has to be satisfactorily explained by any theory for it to be considered viable. No dark matter-based theory currently does that.

^The difference is well within the expected variance for stellar population models. We can reproduce their numbers if we treat ETGs as if they were just red LTGs. I don’t know if that’s what they did, but it ain’t right.

⁺For the record, the isolated fraction of the entire sample is 16%: most galaxies have neighbors. As a function of mass, the isolation criterion leaves a fraction of 8%, 18%, 30%, and 42% of LTG lenses and 9%, 14%, and 22% of ETG lenses, respectively, in each mass bin. The fraction of isolated LTGs is generally higher than ETGs, as expected. There is also a trend for the isolation fraction to increase as mass decreases. In part this is real; more luminous galaxies are more clustered. It may also be that it is easier for objects that exceed 10% of the primary mass (really luminosity) to evade detection as the primaries get fainter so 10% of that is harder to reach.

*Some people take “flat” way too seriously in this context. While it is often true that rotation curves look pretty darn flat over an extended radial range, I say approximately flat because we never measure, and can never measure, exactly a slope of dV/dR = 0.000. As a practical matter, we have adopted a variation of < 5% from point to point as a working definition. The scatter in Tully-Fisher naturally goes up if one adopts a weaker criterion; what one gets for the scatter is all about data quality.

Tully-Fisher from gravitational lensing

Last time, we discussed the remarkable result that gravitational lensing extends the original remarkable result of flat rotation curves much farther out, as far as the data credibly probe. This corroborates and extends the result of Brouwer et al. They did a thorough job, but one thing they did not consider was Tully-Fisher. If the circular speed inferred from gravitational lensing remains constant, does this flat velocity fall on the same Tully-Fisher relation that is seen in kinematic data?

We set out to answer this question. Along the way, we did three new things: 1. Dr. Mistele derived an improved method for doing the lensing analysis, extending the radial range over which the data were credible. 2. He explored the criteria by which galaxies were judged to be isolated, finding a morphological type dependence on how far out one had to exclude. 3. We reanalyzed the stellar masses of the KiDS sample to be consistent with those we used when analyzing the kinematic data of SPARC galaxies. The first two are connected, as how far out we can trust the data depends on how well we can define a clean sample of isolated galaxies. The third resolved an apparent offset between early type galaxies (ETGs, aka ellipticals) and late type galaxies (LTGs, aka spirals) seen by Brouwer et al. That appears to be an artifact of stellar population modeling, as I suspected when I first discussed their result. We don’t need to do any fitting of the mass-to-light ratio; the the apparent offset between types disappears when we use use the same population models for both kinematic and lensing data.

I could write a lot about each of these, but most of it is the stuff of technical details that would be dull to many people. If you’re into that sort of thing, go and read the long science paper which is where such details reside. Here I just want to describe the Tully-Fisher result. Spoiler alert: it is the same as that from kinematics.

First off, I’m talking strictly about the Baryonic Tully-Fisher relation: the scaling between baryonic mass and the flat rotation speed. To address this, we bin the lensing data by mass. The mass of each bin is well defined by the average of the many thousands of galaxies within the bin. By far the dominant uncertainty is the systematic in stellar mass caused by stellar population modeling. We went through this with a fine tooth comb, and I’m confident we have an internally self-consistent result. That doesn’t preclude it being wrong in an absolute sense – such is the nature of astronomy – but we can at least make a straight comparison between kinematic and lensing data using the same best-effort stellar mass estimates.

For the velocity, we estimate the average effective rotation curve for each mass bin from the lensing data. We also split the data into morphological types to look for differences. The statistics go down when one divvies up the data like this, so the uncertainties go up, but there are enough KiDS galaxies to define four mass bins. Here are their inferred rotation curves:

**Figure 1** from Mistele et al. (2024). Circular velocities implied by weak lensing for four baryonic mass bins (most to least massive from the top row to the bottom) for the whole sample (left column), for LTGs (middle column), and for ETGs (right column). The lowest ETG mass bin is not shown because it contains too few lenses. Instead we show results for lenses with spectroscopic redshifts from GAMA, without splitting by mass or type due to the small sample size (gray and white symbols). For comparison, we also show results for KiDS without splitting by mass or type (small yellow symbols). Open symbols at small radii indicate where lenses are not yet effective point masses. Light-colored symbols (not-outlined) at large radii indicate data points that may still be reliable but where the isolation criterion is less certain. The error bars show the statistical errors. Horizontal lines and the corresponding shaded regions indicate the inferred V_flat values and uncertainties that we use for the BTFR. The extent of the horizontal lines indicates the radial range we consider when calculating V_flat.

