The halo mass function

I haven’t written much here of late. This is mostly because I have been busy, but also because I have been actively refraining from venting about some of the sillier things being said in the scientific literature. I went into science to get away from the human proclivity for what is nowadays called “fake news,” but we scientists are human too, and are not immune from the same self-deception one sees so frequently exercised in other venues.

So let’s talk about something positive. Current grad student Pengfei Li recently published a paper on the halo mass function. What is that and why should we care?

One of the fundamental predictions of the current cosmological paradigm, ΛCDM, is that dark matter clumps into halos. Cosmological parameters are known with sufficient precision that we have a very good idea of how many of these halos there ought to be. Their number per unit volume as a function of mass (so many big halos, so many more small halos) is called the halo mass function.

An important test of the paradigm is thus to measure the halo mass function. Does the predicted number match the observed number? This is hard to do, since dark matter halos are invisible! So how do we go about it?

Galaxies are thought to form within dark matter halos. Indeed, that’s kinda the whole point of the ΛCDM galaxy formation paradigm. So by counting galaxies, we should be able to count dark matter halos. Counting galaxies was an obvious task long before we thought there was dark matter, so this should be straightforward: all one needs is the measured galaxy luminosity function – the number density of galaxies as a function of how bright they are, or equivalently, how many stars they are made of (their stellar mass). Unfortunately, this goes tragically wrong.

Galaxy stellar mass function and the predicted halo mass function — Fig. 5 from the review by Bullock & Boylan-Kolchin. The number density of objects is shown as a function of their mass. Colored points are galaxies. The solid line is the predicted number of dark matter halos. The dotted line is what one would expect for galaxies if all the normal matter associated with each dark matter halo turned into stars.

This figure shows a comparison of the observed stellar mass function of galaxies and the predicted halo mass function. It is from a recent review, but it illustrates a problem that goes back as long as I can remember. We extragalactic astronomers spent all of the ’90s obsessing over this problem. [I briefly thought that I had solved this problem, but I was wrong.] The observed luminosity function is nearly flat while the predicted halo mass function is steep. Consequently, there should be lots and lots of faint galaxies for every bright one, but instead there are relatively few. This discrepancy becomes progressively more severe to lower masses, with the predicted number of halos being off by a factor of many thousands for the faintest galaxies. The problem is most severe in the Local Group, where the faintest dwarf galaxies are known. Locally it is called the missing satellite problem, but this is just a special case of a more general problem that pervades the entire universe.

Indeed, the small number of low mass objects is just one part of the problem. There are also too few galaxies at large masses. Even where the observed and predicted numbers come closest, around the scale of the Milky Way, they still miss by a large factor (this being a log-log plot, even small offsets are substantial). If we had assigned “explain the observed galaxy luminosity function” as a homework problem and the students had returned as an answer a line that had the wrong shape at both ends and at no point intersected the data, we would flunk them. This is, in effect, what theorists have been doing for the past thirty years. Rather than entertain the obvious interpretation that the theory is wrong, they offer more elaborate interpretations.

Faced with the choice between changing one’s mind and proving that there is no need to do so, almost everybody gets busy on the proof.
J. K. Galbraith

Theorists persist because this is what CDM predicts, with or without Λ, and we need cold dark matter for independent reasons. If we are unwilling to contemplate that ΛCDM might be wrong, then we are obliged to pound the square peg into the round hole, and bend the halo mass function into the observed luminosity function. This transformation is believed to take place as a result of a variety of complex feedback effects, all of which are real and few of which are likely to have the physical effects that are required to solve this problem. That’s way beyond the scope of this post; all we need to know here is that this is the “physics” behind the transformation that leads to what is currently called Abundance Matching.

Abundance matching boils down to drawing horizontal lines in the above figure, thus matching galaxies with dark matter halos with equal number density (abundance). So, just reading off the graph, a galaxy of stellar mass M_* = 10⁸ M_☉ resides in a dark matter halo of 10¹¹ M_☉, one like the Milky Way with M_* = 5 x 10¹⁰ M_☉ resides in a 10¹² M_☉ halo, and a giant galaxy with M_* = 10¹² M_☉ is the “central” galaxy of a cluster of galaxies with a halo mass of several 10¹⁴ M_☉. And so on. In effect, we abandon the obvious and long-held assumption that the mass in stars should be simply proportional to that in dark matter, and replace it with a rolling fudge factor that maps what we see to what we predict. The rolling fudge factor that follows from abundance matching is called the stellar mass–halo mass relation. Many of the discussions of feedback effects in the literature amount to a post hoc justification for this multiplication of forms of feedback.

This is a lengthy but insufficient introduction to a complicated subject. We wanted to get away from this, and test the halo mass function more directly. We do so by use of the velocity function rather than the stellar mass function.

The velocity function is the number density of galaxies as a function of how fast they rotate. It is less widely used than the luminosity function, because there is less data: one needs to measure the rotation speed, which is harder to obtain than the luminosity. Nevertheless, it has been done, as with this measurement from the HIPASS survey:

Galaxy velocity function — The number density of galaxies as a function of their rotation speed (Zwaan et al. 2010). The bottom panel shows the raw number of galaxies observed; the top panel shows the velocity function after correcting for the volume over which galaxies can be detected. Faint, slow rotators cannot be seen as far away as bright, fast rotators, so the latter are always over-represented in galaxy catalogs.

The idea here is that the flat rotation speed is the hallmark of a dark matter halo, providing a dynamical constraint on its mass. This should make for a cleaner measurement of the halo mass function. This turns out to be true, but it isn’t as clean as we’d like.

Those of you who are paying attention will note that the velocity function Martin Zwaan measured has the same basic morphology as the stellar mass function: approximately flat at low masses, with a steep cut off at high masses. This looks no more like the halo mass function than the galaxy luminosity function did. So how does this help?

To measure the velocity function, one has to use some readily obtained measure of the rotation speed like the line-width of the 21cm line. This, in itself, is not a very good measurement of the halo mass. So what Pengfei did was to fit dark matter halo models to galaxies of the SPARC sample for which we have good rotation curves. Thanks to the work of Federico Lelli, we also have an empirical relation between line-width and the flat rotation velocity. Together, these provide a connection between the line-width and halo mass:

Halo mass-line width relation — The relation Pengfei found between halo mass (M200) and line-width for the NFW (ΛCDM standard) halo model fit to rotation curves from the SPARC galaxy sample.

Once we have the mass-line width relation, we can assign a halo mass to every galaxy in the HIPASS survey and recompute the distribution function. But now we have not the velocity function, but the halo mass function. We’ve skipped the conversion of light to stellar mass to total mass and used the dynamics to skip straight to the halo mass function:

Empirical halo mass function — The halo mass function. The points are the data; these are well fit by a Schechter function (black line; this is commonly used for the galaxy luminosity function). The red line is the prediction of ΛCDM for dark matter halos.

The observed mass function agrees with the predicted one! Test successful! Well, mostly. Let’s think through the various aspects here.

First, the normalization is about right. It does not have the offset seen in the first figure. As it should not – we’ve gone straight to the halo mass in this exercise, and not used the luminosity as an intermediary proxy. So that is a genuine success. It didn’t have to work out this well, and would not do so in a very different cosmology (like SCDM).

Second, it breaks down at high mass. The data shows the usual Schechter cut-off at high mass, while the predicted number of dark matter halos continues as an unabated power law. This might be OK if high mass dark matter halos contain little neutral hydrogen. If this is the case, they will be invisible to HIPASS, the 21cm survey on which this is based. One expects this, to a certain extent: the most massive galaxies tend to be gas-poor ellipticals. That helps, but only by shifting the turn-down to slightly higher mass. It is still there, so the discrepancy is not entirely cured. At some point, we’re talking about large dark matter halos that are groups or even rich clusters of galaxies, not individual galaxies. Still, those have HI in them, so it is not like they’re invisible. Worse, examining detailed simulations that include feedback effects, there do seem to be more predicted high-mass halos that should have been detected than actually are. This is a potential missing gas-rich galaxy problem at the high mass end where galaxies are easy to detect. However, the simulations currently available to us do not provide the information we need to clearly make this determination. They don’t look right, so far as we can tell, but it isn’t clear enough to make a definitive statement.