Note that the average over all KiDS data shown in the lower right bin is the data shown in the press release image in the previous post, but the x-axis is logarithmic here. The GAMA data in that bin provide an important cross-check, as these galaxies have spectroscopic redshifts. They give the same answer as the larger KiDS sample, which relies on photometric redshifts. We need the larger sample to consider finer bins in mass, which is the rest of the plot.

Another thing to note here is that all the data in all the bins are consistent with remaining flat. There are some hints of a turn down at very large radii, particularly for LTGs in the second and third row, but these are not statistically significant, and only happen where the data start to become untrustworthy. Where exactly that happens is a judgement call.

Let’s take a closer look, with a comparison to radio data:

**Figure 2** from Mistele et al. (2024). The circular velocities from weak lensing (circles) compared with those from gas kinematics (diamonds). The individual galaxies illustrated here have among the most extended 21 cm rotation curves in their mass bins; the lensing data continue to much larger radii still. The error bars show the statistical error, while the gray band indicates the systematic uncertainty in the radial accelerations. Symbol colors are as in Figure 1. Open symbols at large radii indicate where lenses are not sufficiently isolated. The solid green lines indicate the circular velocities of NFW halos and baryonic point masses appropriate for each mass bin. Green crosses indicate each NFW halo’s virial radius. The light green band adds a qualitative estimate of a two-halo term contribution to the NFW halo, which may become important at large radii in case our isolation criterion is imperfect there.

Again we see that the lensing data, averaged over many galaxies, extend much further out than the rotation curve of any one individual. The x-axis is again logarithmic, so the lensing data go way further out. They trace to 1 Mpc, which is crazy far beyond the observed ends of the most extended individual galaxies. A more conservative limit is the 300 kpc estimated by Brouwer et al. Surely we can go further than that, but how much further remains a judgement call.

What should we expect? The green lines show the rotation curve we’d expect for galaxy in an NFW halo with parameters specified by the stellar mass-halo mass relation of Kravtsov et al. (2018). Not all such relations agree well with kinematic data; this is the case that agrees most closely. We have intentionally cherry-picked the relation that makes LCDM look best. And it does look good up to a point, for example in the top two mass bins out to the virial radius of the halo (tick marks). Beyond that, not so much, and not at all for the two lower mass bins. The data extend far enough out that we should see the predicted decline. We do not.

The green line only represents the expected halo of the primary galaxy. When one gets so far out, one has to worry about all the other stuff out there. We’ve selected galaxies to be isolated, so there isn’t much that is luminous. But we can only exclude down to some sensitivity limit, there might be lots of tiny dwarf galaxies whose mass adds up and starts to affect the result. And of course there can be completely invisible dark matter. The green band attempts to account for this extra stuff in the so-called 2-halo term. This is hard to do, but we’ve made our best estimate based on the LCDM power spectrum. I’m sure the 2-halo term can be adjusted, but the shape is wrong. It will take some fine-tuning to get an effectively flat rotation curve out of the 1-halo+2 halo terms. They don’t naturally do that.

Something that is easy to do is define a flat value of the rotation speed. That’s just the average over the lensing data. We exclude the points at R < 50 kpc, as the assumption of a spherical mass that we make in the lensing analysis isn’t really valid at those comparatively small scales. We tried averaging over a bunch of different ranges, all of which gave pretty much the same answer. For illustration, we show two cases: a conservative one that only uses the data at R < 300 kpc, and another that goes out to 1 Mpc. Having measured V_flat over these ranges, we can plot Tully-Fisher:

**Figure 3** from Mistele et al. (2024). The baryonic Tully–Fisher relation implied by weak lensing for the entire sample (yellow symbols, left column) and for ETGs and LTGs separately (red and blue symbols, right column). The V_flat values are weighted averages of the V_c values shown in Figure 1 for 50 kpc < R < 300 kpc (first row) and 50 kpc < R < 1000 kpc (second row). Vertical error bars represent a 0.1 dex systematic uncertainty on M_*/L. For comparison, we also show the best fit to the kinematic data from Lelli et al. (2019; solid gray line) and the corresponding binned kinematic data (white diamonds).

Lo and behold, we find the same Baryonic Tully-Fisher relation from lensing data as we find with kinematics. This does not surprise me, but it didn’t have to be true. It shouldn’t be true in LCDM: if we can measure out to the virial radius, we should see some indication of a decline in velocity. We have and we don’t.

We also see no statistically significant separation between ETGs and LTGs. This is important, as a theory like MOND predicts that there should be no morphology dependence: only the baryonic mass matters. Brouwer et al. did see an indication of such a split, but it was small compared to the uncertainty in stellar population models. We don’t see it when we use our own stellar mass estimates. This is particularly true in the more conservative (300 kpc) case. There is a hint of a segregation when we average out to 1000 kpc, but the statistics say this isn’t significant. Since the lowest mass bin is most affected, I suspect this is a hint that the isolation criterion is failing first for the smallest galaxies. That makes sense, as the sensitivity limit on interlopers makes the lowest mass bin most susceptible to having its signal inappropriately boosted. It also makes sense that ETGs would be affected first, as ETGs are known to be more clustered than LTGs. It is really hard to define an isolated sample of ETGs, as discussed at length by Mistele et al.

The lensing data corroborate previous kinematic work. Rotation curves are flat. The amplitude of the flat rotation speed correlates with baryonic mass as M_b ∝ V_f⁴. The radial acceleration relation extends to very low accelerations. These are all predictions of MOND. Moreover they are unique predictions: predictions made a priori by MOND and only by MOND. Dark matter models so far provide no satisfactory explanation*.

That hasn’t prevented people from overlooking these basic facts in order to get to the apparent if statistically meaningless difference between ETGs and LTGs. Nevermind the successes! The slight offset between ETGs and LTGs falsify MOND! Seriously: other scientists have already made this argument to me while completely eliding the successes of MOND. It’s a case of refusing to see the forest for a tree that’s a little away from the others.

I think I said something about how this would happen when I first wrote about Brouwer et al‘s lensing result. Ah yes, here it is:

MOND predicted this behavior well in advance of the observation, so one would have to bend over backwards, rub one’s belly, and simultaneously punch oneself in the face to portray this as anything short of a fantastic success of MOND.

I say that because I’m sure people will line up to punch themselves in the face in exactly this fashion.

And so it has come to pass. Sometimes human behavior is as predictable as galaxy dynamics.

*There are many claims to explain limited portions of these results, but none are satisfactory. There is no LCDM model that matches the entire dynamic range of the radial acceleration relation. See, for example, Fig. 5 of Brouwer et al. (reproduced below), which shows the MICE and BAHAMAS simulations. Neither extend into the regime that is well-constrained by kinematic data; there is no reason to think they would successfully do so and good reason to think otherwise. MICE comes nowhere close to this regime and has no baryonic physics that would allow it do even address this question. BAHAMAS comes close but appears to turn away from the kinematic data before it gets there. We’ve built our own LCDM models; they don’t work either. We can make them come close, but only over a limited dynamic range, not over the full span of the data. It isn’t good enough to only explain a limited range of the data. One has to explain the full range, and the only theory that does that so far is MOND.

Fig. 5 from Brouwer et al. showing the radial acceleration relation inferred from the MICE (red band) and BAHAMAS (orange band) simulations. Not also that in our assessment of stellar masses, the lower acceleration points translate a bit to the right.

Rotation curves: still flat after a million light-years

That rotation curves become flat at large radii is one of the most famous results in extragalactic astronomy. This had been established by Vera Rubin and her collaborators by the late 1970s. There were a few earlier anecdotal cases to this effect, but these seemed like mild curiosities until Rubin showed that the same thing was true over and over again for a hundred spiral galaxies. Flat rotation curves took on the air of a de facto natural law and precipitated the modern dark matter paradigm.

Optical and radio data

Rotation curves shouldn’t be flat. If what we saw was what we got, the rotation curve would reach a peak within the light distribution and decline further out. Perhaps an illustration is in order:

The rotation curve (data points, left) of NGC 6946 (right). The red line shows the expected rotation curve for the detected normal matter, which includes both the stars (yellow, from 2MASS) and atomic gas (blue, from THINGS). This provides a good description of the inner rotation curve but falls short further out. The excess observed rotation leads to the need for dark matter or MOND. Also noted is the extent of the rotation curve measured optically to the effective edge of the stars (Daigle et al. 2006; Epinat et al. 2008) and that measured with radio interferometric observations of the gas (Boomsma et al. 2008).

An obvious question is how far out rotation curves remain flat. In the rotation curves traced with optical observations by Rubin et al., the discrepancy was clear but modest – typically a factor of two in mass. It was possible to imagine that the mass-to-light ratios of stars increased with radius in a systematic way, bending the red line above to match the data out to the edge of the stars. This seemed unlikely, but neither did it seem like a huge ask.

Once one gets to the edge of the stellar distribution, most of the mass has been encompassed, and the rotation curve really should start to decline. Increasing the mass-to-light ratio of the stars ceases to be an option once we run out of stars*. Fortunately, the atomic gas typically extends to larger radii, so provides a tracer further out. Albert Bosma pursued this until there were again enough examples to establish that yes, flat rotation curves were the rule. They extended much further out, well beyond where the mass of the observed stars and gas could explain the data.