Finally, the faint-end slope is about right. That’s amazing. The problem we’ve struggled with for decades is that the observed slope is too flat. Here a steep slope just falls out. It agrees with the ΛCDM down to the lowest mass bin. If there is a missing satellite-type problem here, it is at lower masses than we probe.

That sounds great, and it is. But before we get too excited, I hope you noticed that the velocity function from the same survey is flat like the luminosity function. So why is the halo mass function steep?

When we fit rotation curves, we impose various priors. That’s statistics talk for a way of keeping parameters within reasonable bounds. For example, we have a pretty good idea of what the mass-to-light ratio of a stellar population should be. We can therefore impose as a prior that the fit return something within the bounds of reason.

One of the priors we imposed on the rotation curve fits was that they be consistent with the stellar mass-halo mass relation. Abundance matching is now part and parcel of ΛCDM, so it made sense to apply it as a prior. The total mass of a dark matter halo is an entirely notional quantity; rotation curves (and other tracers) pretty much never extend far enough to measure this. So abundance matching is great for imposing sense on a parameter that is otherwise ill-constrained. In this case, it means that what is driving the slope of the halo mass function is a prior that builds-in the right slope. That’s not wrong, but neither is it an independent test. So while the observationally constrained halo mass function is consistent with the predictions of ΛCDM; we have not corroborated the prediction with independent data. What we really need at low mass is some way to constrain the total mass of small galaxies out to much larger radii that currently available. That will keep us busy for some time to come.

A personal recollection of how we learned to stop worrying and love the Lambda

There is a tendency when teaching science to oversimplify its history for the sake of getting on with the science. How it came to be isn’t necessary to learn it. But to do science requires a proper understanding of the process by which it came to be.

The story taught to cosmology students seems to have become: we didn’t believe in the cosmological constant (Λ), then in 1998 the Type Ia supernovae (SN) monitoring campaigns detected accelerated expansion, then all of a sudden we did believe in Λ. The actual history was, of course, rather more involved – to the point where this oversimplification verges on disingenuous. There were many observational indications of Λ that were essential in paving the way.

Modern cosmology starts in the early 20th century with the recognition that the universe should be expanding or contracting – a theoretical inevitability of General Relativity that Einstein initially tried to dodge by inventing the cosmological constant – and is expanding in fact, as observationally established by Hubble and Slipher and many others since. The Big Bang was largely considered settled truth after the discovery of the existence of the cosmic microwave background (CMB) in 1964.

The CMB held a puzzle, as it quickly was shown to be too smooth. The early universe was both isotropic and homogeneous. Too homogeneous. We couldn’t detect the density variations that could grow into galaxies and other immense structures. Though such density variations are now well measured as temperature fluctuations that are statistically well described by the acoustic power spectrum, the starting point was that these fluctuations were a disappointing no-show. We should have been able to see them much sooner, unless something really weird was going on…

That something weird was non-baryonic cold dark matter (CDM). For structure to grow, it needed the helping hand of the gravity of some unseen substance. Normal matter matter did not suffice. The most elegant cosmology, the Einstein-de Sitter universe, had a mass density Ω_m= 1. But the measured abundances of the light elements were only consistent with the calculations of big bang nucleosynthesis if normal matter amounted to only 5% of Ω_m = 1. This, plus the need to grow structure, led to the weird but seemingly unavoidable inference that the universe must be full of invisible dark matter. This dark matter needed to be some slow moving, massive particle that does not interact with light nor reside within the menagerie of particles present in the Standard Model of Particle Physics.

CDM and early universe Inflation were established in the 1980s. Inflation gave a mechanism that drove the mass density to exactly one (elegant!), and CDM gave us hope for enough mass to get to that value. Together, they gave us the Standard CDM (SCDM) paradigm with Ω_m = 1.000 and H₀ = 50 km/s/Mpc.

elrondwasthere — I was there when SCDM failed.

It is hard to overstate the ferver with which the SCDM paradigm was believed. Inflation required that the mass density be exactly one; Ω_m < 1 was inconceivable. For an Einstein-de Sitter universe to be old enough to contain the oldest stars, the Hubble constant had to be the lower of the two (50 or 100) commonly discussed at that time. That meant that H₀ > 50 was Right Out. We didn’t even discuss Λ. Λ was Unmentionable. Unclean.

SCDM was Known, Khaleesi.

Λ had attained unmentionable status in part because of its origin as Einstein’s greatest blunder, and in part through its association with the debunked Steady State model. But serious mention of it creeps back into the literature by 1990. The first time I personally heard Λ mentioned as a serious scientific possibility was by Yoshii at a conference in 1993. Yoshii based his argument on a classic cosmological test, N(m) – the number of galaxies as a function of how faint they appeared. The deeper you look, the more you see, in a way that depends on the intrinsic luminosity of galaxies, and how they fill space. Look deep enough, and you begin to trace the geometry of the cosmos.

At this time, one of the serious problems confronting the field was the faint blue galaxies problem. There were so many faint galaxies on the sky, it was incredibly difficult to explain them all. Yoshii made a simple argument. To get so many galaxies, we needed a big volume. The only way to do that in the context of the Robertson-Walker metric that describes the geometry of the universe is if we have a large cosmological constant, Λ. He was arguing for ΛCDM five years before the SN results.

gold_hat_portrayed_by_alfonso_bedoya — Lambda? We don’t need no stinking Lambda!

Yoshii was shouted down. NO! Galaxies evolve! We don’t need no stinking Λ! In retrospect, Yoshii & Peterson (1995) looks like a good detection of Λ. Perhaps Yoshii & Peterson also deserve a Nobel prize?

Indeed, there were many hints that Λ (or at least low Ω_m) was needed, e.g., the baryon catastrophe in clusters, the power spectrum of IRAS galaxies, the early appearance of bound structures, the statistics of gravitational lenses, and so on. Certainly by the mid-90s it was clear that we were not going to make it to Ω_m = 1. Inflation was threatened – it requires Ω_m = 1 – or at least a flat geometry: Ω_m+Ω_Λ = 1.

SCDM was in crisis.

A very influential 1995 paper by Ostriker & Steinhardt did a lot to launch ΛCDM. I was impressed by the breadth of data Ostriker & Steinhardt discussed, all of which demanded low Ω_m. I thought the case for Λ was less compelling, as it hinged on the age problem in a way that might also have been solved, at that time, by simply having an open universe (low Ω_m with no Λ). This would ruin Inflation, but I wasn’t bothered by that. I expect they were. Regardless, they definitely made that case for ΛCDM three years before the supernovae results. Their arguments were accepted by almost everyone who was paying attention, including myself. I heard Ostriker give a talk around this time during which he was asked “what cosmology are you assuming?” to which he replied “the right one.” Called the “concordance” cosmology by Ostriker & Steinhardt, ΛCDM had already achieved the status of most-favored cosmology by the mid-90s.

omhannotated — A simplified version of the diagram of Ostriker & Steinhardt (1995) illustrating just a few of the constraints they discussed. Direct measurements of the expansion rate, mass density, and ages of the oldest stars excluded SCDM, instead converging on a narrow window – what we now call ΛCDM.

Ostriker & Steinhardt neglected to mention an important prediction of Λ: not only should the universe expand, but that expansion rate should accelerate! In 1995, that sounded completely absurd. People had looked for such an effect, and claimed not to see it. So I wrote a brief note pointing out the predicted acceleration of the expansion rate. I meant it in a bad way: how crazy would it be if the expansion of the universe was accelerating?! This was an obvious and inevitable consequence of ΛCDM that was largely being swept under the rug at that time.

I mean[t], surely we could live with Ω_m < 1 but no Λ. Can’t we all just get along? Not really, as it turned out. I remember Mike Turner pushing the SN people very hard in Aspen in 1997 to Admit Λ. He had an obvious bias: as an Inflationary cosmologist, he had spent the previous decade castigating observers for repeatedly finding Ω_m < 1. That’s too little mass, you fools! Inflation demands Ω_m = 1.000! Look harder!

By 1997, Turner had, like many cosmologists, finally wrapped his head around the fact that we weren’t going to find enough mass for Ω_m = 1. This was a huge problem for Inflation. The only possible solution, albeit an ugly one, was if Λ made up the difference. So there he was at Aspen, pressuring the people who observed supernova to Admit Λ. One, in particular, was Richard Ellis, a great and accomplished astronomer who had led the charge in shouting down Yoshii. They didn’t yet have enough data to Admit Λ. Not.Yet.

By 1998, there were many more high redshift SNIa. Enough to see Λ. This time, after the long series of results only partially described above, we were intellectually prepared to accept it – unlike in 1993. Had the SN experiments been conducted five years earlier, and obtained exactly the same result, they would not have been awarded the Nobel prize. They would instead have been dismissed as a trick of astrophysics: the universe evolves, metallicity was lower at earlier times, that made SN then different from now, they evolve and so cannot be used as standard candles. This sounds silly now, as we’ve figured out how to calibrate for intrinsic variations in the luminosities of Type Ia SN, but that is absolutely how we would have reacted in 1993, and no amount of improvements in the method would have convinced us. This is exactly what we did with faint galaxy counts: galaxies evolve; you can’t hope to understand that well enough to constrain cosmology. Do you ever hear them cited as evidence for Λ?

Great as the supernovae experiments to measure the metric genuinely were, they were not a discovery so much as a confirmation of what cosmologists had already decided to believe. There was no singular discovery that changed the way we all thought. There was a steady drip, drip, drip of results pointing towards Λ all through the ’90s – the age problem in which the oldest stars appeared to be older than the universe in which they reside, the early appearance of massive clusters and galaxies, the power spectrum of galaxies from redshift surveys that preceded Sloan, the statistics of gravitational lenses, and the repeated measurement of 1/4 < Ω_m < 1/3 in a large variety of independent ways – just to name a few. By the mid-90’s, SCDM was dead. We just refused to bury it until we could accept ΛCDM as a replacement. That was what the Type Ia SN results really provided: a fresh and dramatic reason to accept the accelerated expansion that we’d already come to terms with privately but had kept hidden in the closet.

Note that the acoustic power spectrum of temperature fluctuations in the cosmic microwave background (as opposed to the mere existence of the highly uniform CMB) plays no role in this history. That’s because temperature fluctuations hadn’t yet been measured beyond their rudimentary detection by COBE. COBE demonstrated that temperature fluctuations did indeed exist (finally!) as they must, but precious little beyond that. Eventually, after the settling of much dust, COBE was recognized as one of many reasons why Ω_m ≠ 1, but it was neither the most clear nor most convincing reason at that time. Now, in the 21st century, the acoustic power spectrum provides a great way to constrain what all the parameters of ΛCDM have to be, but it was a bit player in its development. The water there was carried by traditional observational cosmology using general purpose optical telescopes in a great variety of different ways, combined with a deep astrophysical understanding of how stars, galaxies, quasars and the whole menagerie of objects found in the sky work. All the vast knowledge incorporated in textbooks like those by Harrison, by Peebles, and by Peacock – knowledge that often seems to be lacking in scientists trained in the post-WMAP era.

Despite being a late arrival, the CMB power spectrum measured in 2000 by Boomerang and 2003 by WMAP did one important new thing to corroborate the ΛCDM picture. The supernovae data didn’t detect accelerated expansion so much as exclude the deceleration we had nominally expected. The data were also roughly consistent with a coasting universe (neither accelerating nor decelerating); the case for acceleration only became clear when we assumed that the geometry of the universe was flat (Ω_m+Ω_Λ = 1). That didn’t have to work out, so it was a great success of the paradigm when the location of the first peak of the power spectrum appeared in exactly the right place for a flat FLRW geometry.

The consistency of these data have given ΛCDM an air of invincibility among cosmologists. But a modern reconstruction of the Ostriker & Steinhardt diagram leaves zero room remaining – hence the tension between H₀ = 73 measured directly and H₀ = 67 from multiparameter CMB fits.

omhannotated_cmb — Constraints from the acoustic power spectrum of the CMB overplotted on the direct measurements from the plot above. Initially in great consistency with those measurement, the best fit CMB values have steadily wandered away from the most-favored region of parameter space that established ΛCDM in the first place. This is most apparent in the tension with H₀.

In cosmology, we are accustomed to having to find our way through apparently conflicting data. The difference between an expansion rate of 67 and 73 seems trivial given that the field was long riven – in living memory – by the dispute between 50 and 100. This gives rise to the expectation that the current difference is just a matter of some subtle systematic error somewhere. That may well be correct. But it is also conceivable that FLRW is inadequate to describe the universe, and we have been driven to the objectively bizarre parameters of ΛCDM because it happens to be the best approximation that can be obtained to what is really going on when we insist on approximating it with FLRW.

Though a logical possibility, that last sentence will likely drive many cosmologists to reach for their torches and pitchforks. Before killing the messenger, we should remember that we once endowed SCDM with the same absolute certainty we now attribute to ΛCDM. I was there, 3,000 internet years ago, when SCDM failed. There is nothing so sacred in ΛCDM that it can’t suffer the same fate, as has every single cosmology ever devised by humanity.

Today, we still lack definitive knowledge of either dark matter or dark energy. These add up to 95% of the mass-energy of the universe according to ΛCDM. These dark materials must exist.

It is Known, Khaleesi.

Hypothesis testing with gas rich galaxies

This Thanksgiving, I’d highlight something positive. Recently, Bob Sanders wrote a paper pointing out that gas rich galaxies are strong tests of MOND. The usual fit parameter, the stellar mass-to-light ratio, is effectively negligible when gas dominates. The MOND prediction follows straight from the gas distribution, for which there is no equivalent freedom. We understand the 21 cm spin-flip transition well enough to relate observed flux directly to gas mass.

In any human endeavor, there are inevitably unsung heroes who carry enormous amounts of water but seem to get no credit for it. Sanders is one of those heroes when it comes to the missing mass problem. He was there at the beginning, and has a valuable perspective on how we got to where we are. I highly recommend his books, The Dark Matter Problem: A Historical Perspective and Deconstructing Cosmology.

In bright spiral galaxies, stars are usually 80% or so of the mass, gas only 20% or less. But in many dwarf galaxies, the mass ratio is reversed. These are often low surface brightness and challenging to observe. But it is a worthwhile endeavor, as their rotation curve is predicted by MOND with extraordinarily little freedom.

Though gas rich galaxies do indeed provide an excellent test of MOND, nothing in astronomy is perfectly clean. The stellar mass-to-light ratio is an irreducible need-to-know parameter. We also need to know the distance to each galaxy, as we do not measure the gas mass directly, but rather the flux of the 21 cm line. The gas mass scales with flux and the square of the distance (see equation 7E7), so to get the gas mass right, we must first get the distance right. We also need to know the inclination of a galaxy as projected on the sky in order to get the rotation to which we’re fitting right, as the observed line of sight Doppler velocity is only sin(i) of the full, in-plane rotation speed. The 1/sin(i) correction becomes increasingly sensitive to errors as i approaches zero (face-on galaxies).

The mass-to-light ratio is a physical fit parameter that tells us something meaningful about the amount of stellar mass that produces the observed light. In contrast, for our purposes here, distance and inclination are “nuisance” parameters. These nuisance parameters can be, and generally are, measured independently from mass modeling. However, these measurements have their own uncertainties, so one has to be careful about taking these measured values as-is. One of the powerful aspects of Bayesian analysis is the ability to account for these uncertainties to allow for the distance to be a bit off the measured value, so long as it is not too far off, as quantified by the measurement uncertainties. This is what current graduate student Pengfei Li did in Li et al. (2018). The constraints on MOND are so strong in gas rich galaxies that often the nuisance parameters cannot be ignored, even when they’re well measured.

To illustrate what I’m talking about, let’s look at one famous example, DDO 154. This galaxy is over 90% gas. The stars (pictured above) just don’t matter much. If the distance and inclination are known, the MOND prediction for the rotation curve follows directly. Here is an example of a MOND fit from a recent paper:

DDO154_MOND_180805695 — The MOND fit to DDO 154 from Ren et al. (2018). The black points are the rotation curve data, the green line is the Newtonian expectation for the baryons, and the red line is their MOND fit.

This is terrible! The MOND fit – essentially a parameter-free prediction – misses all of the data. MOND is falsified. If one is inclined to hate MOND, as many seem to be, then one stops here. No need to think further.

If one is familiar with the ups and downs in the history of astronomy, one might not be so quick to dismiss it. Indeed, one might notice that the shape of the MOND prediction closely tracks the shape of the data. There’s just a little difference in scale. That’s kind of amazing for a theory that is wrong, especially when it is amplifying the green line to predict the red one: it needn’t have come anywhere close.