How much further out? It depends on the galaxy. A convenient metric is the scale length of the disk, which is a measure of the extent of the light distribution. Some galaxies are bigger than others. The peak of the contribution of the stars to the rotation curve occurs around 2.2 scale lengths. The rotation curve of NGC 6946 extends to about 7 scale lengths, far enough to make the discrepancy clear. For a long time, the record holder was NGC 2403, with a rotation curve that remains flat for 20 scale lengths.

Twenty scale lengths is a long way out. It is observations like this that demanded dark matter halos that are much larger than the galaxies they contain. They also posed a puzzle, since we were still nowhere near finding the edge of the mass distribution. Rotation curves seemed to persist in being flat indefinitely.

Results from gravitational lensing

Weak gravitational lensing provides a statistical technique to probe the gravitational potential of galaxies. Brouwer et al. did pioneering work with data from the KiDS survey, and found that the radial acceleration relation extended to much lower accelerations than probed by the types of kinematic data discussed above. That implies that rotation curves remain flat way far out. How far?

Postdoc Tobias Mistele worked out an elegant technique to improve the analysis of lensing data. His analysis corroborates the findings of Brouwer et al. It also provides the opportunity to push further out.

Weak gravitational lensing is a subtle effect – so subtle that one must coadd thousands of galaxies to get a signal. Beyond that, the limiting effect on the result is how isolated the galaxies are. Lensing is sensitive to all mass; if you go far enough out you start to run into other galaxies whose mass contributes to the signal. So one key is to identify isolated galaxies, and restrict the sample to them. KiDS is large enough to do this. Indeed, Mistele was able to show that while neighbors⁺ were a definite concern for elliptical galaxies, they were much less of a problem for spirals. Consequently, we can trace the implied rotation curve way far out.

How far out? In a new paper, Mistele shows that rotation curves continue way far out. Way way way far out. I mean, damn.

The average rotation curve of isolated galaxies (blue points) inferred from KiDS gravitational lensing data. This remains flat well beyond a million light-years with no end in sight. The width of the figure is the distance between the Milky Way and Andromeda. For comparison, the rotation curve of a single galaxy, UGC 6614, is shown in red. An image of the galaxy is shown to scale centered at the origin. UGC 6614 was selected for this illustration because it has a comparable rotation speed to the KiDS average and because it is one of the largest galaxies known: the red points are already a very extended rotation curve. Image credit: Mistele, Lelli, & McGaugh 2024.

Optical rotation curves typically extend to the edge of the stellar disk. That’s about 8 kpc in the example of NGC 6946 given above. Radio observations of the atomic gas of that galaxy extend to 17 kpc. That fits within the first two tick marks on the graph with the lensing rotation curve.

UGC 6614 is a massive galaxy with a very extended low surface brightness disk. Its rotation curve is traced by radio data to over 60 kpc. It is one of the most extended individual rotation curves known. The statistical lensing data push this out by a factor of ten, and more, with no end in sight. The flat rotation curves found by Rubin and Bosma and everyone else appear to persist indefinitely.

So what does it mean? First, flat rotation curves really are a law of nature, in the same sense of Kepler’s laws of planetary motion. Galaxies don’t obey those planetary rules, they have their own set of rules. This is what nature does.

In terms of dark matter halos, the extent of isolated galaxy rotation curves is surprisingly large. Just as we come to the edge of the stellar disk, then the gas disk, we should eventually hit the edge of the dark matter halo. In principle we can imagine this to be arbitrarily large, but in practice there are other galaxies in the universe so this cannot go one forever.

In the context of LCDM, we now have a pretty good idea of how extended halos should be from abundance matching. A galaxy of the mass of UGC 6614 should live in a halo with a virial radius of about 300 kpc or less. There is some uncertainty in this, of course, but we really should have hit the edge with the lensing data. There should be some sign of it, but we see none.

One complication is the so-called 2-halo term. In addition to the primary dark matter halo that hosts a galaxy, when you get very far out, you run into other halos. Isolated galaxies are selected to avoid this to the extent possible, but eventually there will be some extra mass that causes extra lensing signal that would cause an overestimate of the rotation speed. I’ll forgo a detailed discussion of this for now (see Mistele et al. if you’re eager), but the bottom line is that it would require some unnatural fine-tuning for the 1+2 halo terms to add up to such flat rotation curves. There ought to be a perceptible feature in the transition from the primary halo to the surrounding environment. We don’t see that.

In the context of MOND, a flat rotation curve that persists indefinitely is completely natural. That’s what an isolated galaxy should do. Even in MOND there should be an environmental effect: the mass of everything else in the universe should impose an external field effect that eventually limits the extent of the rotation curve. How this transition happens depends on the density of other galaxies; by selecting isolated galaxies this effect is put off as much as possible. Hopefully it will be detected as the data improve from projects like Euclid.