Here is the fit to the same galaxy using the same data [already] published in Li et al.:

DDO154_RAR_Li2018 — The MOND fit to DDO 154 from Li et al. (2018) using the same data as above, as tabulated in SPARC.

Now we have a good fit, using the same data! How can this be so?

I have not checked what Ren et al. did to obtain their MOND fits, but having done this exercise myself many times, I recognize the slight offset they find as a typical consequence of holding the nuisance parameters fixed. What if the measured distance is a little off?

Distance estimates to DDO 154 in the literature range from 3.02 Mpc to 6.17 Mpc. The formally most accurate distance measurement is 4.04 ± 0.08 Mpc. In the fit shown here, we obtained 3.87 ± 0.16 Mpc. The error bars on these distances overlap, so they are the same number, to measurement accuracy. These data do not falsify MOND. They demonstrate that it is sensitive enough to tell the difference between 3.8 and 4.1 Mpc.

One will never notice this from a dark matter fit. Ren et al. also make fits with self-interacting dark matter (SIDM). The nifty thing about SIDM is that it makes quasi-constant density cores in dark matter halos. Halos of this form are not predicted by “ordinary” cold dark matter (CDM), but often give better fits than either MOND of the NFW halos of dark matter-only CDM simulations. For this galaxy, Ren et al. obtain the following SIDM fit.

DDO154_SIDM_180805695 — The SIDM fit to DDO 154 from Ren et al.

This is a great fit. Goes right through the data. That makes it better, right?

Not necessarily. In addition to the mass-to-light ratio (and the nuisance parameters of distance and inclination), dark matter halo fits have [at least] two additional free parameters to describe the dark matter halo, such as its mass and core radius. These parameters are highly degenerate – one can obtain equally good fits for a range of mass-to-light ratios and core radii: one makes up for what the other misses. Parameter degeneracy of this sort is usually a sign that there is too much freedom in the model. In this case, the data are adequately described by one parameter (the MOND fit M*/L, not counting the nuisances in common), so using three (M*/L, Mhalo, Rcore) is just an exercise in fitting a French curve. There is ample freedom to fit the data. As a consequence, you’ll never notice that one of the nuisance parameters might be a tiny bit off.

In other words, you can fool a dark matter fit, but not MOND. Erwin de Blok and I demonstrated this 20 years ago. A common myth at that time was that “MOND is guaranteed to fit rotation curves.” This seemed patently absurd to me, given how it works: once you stipulate the distribution of baryons, the rotation curve follows from a simple formula. If the two don’t match, they don’t match. There is no guarantee that it’ll work. Instead, it can’t be forced.

As an illustration, Erwin and I tried to trick it. We took two galaxies that are identical in the Tully-Fisher plane (NGC 2403 and UGC 128) and swapped their mass distribution and rotation curve. These galaxies have the same total mass and the same flat velocity in the outer part of the rotation curve, but the detailed distribution of their baryons differs. If MOND can be fooled, this closely matched pair ought to do the trick. It does not.

NGC2403UGC128trickMOND — An attempt to fit MOND to a hybrid galaxy with the rotation curve of NGC 2403 and the baryon distribution of UGC 128. The mass-to-light ratio is driven to unphysical values (6 in solar units), but an acceptable fit is not obtained.

Our failure to trick MOND should not surprise anyone who bothers to look at the math involved. There is a one-to-one relation between the distribution of the baryons and the resulting rotation curve. If there is a mismatch between them, a fit cannot be obtained.

We also attempted to play this same trick on dark matter. The standard dark matter halo fitting function at the time was the pseudo-isothermal halo, which has a constant density core. It is very similar to the halos of SIDM and to the cored dark matter halos produced by baryonic feedback in some simulations. Indeed, that is the point of those efforts: they are trying to capture the success of cored dark matter halos in fitting rotation curve data.

NGC2403UGC128trickDM — A fit to the hybrid galaxy with a cored (pseudo-isothermal) dark matter halo. A satisfactory fit is readily obtained.

Dark matter halos with a quasi-constant density core do indeed provide good fits to rotation curves. Too good. They are easily fooled, because they have too many degrees of freedom. They will fit pretty much any plausible data that you throw at them. This is why the SIDM fit to DDO 154 failed to flag distance as a potential nuisance. It can’t. You could double (or halve) the distance and still find a good fit.

This is why parameter degeneracy is bad. You get lost in parameter space. Once lost there, it becomes impossible to distinguish between successful, physically meaningful fits and fitting epicycles.

Astronomical data are always subject to improvement. For example, the THINGS project obtained excellent data for a sample of nearby galaxies. I made MOND fits to all the THINGS (and other) data for the MOND review Famaey & McGaugh (2012). Here’s the residual diagram, which has been on my web page for many years:

rcresid_mondfits — Residuals of MOND fits from Famaey & McGaugh (2012).

These are, by and large, good fits. The residuals have a well defined peak centered on zero. DDO 154 was one of the THINGS galaxies; lets see what happens if we use those data.

DDO154mond_i66 — The rotation curve of DDO 154 from THINGS (points with error bars). The Newtonian expectation for stars is the green line; the gas is the blue line. The red line is the MOND prediction. Not that the gas greatly outweighs the stars beyond 1.5 kpc; the stellar mass-to-light ratio has extremely little leverage in this MOND fit.

The first thing one is likely to notice is that the THINGS data are much better resolved than the previous generation used above. The first thing I noticed was that THINGS had assumed a distance of 4.3 Mpc. This was prior to the measurement of 4.04, so lets just start over from there. That gives the MOND prediction shown above.

And it is a prediction. I haven’t adjusted any parameters yet. The mass-to-light ratio is set to the mean I expect for a star forming stellar population, 0.5 in solar units in the Sptizer 3.6 micron band. D=4.04 Mpc and i=66 as tabulated by THINGS. The result is pretty good considering that no parameters have been harmed in the making of this plot. Nevertheless, MOND overshoots a bit at large radii.

Constraining the inclinations for gas rich dwarf galaxies like DDO 154 is a bit of a nightmare. Literature values range from 20 to 70 degrees. Seriously. THINGS itself allows the inclination to vary with radius; 66 is just a typical value. Looking at the fit Pengfei obtained, i=61. Let’s try that.

DDO154mond_i61 — MOND fit to the THINGS data for DDO 154 with the inclination adjusted to the value found by Li et al. (2018).

The fit is now satisfactory. One tweak to the inclination, and we’re done. This tweak isn’t even a fit to these data; it was adopted from Pengfei’s fit to the above data. This tweak to the inclination is comfortably within any plausible assessment of the uncertainty in this quantity. The change in sin(i) corresponds to a mere 4% in velocity. I could probably do a tiny bit better with further adjustment – I have left both the distance and the mass-to-light ratio fixed – but that would be a meaningless exercise in statistical masturbation. The result just falls out: no muss, no fuss.

Hence the point Bob Sanders makes. Given the distribution of gas, the rotation curve follows. And it works, over and over and over, within the bounds of the uncertainties on the nuisance parameters.

One cannot do the same exercise with dark matter. It has ample ability to fit rotation curve data, once those are provided, but zero power to predict it. If all had been well with ΛCDM, the rotation curves of these galaxies would look like NFW halos. Or any number of other permutations that have been discussed over the years. In contrast, MOND makes one unique prediction (that was not at all anticipated in dark matter), and that’s what the data do. Out of the huge parameter space of plausible outcomes from the messy hierarchical formation of galaxies in ΛCDM, Nature picks the one that looks exactly like MOND.

star_trek_tv_spock_3_copy_-_h_2018 — This outcome is illogical.

It is a bad sign for a theory when it can only survive by mimicking its alternative. This is the case here: ΛCDM must imitate MOND. There are now many papers asserting that it can do just this, but none of those were written before the data were provided. Indeed, I consider it to be problematic that clever people can come with ways to imitate MOND with dark matter. What couldn’t it imitate? If the data had all looked like technicolor space donkeys, we could probably find a way to make that so as well.

Cosmologists will rush to say “microwave background!” I have some sympathy for that, because I do not know how to explain the microwave background in a MOND-like theory. At least I don’t pretend to, even if I had more predictive success there than their entire community. But that would be a much longer post.