The primary prediction of MOND is an indefinitely extended rotation curve; the external field effect is a subtle detail. Yet again, that is what we see: MOND gets it right without really trying, and in a way that makes little sense in terms of dark matter. Sometimes I wish MOND had never been invented so we could claim to have discovered something profoundly new, or at least discuss the empirical result without concern that the data would get confused with the theory. MOND predictions keep being corroborated, yet the community persists in ignoring its implications, even in terms of dark matter. It’s gotta be telling us something.

We have a press release about this result, so perhaps you will see it kicking around your news feed.

*We could, of course, invoke dark stars, but that’s just an invisible horse of a different color.

⁺There is a well known correlation between morphology and density such that elliptical galaxies tend to live in the densest environments. This means that they are more likely to have neighbors that interfere with the lensing measurement, so finding that identifying isolated ellipticals with a clean lensing signal is more challenging that finding isolated spirals comes as no surprise. Isolated ellipticals do exist so it is possible, but one has to be very restrictive with the sample.

Updated WIMP Exclusion Diagram

This is an update to a post from a few years ago, which itself was an update to a webpage I wrote in 2008, with many updates in between. At that time, the goalposts for detecting WIMPs had already moved repeatedly. I felt some need then to write down a brief synopsis of the history of a beloved hypothesis (including by myself) that had obviously failed as the goalposts were in motion again. That was sixteen years ago.

It is important to remember where we started from, which is now ancient history lost in the myths of time to most who are now working in the field. Indeed, when I search for mention of the WIMP miracle, the theoretical argument that launched a thousand underground detection experiments, little comes up: this essential element of the field has been memory-holed after its failure. I suppose that’s to be expected, as the same thing happened with the decay of the B₀ meson: once heralded as the “golden test” for supersymmetry, it simply stopped getting mentioned after it didn’t work out.

The original expectation for WIMPs was a particle of mass around 100 GeV/c² with an interaction cross-section of about 10^-39 cm². While I remember this, it is getting rare to find this statement, so let me quote a particle physicist:

“The most appealing possibility – a weak scale dark matter particle interacting with matter via Z-boson exchange – leads to the cross section of order 10^-39 cm²“

14 April 2011 Resonaances

To translate a little bit, the Z-boson is a carrier of the weak nuclear force (as photons are for electromagnetism), so this envisions an otherwise normal interaction that involves a new particle, the WIMP. The weak force is, well, weak, so the interaction probability is small, as quantified by the tiny cross section of 10^-39 cm². That makes such interactions rare, but particle physicists are talented at detecting such phenomena. It helps to have a lot of target material in your detector in a place that is well-shielded from background interference, hence all the giant underground WIMP experiments. Consequently, to continue the quote above,

“the cross section of order 10^-39 cm² … was excluded back in the 80s by the first round of dark matter experiments.”

And so the goalposts were set in motion. There were many steps along this path, so I’ll highlight only one, circa 2008. To complete the quote from Resonaances,

“There exists another natural possibility for WIMP dark matter: a particle interacting via Higgs boson exchange. This would lead to the cross section in the 10^-42 – 10^-46 cm² ballpark (depending on the Higgs mass and on the coupling of dark matter to the Higgs).”

So the interaction via the Z-boson had been excluded, but one can have other interactions, this one via the Higgs (which had not quite yet been detected: discovery was in 2012; the Resonaances quote is from 2011. Since then, the Higgs might be said to be “too normal” to make room for any of this.) The possibility of Higgs exchange leads to the blue-green predicted region of Trotta et al. (2008) in the exclusion diagram shown below. If one looks for such plots in the literature, one finds a natural tendency for their upper limits to migrate downwards along with the limits they portray. I thought it might be instructive to update the plot to show the full range of progress:

The interaction cross section as a function of WIMP mass. The original expectation of 10^-39 cm² is at top. Gray areas are regions that were experimentally excluded by 2008 (before the blue-green prediction) and by 2022, which is the most recent update as of this writing. The most sensitive limit is 10^-47 cm², eight orders of magnitude below the original prediction.

I call out the 2008 threshold because we had a conference here at CWRU in 2009 (while I was at the University of Maryland) at which the Trotta et al. prediction was presented. I had already become skeptical of the moving goalposts, so I wondered how much of the probability density was in the tail to low cross-section. A low-likelihood tail seems a lot more probable once the head is lopped off! I made this point at the time, and asked how important the tail was. The answer was about 2% or the probability. The speaker went on to express the usual overconfidence that WIMPs would be detected in the more likely region (marked by an X in the blue region with the handy arrow pointing to it).

The experimentalists have done a fabulous job in increasing the sensitivity of their experiments so that they can see to ever lower interaction cross section. Had WIMPs existed as predicted initially, or subsequently, they would have been detected by now. These experiments have succeeded in failing quite brilliantly. I had long before shown that the astronomical data did not add up for any flavor of dark matter. Maybe WIMPs don’t live in this universe?

While we’d be happy to detect dark matter anywhere in parameter space, the WIMP does have sweet spots: first 10^-39 cm² then 10^-44 cm². Now that those are gone, what’s next? From the particle physics perspective, I’ve heard it said that the next logical expectation for the cross-section is around 10^-48 cm². This apparently follows from “two-loop corrections.” I have only a vague idea of what that means, but in my practical experience it translates to “a difficult-to-compute effect so exotic that it likely has no bearing on reality, except maybe in the sixth place of decimals.”

More generally, this continual moving of the cross section goalpost is what I meant back in 2008 by the scientific version of the express elevator to hell. It just keeps going down, and can do so forever. I keep warning my colleagues about these things, and they keep not heeding the warnings. Being a scientific Cassandra is getting old.

The problem with pushing detection limits to still lower cross-sections like 10^-48 cm² is that the universe is indeed full of weakly interacting particles with at least a little bit of mass: neutrinos. These are not as massive as WIMPs, and should not be confused with them: neutrinos are Standard Model particles that are known to exist and to have a very small mass (< 1 eV) while WIMPs are expected to be hundreds of GeV and require entirely new physics beyond the Standard Model. I shouldn’t need to say this, but WIMPs and neutrinos are very different beasts. However, they do both have mass and interact weakly, so I’ve noticed that some of the more rabid advocates of dark matter mix these two in order to claim that we know weakly interacting dark matter exists. That much is technically true, but in technical parlance it is also some bold bullshit. Hmmm, actually, I think it is worse than ordinary bullshit. It is willful scientific disinformation that intentionally sews confusion by conflating the unconfirmed existence of WIMPs with the known existence of neutrinos in order to lend an air of certainty to a failed hypothesis.

*WIMP experimental limits (via Hamdan 2021) with the expected neutrino background in orange. Once this sensitivity is reached, any WIMP signal becomes obscured by the neutrino background.*

Meanwhile, experimental progress proceeds apace. The coming generation of WIMP detectors should be sensitive to the solar and atmospheric neutrino background. That is astrophysically interesting, as it can probe nuclear reactions in the sun and, in principle, those in every supernova that have ever exploded. This has bugger all to do with dark matter. However, since that’s what people are looking for, what they built these detectors to find, and they’re completely convinced dark matter exists, and a Nobel prize awaits whoever detects it first, I expect that the first neutrino detections will be misinterpreted as WIMP detections. There will be much arguing between groups, claims and counterclaims, and after a few years it will be recognized that these coming detections are neutrinos not WIMPs. First there will probably be many over-hyped claims that mislead the public into thinking dark matter has been detected.

But there I go being a scientific Cassandra again.

Aurora Over Ohio

And pretty much everywhere else

First, a pretty picture:

*Aurora over my house in Cleveland Heights, Ohio, USA, the evening of Friday, May 10.*

The sun is nearing the peak of its eleven year sunspot cycle. That means lots of sunspots and associated activity. Solar prominences, visible to the naked eye during the eclipse, are bands of plasma entrained in the magnetic field connecting pairs of sunspots. Once in a while, these break out in solar flares. Lately, the sun has produced a series of X-class flares (the largest type) with associated coronal mass ejections (CMEs) that send huge blobs of plasma hurtling out into space.

Space is big, so CMEs usually don’t impact Earth. But sometimes they do, and they have a number of effects. The plasma impinges on Earth’s magnetic field, which funnels charged particles towards the poles. When these high-speed particles hit atoms and molecules high up in the atmosphere, they transfer energy that excites quantum states. The relaxation of these states leads to the emission of the light we perceive as aurora.

I heard there was a possibility of aurora being visible at our latitude Friday night. I didn’t expect much – the northern lights are notoriously fickle, and usually only appear much further to the north – hence the name. It has to be fully dark to see them at all, so I walked out at about 10 PM and looked up. Not much. Maybe some thin clouds. Only that’s a strangely shaped cloud. And, as my eyes adjusted, one shone red, the other green. The northern lights had come to me.

*Aurora wax and wane with the plasma breeze; this is the view a few minutes later.*

NOAA has a good explainer. The greens and reds are from excited atomic oxygen, at different altitudes owing to the different lifetimes of the associated quantum states. Atoms can be de-excited as well as excited by collisions, so we only get emission lines when the density* of surrounding atoms is low enough that light gets emitted before collisional de-excitation. That means the green comes from oxygen over 100 km up; the red comes from even higher, more like 300 km. There is barely any atmosphere at all at these altitudes.