For now, note that the situation is even worse for dark matter than I have so far made it sound. In many dwarf galaxies, the rotation velocity exceeds that attributable to the baryons (with Newton alone) at practically all radii. By a lot. DDO 154 is a very dark matter dominated galaxy. The baryons should have squat to say about the dynamics. And yet, all you need to know to predict the dynamics is the baryon distribution. The baryonic tail wags the dark matter dog.

But wait, it gets better! If you look closely at the data, you will note a kink at about 1 kpc, another at 2, and yet another around 5 kpc. These kinks are apparent in both the rotation curve and the gas distribution. This is an example of Sancisi’s Law: “For any feature in the luminosity profile there is a corresponding feature in the rotation curve and vice versa.” This is a general rule, as Sancisi observed, but it makes no sense when the dark matter dominates. The features in the baryon distribution should not be reflected in the rotation curve.

The observed baryons orbit in a disk with nearly circular orbits confined to the same plane. The dark matter moves on eccentric orbits oriented every which way to provide pressure support to a quasi-spherical halo. The baryonic and dark matter occupy very different regions of phase space, the six dimensional volume of position and momentum. The two are not strongly coupled, communicating only by the weak force of gravity in the standard CDM paradigm.

One of the first lessons of galaxy dynamics is that galaxy disks are subject to a variety of instabilities that grow bars and spiral arms. These are driven by disk self-gravity. The same features do not appear in elliptical galaxies because they are pressure supported, 3D blobs. They don’t have disks so they don’t have disk self-gravity, much less the features that lead to the bumps and wiggles observed in rotation curves.

Elliptical galaxies are a good visual analog for what dark matter halos are believed to be like. The orbits of dark matter particles are unable to sustain features like those seen in baryonic disks. They are featureless for the same reasons as elliptical galaxies. They don’t have disks. A rotation curve dominated by a spherical dark matter halo should bear no trace of the features that are seen in the disk. And yet they’re there, often enough for Sancisi to have remarked on it as a general rule.

It gets worse still. One of the original motivations for invoking dark matter was to stabilize galactic disks: a purely Newtonian disk of stars is not a stable configuration, yet the universe is chock full of long-lived spiral galaxies. The cure was to place them in dark matter halos.

The problem for dwarfs is that they have too much dark matter. The halo stabilizes disks by suppressing the formation of structures that stem from disk self-gravity. But you need some disk self-gravity to have the observed features. That can be tuned to work in bright spirals, but it fails in dwarfs because the halo is too massive. As a practical matter, there is no disk self-gravity in dwarfs – it is all halo, all the time. And yet, we do see such features. Not as strong as in big, bright spirals, but definitely present. Whenever someone tries to analyze this aspect of the problem, they inevitably come up with a requirement for more disk self-gravity in the form of unphysically high stellar mass-to-light ratios (something I predicted would happen). In contrast, this is entirely natural in MOND (see, e.g., Brada & Milgrom 1999 and Tiret & Combes 2008), where it is all disk self-gravity since there is no dark matter halo.

The net upshot of all this is that it doesn’t suffice to mimic the radial acceleration relation as many simulations now claim to do. That was not a natural part of CDM to begin with, but perhaps it can be done with smooth model galaxies. In most cases, such models lack the resolution to see the features seen in DDO 154 (and in NGC 1560 and in IC 2574, etc.) If they attain such resolution, they better not show such features, as that would violate some basic considerations. But then they wouldn’t be able to describe this aspect of the data.

Simulators by and large seem to remain sanguine that this will all work out. Perhaps I have become too cynical, but I recall hearing that 20 years ago. And 15. And ten… basically, they’ve always assured me that it will work out even though it never has. Maybe tomorrow will be different. Or would that be the definition of insanity?

It Must Be So. But which Must?

In the last post, I noted some of the sociological overtones underpinning attitudes about dark matter and modified gravity theories. I didn’t get as far as the more scientifically interesting part, which illustrates a common form of reasoning in physics.

About modified gravity theories, Bertone & Tait state

“the only way these theories can be reconciled with observations is by effectively, and very precisely, mimicking the behavior of cold dark matter on cosmological scales.”

Leaving aside just which observations need to be mimicked so precisely (I expect they mean power spectrum; perhaps they consider this to be so obvious that it need not be stated), this kind of reasoning is both common and powerful – and frequently correct. Indeed, this is exactly the attitude I expressed in my review a few years ago for the Canadian Journal of Physics, quoted in the image above. I get it. There are lots of positive things to be said for the standard cosmology.

This upshot of this reasoning is, in effect, that “cosmology works so well that non-baryonic dark matter must exist.” I have sympathy for this attitude, but I also remember many examples in the history of cosmology where it has gone badly wrong. There was a time, not so long ago, that the matter density had to be the critical value, and the Hubble constant had to be 50 km/s/Mpc. By and large, it is the same community that insisted on those falsehoods with great intensity that continues to insist on conventionally conceived cold dark matter with similarly fundamentalist insistence.

I think it is an overstatement to say that the successes of cosmology (as we presently perceive them) prove the existence of dark matter. A more conservative statement is that the ΛCDM cosmology is correct if, and only if, dark matter exists. But does it? That’s a separate question, which is why laboratory searches are so important – including null results. It was, after all, the null result of Michelson & Morley that ultimately put an end to the previous version of an invisible aetherial medium, and sparked a revolution in physics.

Here I point out that the same reasoning asserted by Bertone & Tait as a slam dunk in favor of dark matter can just as accurately be asserted in favor of MOND. To directly paraphrase the above statement:

“the only way ΛCDM can be reconciled with observations is by effectively, and very precisely, mimicking the behavior of MOND on galactic scales.”

This is a terrible problem for dark matter. Even if it were true, as is often asserted, that MOND only fits rotation curves, this would still be tantamount to a falsification of dark matter by the same reasoning applied by Bertone & Tait.

Lets look at just one example, NGC 1560:

ngc1560mond — The rotation curve of NGC 1560 (points) together with the Newtonian expectation (black line) and the MOND fit (blue line). Data from Begeman et al. (1991) and Gentile et al. (2010).

MOND fits the details of this rotation curve in excruciating detail. It provides just the right amount of boost over the Newtonian expectation, which varies from galaxy to galaxy. Features in the baryon distribution are reflected in the rotation curve. That is required in MOND, but makes no sense in dark matter, where the excess velocity over the Newtonian expectation is attributed to a dynamically hot, dominant, quasi-spherical dark matter halo. Such entities cannot support the features commonly seen in thin, dynamically cold disks. Even if they could, there is no reason that features in the dominant dark matter halo should align with those in the disk: a sphere isn’t a disk. In short, it is impossible to explain this with dark matter – to the extent that anything is ever impossible for the invisible.

NGC 1560 is a famous case because it has such an obvious feature. It is common to dismiss this as some non-equilibrium fluke that should simply be ignored. That is always a dodgy path to tread, but might be OK if it were only this galaxy. But similar effects are seen over and over again, to the point that they earned an empirical moniker: Renzo’s Rule. Renzo’s rule is known to every serious student of rotation curves, but has not informed the development of most dark matter theory. Ignoring this information is like leaving money on the table.

MOND fits not just NGC 1560, but very nearly* every galaxy we measure. It does so with excruciatingly little freedom. The only physical fit parameter is the stellar mass-to-light ratio. The gas fraction of NGC 1560 is 75%, so M*/L plays little role. We understand enough about stellar populations to have an idea what to expect; MOND fits return mass-to-light ratios that compare well with the normalization, color dependence, and band-pass dependent scatter expected from stellar population synthesis models.

MLBV_MOND — The mass-to-light ratio from MOND fits (points) in the blue (left panel) and near-infrared (right panel) pass-bands plotted against galaxy color (blue to the left, red to the right). From the perspective of stellar populations, one expects more scatter and a steeper color dependence in the blue band, as observed. The lines are stellar population models from Bell et al. (2003). These are completely independent, and have not been fit to the data in any way. One could hardly hope for better astrophysical agreement.

One can also fit rotation curve data with dark matter halos. These require a minimum of three parameters to the one of MOND. In addition to M*/L, one also needs at least two parameters to describe the dark matter halo of each galaxy – typically some characteristic mass and radius. In practice, one finds that such fits are horribly degenerate: one can not cleanly constrain all three parameters, much less recover a sensible distribution of M*/L. One cannot construct the plot above simply by asking the data what it wants as one can with MOND.

The “disk-halo degeneracy” in dark matter halo fits to rotation curves has been much discussed in the literature. Obsessed over, dismissed, revived, and ultimately ignored without satisfactory understanding. Well, duh. This approach uses three parameters per galaxy when it takes only one to describe the data. Degeneracy between the excess fit parameters is inevitable.

From a probabilistic perspective, there is a huge volume of viable parameter space that could (and should) be occupied by galaxies composed of dark matter halos plus luminous galaxies. Two identical dark matter halos might host very different luminous galaxies, so would have rotation curves that differed with the baryonic component. Two similar looking galaxies might reside in rather different dark matter halos, again having rotation curves that differ.

The probabilistic volume in MOND is much smaller. Absolutely tiny by comparison. There is exactly one and only one thing each rotation curve can do: what the particular distribution of baryons in each galaxy says it should do. This is what we observe in Nature.

The only way ΛCDM can be reconciled with observations is by effectively, and very precisely, mimicking the behavior of MOND on galactic scales. There is a vast volume of parameter space that the rotation curves of galaxies could, in principle, inhabit. The naive expectation was exponential disks in NFW halos. Real galaxies don’t look like that. They look like MOND. Magically, out of the vast parameter space available to galaxies in the dark matter picture, they only ever pick the tiny sub-volume that very precisely mimics MOND.

The ratio of probabilities is huge. So many dark matter models are possible (and have been mooted over the years) that it is indefinably huge. The odds of observing MOND-like phenomenology in a ΛCDM universe is practically zero. This amounts to a practical falsification of dark matter.

I’ve never said dark matter is falsified, because I don’t think it is a falsifiable concept. It is like epicycles – you can always fudge it in some way. But at a practical level, it was falsified a long time ago.

That is not to say MOND has to be right. That would be falling into the same logical trap that says ΛCDM has to be right. Obviously, both have virtues that must be incorporated into whatever the final answer may be. There are some efforts in this direction, but by and large this is not how science is being conducted at present. The standard script is to privilege those data that conform most closely to our confirmation bias, and pour scorn on any contradictory narrative.

In my assessment, the probability of ultimate success through ignoring inconvenient data is practically zero. Unfortunately, that is the course upon which much of the field is currently set.

*There are of course exceptions: no data are perfect, so even the right theory will get it wrong once in a while. The goof rate for MOND fits is about what I expect: rare, but more frequent for lower quality data. Misfits are sufficiently rare that to obsess over them is to refuse to see the forest for a few outlying trees.

Here’s a residual plot of MOND fits. See the peak at right? That’s the forest. See the tiny tail to one side? That’s an outlying tree.

Dwarf Satellite Galaxies. III. The dwarfs of Andromeda

Like the Milky Way, our nearest giant neighbor, Andromeda (aka M31), has several dozen dwarf satellite galaxies. A few of these were known and had measured velocity dispersions at the time of my work with Joe Wolf, as discussed previously. Also like the Milky Way, the number of known objects has grown rapidly in recent years – thanks in this case largely to the PAndAS survey.

PAndAS imaged the area around M31 and M33, finding many individual red giant stars. These trace out the debris from interactions and mergers as small dwarfs are disrupted and consumed by their giant host. They also pointed up the existence of previously unknown dwarf satellites.

M31fromPANDAS_ McC2012_EPJ_19_01003 — The PAndAS survey field. Dwarf satellites are circled.

As the PAndAS survey started reporting the discovery of new dwarf satellites around Andromeda, it occurred to me that this provided the opportunity to make genuine a priori predictions. These are the gold standard of the scientific method. We could use the observed luminosity and size of the newly discovered dwarfs to predict their velocity dispersions.

I tried to do this for both ΛCDM and MOND. I will not discuss the ΛCDM case much, because it can’t really be done. But it is worth understanding why this is.

In ΛCDM, the velocity dispersion is determined by the dark matter halo. This has only a tenuous connection to the observed stars, so just knowing how big and bright a dwarf is doesn’t provide much predictive power about the halo. This can be seen from this figure by Tollerud et al (2011):

Tollerud2011_ml_scatter — Virial mass of the dark matter halo as a function of galaxy luminosity. Dwarfs satellites reside in the wide colored band of low luminosities.

This graph is obtained by relating the number density of galaxies (an observed quantity) to that of the dark matter halos in which they reside (a theoretical construct). It is highly non-linear, deviating strongly from the one-to-one line we expected early on. There is no reason to expect this particular relation; it is imposed on us by the fact that the observed luminosity function of galaxies is rather flat while the predicted halo mass function is steep. Nowadays, this is usually called the missing satellite problem, but this is a misnomer as it pervades the field.

Addressing the missing satellites problem would be another long post, so lets just accept that the relation between mass and light has to follow something like that illustrated above. If a dwarf galaxy has a luminosity of a million suns, one can read off the graph that it should live in a dark halo with a mass of about 10¹⁰ M_☉. One could use this to predict the velocity dispersion, but not very precisely, because there’s a big range corresponding to that luminosity (the bands in the figure). It could be as much as 10¹¹ M_☉ or as little as 10⁹ M_☉. This corresponds to a wide range of velocity dispersions. This wide range is unavoidable because of the difference in the luminosity function and halo mass function. Small variations in one lead to big variations in the other, and some scatter in dark halo properties is unavoidable.

Consequently, we only have a vague range of expected velocity dispersions in ΛCDM. In practice, we never make this prediction. Instead, we compare the observed velocity dispersion to the luminosity and say “gee, this galaxy has a lot of dark matter” or “hey, this one doesn’t have much dark matter.” There’s no rigorously testable prior.

In MOND, what you see is what you get. The velocity dispersion has to follow from the observed stellar mass. This is straightforward for isolated galaxies: M_* ∝ σ⁴ – this is essentially the equivalent of the Tully-Fisher relation for pressure supported systems. If we can estimate the stellar mass from the observed luminosity, the predicted velocity dispersion follows.

Many dwarf satellites are not isolated in the MONDian sense: they are subject to the external field effect (EFE) from their giant hosts. The over-under for whether the EFE applies is the point when the internal acceleration from all the stars of the dwarf on each other is equal to the external acceleration from orbiting the giant host. The amplitude of the discrepancy in MOND depends on how low the total acceleration is relative to the critical scale a₀. The external field in effect adds some acceleration that wouldn’t otherwise be there, making the discrepancy less than it would be for an isolated object. This means that two otherwise identical dwarfs may be predicted to have different velocity dispersions is they are or are not subject to the EFE. This is a unique prediction of MOND that has no analog in ΛCDM.

It is straightforward to derive the equation to predict velocity dispersions in the extreme limits of isolated (a_ex ≪ a_in < a₀) or EFE dominated (a_in ≪ a_ex < a₀) objects. In reality, there are many objects for which a_in ≈ a_ex, and no simple formula applies. In practice, we apply the formula that more nearly applies, and pray that this approximation is good enough.

There are many other assumptions and approximations that must be made in any theory: that an object is spherical, isotropic, and in dynamical equilibrium. All of these must fail at some level, but it is the last one that is the most serious concern. In the case of the EFE, one must also make the approximation that the object is in equilibrium at the current level of the external field. That is never true, as both the amplitude and the vector of the external field vary as a dwarf orbits its host. But it might be an adequate approximation if this variation is slow. In the case of a circular orbit, only the vector varies. In general the orbits are not known, so we make the instantaneous approximation and once again pray that it is good enough. There is a fairly narrow window between where the EFE becomes important and where we slip into the regime of tidal disruption, but lets plow ahead and see how far we can get, bearing in mind that the EFE is a dynamical variable of which we only have a snapshot.

To predict the velocity dispersion in the isolated case, all we need to know is the luminosity and a stellar mass-to-light ratio. Assuming the dwarfs of Andromeda to be old stellar populations, I adopted a V-band mass-to-light ratio of 2 give or take a factor of 2. That usually dominates the uncertainty, though the error in the distance can sometimes impact the luminosity at a level that impacts the prediction.