Similar views were reported all over the planet. Aurora are usually restricted to very northerly latitudes, hence the moniker northern lights. A big CME floods the Earth’s magnetic field (and can distort it), leading to the appearance of aurora at lower latitudes. I had only seen them once before, in Ann Arbor in 1989, and then only as a ghostly grey wisp on the northern horizon. It takes a big event to produce colorful aurora overhead in Ohio.

*The blues and purples are from molecular nitrogen, the predominant component of our atmosphere.*

It wasn’t just Ohio! Bright aurora were reported at all longitudes – I’ve seen lots of great pictures from Europe – to remarkably southerly latitudes, extending even to Florida and the Caribbean. This southerly reach is remarkable, but not uniform. One could see aurora overhead at the Apache Point Observatory in New Mexico, but they only appeared on the northern horizon at Kitt Peak in Arizona. Even that is an incredibly rare event!

Flares and CMEs have effects besides auroras – so much so that there is an entire field of space weather. The weather in space is particularly relevant to satellite operations, as big flares can blind or even damage sensors on satellites. It also affects their orbits. The radiation is also a hazard to would-be space travelers: you don’t want to get caught in a CME during a multi-month trip to Mars.

The sun is especially active right now. Usually rare, there have been multiple X-class flares. The space weather page sounds a bit exhausted, with stories like Region 3664 Remains Relentless and Another X-flare from Another Region! It seems a little like the weathermen they send to report on major storms by standing out in them for the entertainment of the audience. Only don’t try this in space.

Solar activity has not yet reached its peak, so hopefully we’ll get more opportunities to see aurora from the convenience of home.

*I have to admit to being impressed at how well a 3 second exposure with an iPhone does at capturing the aurora.*

Will wonders never cease? An eclipse in April, aurora in May; one could get spoiled: what will June bring? Hopefully not a Carrington level geomagnetic storm, which would make great aurora but the associated fluctuations in magnetic field would drive currents in electrical lines that could have an adverse effect on the power grid. Potentially very adverse. OK, maybe not that adverse, but I do appreciate having electricity.

*The atomic lines we see in aurora are from neutral oxygen, [O I] in the parlance of spectroscopy. This is strange to me, as I’ve worked on nebular spectra, where prominent emission lines are due to singly and doubly ionized oxygen – [O II] and [O III] in the parlance of spectroscopy. These lines thrive only in the extremely low density, practical vacuum of space (densities of tens or maybe hundreds of atoms per cubic centimeter), and were unknown in the laboratory when first observed astronomically. For a time, it was thought that, like helium in the sun, they represented a new element, nebulium – the stuff of which nebula were made.

The MHONGOOSE survey of atomic gas in and around galaxies

I have been spending a lot of time lately writing up a formal paper on high redshift galaxies, so haven’t had much time to write here. The paper is a lot more involved than I told you so, but yeah, I did. Repeatedly. I do have a start on a post on self-interacting dark matter that I hope eventually to get back to. Today, I want to give a quick note about the MHONGOOSE survey. But first, a non-commercial interruption.

Triton Station joins Rogue Scholar

In internet news, Triton Station has joined Rogue Scholar. The blog itself hasn’t moved; Rogue Scholar is a community of science blogs. It provides some important capabilities, including full-text search, long-term archiving, DOIs, and metadata. The DOIs (Digital Object Identifiers) were of particular interest to me, as they have become the standard for identifying unique articles in regular academic journals now that these have mostly (entirely?) gone on-line. I had not envisioned ever citing this blog in a refereed journal, but a DOI makes it possible to do so. Any scientists who find a post useful are welcome to make use of this feature. I’m inclined to follow the example of JCAP and make the format volume, page be yearmonth, date (YYMM, DD), which comes out to Triton Station (2022), 2201, 03 in the standard astronomy journal format. I do not anticipate continuing to publish in the twenty second century, so no need for YYYYMM, Y2K experience notwithstanding.

For everyone interested in science, Rogue Scholar is a great place to find new blogs.

MHONGOOSE

In science news, the MHONGOOSE collaboration has released its big survey summary paper. Many survey science papers are in the pipeline. Congratulations to all involved, especially PI Erwin de Blok.