To predict the velocity dispersion in the EFE case, we again need the stellar mass, but now also need to know the size of the stellar system and the intensity of the external field to which it is subject. The latter depends on the mass of the host galaxy and the distance from it to the dwarf. This latter quantity is somewhat fraught: it is straightforward to measure the projected distance on the sky, but we need the 3D distance – how far in front or behind each dwarf is as well as its projected distance from the host. This is often a considerable contributor to the error budget. Indeed, some dwarfs may be inferred to be in the EFE regime for the low end of the range of adopted stellar mass-to-light ratio, and the isolated regime for the high end.

In this fashion, we predicted velocity dispersions for the dwarfs of Andromeda. We in this case were Milgrom and myself. I had never collaborated with him before, and prefer to remain independent. But I also wanted to be sure I got the details described above right. Though it wasn’t much work to make the predictions once the preliminaries were established, it was time consuming to collect and vet the data. As we were writing the paper, velocity dispersion measurements started to appear. People like Michelle Collins, Erik Tollerud, and Nicolas Martin were making follow-up observations, and publishing velocity dispersion for the objects we were making predictions for. That was great, but they were too good – they were observing and publishing faster than we could write!

Nevertheless, we managed to make and publish a priori predictions for 10 dwarfs before any observational measurements were published. We also made blind predictions for the other known dwarfs of Andromeda, and checked the predicted velocity dispersions against all measurements that we could find in the literature. Many of these predictions were quickly tested by on-going programs (i.e., people were out to measure velocity dispersions, whether we predicted them or not). Enough data rolled in that we were soon able to write a follow-up paper testing our predictions.

Nailed it. Good data were soon available to test the predictions for 8 of the 10* a priori cases. All 8 were consistent with our predictions. I was particularly struck by the case of And XXVIII, which I had called out as perhaps the best test. It was isolated, so the messiness of the EFE didn’t apply, and the uncertainties were low. Moreover, the predicted velocity dispersion was low – a good deal lower than broadly expected in ΛCDM: 4.3 km/s, with an uncertainty just under 1 km/s. Two independent observations were subsequently reported. One found 4.9 ± 1.6 km/s, the other 6.6 ± 2.1 km/s, both in good agreement within the uncertainties.

We made further predictions in the second paper as people had continued to discover new dwarfs. These also came true. Here is a summary plot for all of the dwarfs of Andromeda:

AndDwarfswithGoldStars.002 — The velocity dispersions of the dwarf satellites of Andromeda. Each numbered box corresponds to one dwarf (x=1 is for And I and so on). Measured velocity dispersions have a number next to them that is the number of stars on which the measurement is based. MOND predictions are circles: green if isolated, open if the EFE applies. Points appear within each box in the order they appeared in the literature, from left to right. The vast majority of Andromeda’s dwarfs are consistent with MOND (large green circles). Two cases are ambiguous (large yellow circles), having velocity dispersions based only a few stars. Only And V appears to be problematic (large red circle).

MOND works well for And I, And II, And III, And VI, And VII, And IX, And X, And XI, And XII, And XIII, And XIV, And XV, And XVI, And XVII, And XVIII, And XIX, And XX, And XXI, And XXII, And XXIII, And XXIV, And XXV, And XXVIII, And XXIX, And XXXI, And XXXII, and And XXXIII. There is one problematic case: And V. I don’t know what is going on there, but note that systematic errors frequently happen in astronomy. It’d be strange if there weren’t at least one goofy case.

Nevertheless, the failure of And V could be construed as a falsification of MOND. It ought to work in every single case. But recall the discussion of assumptions and uncertainties above. Is falsification really the story these data tell?

We do have experience with various systematic errors. For example, we predicted that the isolated dwarfs spheroidal Cetus should have a velocity dispersion in MOND of 8.2 km/s. There was already a published measurement of 17 ± 2 km/s, so we reported that MOND was wrong in this case by over 3σ. Or at least we started to do so. Right before we submitted that paper, a new measurement appeared: 8.3 ± 1 km/s. This is an example of how the data can sometimes change by rather more than the formal error bars suggest is possible. In this case, I suspect the original observations lacked the spectral resolution to resolve the velocity dispersion. At any rate, the new measurement (8.3 km/s) was somewhat more consistent with our prediction (8.2 km/s).

The same predictions cannot even be made in ΛCDM. The velocity data can always be fit once they are in hand. But there is no agreed method to predict the velocity dispersion of a dwarf from its observed luminosity. As discussed above, this should not even be possible: there is too much scatter in the halo mass-stellar mass relation at these low masses.

An unsung predictive success of MOND absent from the graph above is And IV. When And IV was discovered in the general direction of Andromeda, it was assumed to be a new dwarf satellite – hence the name. Milgrom looked at the velocities reported for this object, and said it had to be a background galaxy. No way it could be a dwarf satellite – at least not in MOND. I see no reason why it couldn’t have been in ΛCDM. It is absent from the graph above, because it was subsequently confirmed to be much farther away (7.2 Mpc vs. 750 kpc for Andromeda).

The box for And XVII is empty because this system is manifestly out of equilibrium. It is more of a stellar stream than a dwarf, appearing as a smear in the PAndAS image rather than as a self-contained dwarf. I do not recall what the story with the other missing object (And VIII) is.

While writing the follow-up paper, I also noticed that there were a number of Andromeda dwarfs that were photometrically indistinguishable: basically the same in terms of size and stellar mass. But some were isolated while others were subject to the EFE. MOND predicts that the EFE cases should have lower velocity dispersion than the isolated equivalents.

AndDwarfswithGoldStars.003 — The velocity dispersions of the dwarfs of Andromeda, highlighting photometrically matched pairs – dwarfs that should be indistinguishable, but aren’t because of the EFE.

And XXVIII (isolated) has a higher velocity dispersion than its near-twin And XVII (EFE). The same effect might be acting in And XVIII (isolated) and And XXV (EFE). This is clear if we accept the higher velocity dispersion measurement for And XVIII, but an independent measurements begs to differ. The former has more stars, so is probably more reliable, but we should be cautious. The effect is not clear in And XVI (isolated) and And XXI (EFE), but the difference in the prediction is small and the uncertainties are large.

An aggressive person might argue that the pairs of dwarfs is a positive detection of the EFE. I don’t think the data for the matched pairs warrant that, at least not yet. On the other hand, the appropriate use of the EFE was essential to all the predictions, not just the matched pairs.

The positive detection of the EFE is important, as it is a unique prediction of MOND. I see no way to tune ΛCDM galaxy simulations to mimic this effect. Of course, there was a very recent time when it seemed impossible for them to mimic the isolated predictions of MOND. They claim to have come a long way in that regard.

But that’s what we’re stuck with: tuning ΛCDM to make it look like MOND. This is why a priori predictions are important. There is ample flexibility to explain just about anything with dark matter. What we can’t seem to do is predict the same things that MOND successfully predicts… predictions that are both quantitative and very specific. We’re not arguing that dwarfs in general live in ~15 or 30 km/s halos, as we must in ΛCDM. In MOND we can say this dwarf will have this velocity dispersion and that dwarf will have that velocity dispersion. We can distinguish between 4.9 and 7.3 km/s. And we can do it over and over and over. I see no way to do the equivalent in ΛCDM, just as I see no way to explain the acoustic power spectrum of the CMB in MOND.

This is not to say there are no problematic cases for MOND. Read, Walker, & Steger have recently highlighted the matched pair of Draco and Carina as an issue. And they are – though here I already have reason to suspect Draco is out of equilibrium, which makes it challenging to analyze. Whether it is actually out of equilibrium or not is a separate question.

I am not thrilled that we are obliged to invoke non-equilibrium effects in both theories. But there is a difference. Brada & Milgrom provided a quantitative criterion to indicate when this was an issue before I ran into the problem. In ΛCDM, the low velocity dispersions of objects like And XIX, XXI, XXV and Crater 2 came as a complete surprise despite having been predicted by MOND. Tidal disruption was only invoked after the fact – and in an ad hoc fashion. There is no way to know in advance which dwarfs are affected, as there is no criterion equivalent to that of Brada. We just say “gee, that’s a low velocity dispersion. Must have been disrupted.” That might be true, but it gives no explanation for why MOND predicted it in the first place – which is to say, it isn’t really an explanation at all.

I still do not understand is why MOND gets any predictions right if ΛCDM is the universe we live in, let alone so many. Shouldn’t happen. Makes no sense.

If this doesn’t confuse you, you are not thinking clearly.

*The other two dwarfs were also measured, but with only 4 stars in one and 6 in the other. These are too few for a meaningful velocity dispersion measurement.