Erwin was an early collaborator of mine who played a pivotal role in measuring the atomic gas properties of low surface brightness galaxies, establishing the cusp-core problem, and that low surface brightness galaxies are dark matter dominated (or at least evince large mass discrepancies, as predicted by MOND). He has done a lot more since then, among them playing a leading role in the large VLA survey of nearby galaxies, THINGS. In astronomy we’re always looking forward to the next big survey – its a big universe; there’s always more out there. So after THINGS he conceived and began work on MHONGOOSE. It has been a long road tied to the construction of the MeerKAT array of radio telescopes – a major endeavor on the road to the ambitious Square Kilometer Array.

I was involved in the early phases of the MHONGOOSE project, helping to select the sample of target galaxies (it is really important to cover the full dynamic range of galaxy properties, dwarf to giant) and define the aspirational target sensitivity. HI observations often taper off below a column density of 10²⁰ hydrogen atoms per cm² (about 1 solar mass per square parsec). With work, one can get down to a few times 10¹⁹ cm^-2. We want to go much deeper to see how much farther out the atomic gas extends. It was already known to go further out than the stars, but how far? Is there a hard edge, or just a continuous fall off?

We also hope to detect new dwarf galaxies that are low surface brightness in HI. There could, in theory, be zillions of such things lurking in all the dark matter subhalos that are predicted to exist around big galaxies. Irrespective of theory, are there HI gas-rich galaxies that are entirely devoid of stars? Do such things exist? People have been looking for them a long time, and there are now many examples of galaxies that are well over 95% gas, but there always seem to be at least a few stars associated with them. Is this always true? If we have cases that are 98, 99% gas, why not 100%? Do galaxies with gas always manage to turn at least a little of it into stars? They do have a Hubble time to work on it, so it is also a question why there is so much gas still around in these cases.

And… a lot of other things, but I don’t want to be here all day. So just a few quick highlights from the main survey paper. First, the obligatory sensitivity diagram. This shows how deep the survey reaches (lower column density) as a function of resolution (beam size). You want to see deeply and you want to resolve what you see, so ideally both of these numbers would be small. MHONGOOSE undercuts existing surveys, and is unlikely to be bettered until the full SKA comes on-line, which is still a long way off.

*Sensitivity versus resolution in HI surveys.*

And here are a couple of individual galaxy observations:

Optical images and the HI moment zero, one, and two maps. The moment zero map of the intensity of 21 cm radiation tells us where the atomic gas is, and how much of it there is. The moment one map is the velocity field from which we can construct a rotation curve. The second moment measures the velocity dispersion of the gas.

These are beautiful data. The spiral arms appear in the HI as well as in starlight, and continue in HI to larger radii. The outer edge of the HI disk is pretty hard; there doesn’t seem to be a lot of extra gas at low column densities extending indefinitely into the great beyond. I’m particular struck by the velocity dispersion of NGC 1566 tracking the spiral structure: this means the spiral arms have mass, and any stirring caused by star formation is localized to the spirals where much of the star formation goes on. That’s natural, but the surroundings seem relatively unperturbed: feedback is happening locally, but not globally. The velocity field of NGC 5068 has a big twist in the zero velocity contour (the thick line dividing the red receding side from the blue approaching side); this is a signature of non-circular motion, probably caused in this case by the visible bar. These are two-dimensional examples of Renzo’s rule (Sancisi’s Law), in which features in the visible mass distribution correspond to features in the kinematics.

I’ll end with a quick peak at the environments around some MHONGOOSE target galaxies:

*Fields where additional galaxies (in blue) are present around the central target*.

This is nifty on many levels. First, some (presumptively satellite) dwarf galaxies are detected. That in itself is a treat to me: once upon a time, Renzo Sancisi asked me to smooth the bejeepers out of the LSB galaxy data cubes to look for satellites. After much work, we found nada. Nothing. Zilch. It turns out that LSB galaxies are among the most isolated galaxy types in the universe. So that we detect some things here is gratifying, even in targets that are not LSBs.

Second, there are not a lot of new detections. The halos of big galaxies are not swimming in heretofore unseen swarms of low column density gas clouds. There can always be more at sensitivities yet unreached, but the data sure don’t encourage that perspective. MHONGOOSE is sensitive to very low mass gas clouds. The exact limit is distance-dependent, but a million solar masses of atomic gas should be readily visible. That’s a tiny amount by extragalactic standards, about one globular cluster’s worth of material. There’s just not a lot there.

Disappointing as the absence of zillions of new detections may be discovery-wise, it does teach us some important lessons. Empirically, galaxies look like island universes in gas as well as stars. There may be a few outlying galaxies, but they are not embedded in an obvious cosmic network of ephemeral cold gas. Nor are there thousands of unseen satellites/subhalos suddenly becoming visible – at least not in atomic gas. Theorists can of course imagine other things, but we observers can only measure one thing at a time, as instrumentation and telescope availability allows. This is a big step forward.

Triton Station

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