Dwarf Satellite Galaxies. II. Non-equilibrium effects in ultrafaint dwarfs

I have been wanting to write about dwarf satellites for a while, but there is so much to tell that I didn’t think it would fit in one post. I was correct. Indeed, it was worse than I thought, because my own experience with low surface brightness (LSB) galaxies in the field is a necessary part of the context for my perspective on the dwarf satellites of the Local Group. These are very different beasts – satellites are pressure supported, gas poor objects in orbit around giant hosts, while field LSB galaxies are rotating, gas rich galaxies that are among the most isolated known. However, so far as their dynamics are concerned, they are linked by their low surface density.

Where we left off with the dwarf satellites, circa 2000, Ursa Minor and Draco remained problematic for MOND, but the formal significance of these problems was not great. Fornax, which had seemed more problematic, was actually a predictive success: MOND returned a low mass-to-light ratio for Fornax because it was full of young stars. The other known satellites, Carina, Leo I, Leo II, Sculptor, and Sextans, were all consistent with MOND.

The Sloan Digital Sky Survey resulted in an explosion in the number of satellites galaxies discovered around the Milky Way. These were both fainter and lower surface brightness than the classical dwarfs named above. Indeed, they were often invisible as objects in their own right, being recognized instead as groupings of individual stars that shared the same position in space and – critically – velocity. They weren’t just in the same place, they were orbiting the Milky Way together. To give short shrift to a long story, these came to be known as ultrafaint dwarfs.

Ultrafaint dwarf satellites have fewer than 100,000 stars. That’s tiny for a stellar system. Sometimes they had only a few hundred. Most of those stars are too faint to see directly. Their existence is inferred from a handful of red giants that are actually observed. Where there are a few red giants orbiting together, there must be a source population of fainter stars. This is a good argument, and it is likely true in most cases. But the statistics we usually rely on become dodgy for such small numbers of stars: some of the ultrafaints that have been reported in the literature are probably false positives. I have no strong opinion on how many that might be, but I’d be really surprised if it were zero.

Nevertheless, assuming the ultrafaints dwarfs are self-bound galaxies, we can ask the same questions as before. I was encouraged to do this by Joe Wolf, a clever grad student at UC Irvine. He had a new mass estimator for pressure supported dwarfs that we decided to apply to this problem. We used the Baryonic Tully-Fisher Relation (BTFR) as a reference, and looked at it every which-way. Most of the text is about conventional effects in the dark matter picture, and I encourage everyone to read the full paper. Here I’m gonna skip to the part about MOND, because that part seems to have been overlooked in more recent commentary on the subject.

For starters, we found that the classical dwarfs fall along the extrapolation of the BTFR, but the ultrafaint dwarfs deviate from it.

Fig1_annotated — Fig. 1 from McGaugh & Wolf (2010, annotated). The BTFR defined by rotating galaxies (gray points) extrapolates well to the scale of the dwarf satellites of the Local Group (blue points are the classical dwarf satellites of the Milky Way; red points are satellites of Andromeda) but not to the ultrafaint dwarfs (green points). Two of the classical dwarfs also fall off of the BTFR: Draco and Ursa Minor.

The deviation is not subtle, at least not in terms of mass. The ultrataints had characteristic circular velocities typical of systems 100 times their mass! But the BTFR is steep. In terms of velocity, the deviation is the difference between the 8 km/s typically observed, and the ~3 km/s needed to put them on the line. There are a large number of systematic effects errors that might arise, and all act to inflate the characteristic velocity. See the discussion in the paper if you’re curious about such effects; for our purposes here we will assume that the data cannot simply be dismissed as the result of systematic errors, though one should bear in mind that they probably play a role at some level.

Taken at face value, the ultrafaint dwarfs are a huge problem for MOND. An isolated system should fall exactly on the BTFR. These are not isolated systems, being very close to the Milky Way, so the external field effect (EFE) can cause deviations from the BTFR. However, these are predicted to make the characteristic internal velocities lower than the isolated case. This may in fact be relevant for the red points that deviate a bit in the plot above, but we’ll return to that at some future point. The ultrafaints all deviate to velocities that are too high, the opposite of what the EFE predicts.

The ultrafaints falsify MOND! When I saw this, all my original confirmation bias came flooding back. I had pursued this stupid theory to ever lower surface brightness and luminosity. Finally, I had found where it broke. I felt like Darth Vader in the original Star Wars:

darth-vader-i-have-you-now_1 — I have you now!

The first draft of my paper with Joe included a resounding renunciation of MOND. No way could it escape this!

But…

I had this nagging feeling I was missing something. Darth should have looked over his shoulder. Should I?

Surely I had missed nothing. Many people are unaware of the EFE, just as we had been unaware that Fornax contained young stars. But not me! I knew all that. Surely this was it.

Nevertheless, the nagging feeling persisted. One part of it was sociological: if I said MOND was dead, it would be well and truly buried. But did it deserve to be? The scientific part of the nagging feeling was that maybe there had been some paper that addressed this, maybe a decade before… perhaps I’d better double check.

Indeed, Brada & Milgrom (2000) had run numerical simulations of dwarf satellites orbiting around giant hosts. MOND is a nonlinear dynamical theory; not everything can be approximated analytically. When a dwarf satellite is close to its giant host, the external acceleration of the dwarf falling towards its host can exceed the internal acceleration of the stars in the dwarf orbiting each other – hence the EFE. But the EFE is not a static thing; it varies as the dwarf orbits about, becoming stronger on closer approach. At some point, this variation becomes to fast for the dwarf to remain in equilibrium. This is important, because the assumption of dynamical equilibrium underpins all these arguments. Without it, it is hard to know what to expect short of numerically simulating each individual dwarf. There is no reason to expect them to remain on the equilibrium BTFR.

Brada & Milgrom suggested a measure to gauge the extent to which a dwarf might be out of equilibrium. It boils down to a matter of timescales. If the stars inside the dwarf have time to adjust to the changing external field, a quasi-static EFE approximation might suffice. So the figure of merit becomes the ratio of internal orbits per external orbit. If the stars inside a dwarf are swarming around many times for every time it completes an orbit around the host, then they have time to adjust. If the orbit of the dwarf around the host is as quick as the internal motions of the stars within the dwarf, not so much. At some point, a satellite becomes a collection of associated stars orbiting the host rather than a self-bound object in its own right.

Fig7_annotated — Deviations from the BTFR (left) and the isophotal shape of dwarfs (right) as a function of the number of internal orbits a star at the half-light radius makes for every orbit a dwarf makes around its giant host (Fig. 7 of McGaugh & Wolf 2010).

Brada & Milgrom provide the formula to compute the ratio of orbits, shown in the figure above. The smaller the ratio, the less chance an object has to adjust, and the more subject it is to departures from equilibrium. Remarkably, the amplitude of deviation from the BTFR – the problem I could not understand initially – correlates with the ratio of orbits. The more susceptible a dwarf is to disequilibrium effects, the farther it deviated from the BTFR.

This completely inverted the MOND interpretation. Instead of falsifying MOND, the data now appeared to corroborate the non-equilibrium prediction of Brada & Milgrom. The stronger the external influence, the more a dwarf deviated from the equilibrium expectation. In conventional terms, it appeared that the ultrafaints were subject to tidal stirring: their internal velocities were being pumped up by external influences. Indeed, the originally problematic cases, Draco and Ursa Minor, fall among the ultrafaint dwarfs in these terms. They can’t be in equilibrium in MOND.

If the ultrafaints are out of equilibrium, the might show some independent evidence of this. Stars should leak out, distorting the shape of the dwarf and forming tidal streams. Can we see this?

A definite maybe:

Ell_D_wImages — The shapes of some ultrafaint dwarfs. These objects are so diffuse that they are invisible on the sky; their shape is illustrated by contours or heavily smoothed grayscale pseudo-images.

The dwarfs that are more subject to external influence tend to be more elliptical in shape. A pressure supported system in equilibrium need not be perfectly round, but one departing from equilibrium will tend to get stretched out. And indeed, many of the ultrafaints look Messed Up.

I am not convinced that all this requires MOND. But it certainly doesn’t falsify it. Tidal disruption can happen in the dark matter context, but it happens differently. The stars are buried deep inside protective cocoons of dark matter, and do not feel tidal effects much until most of the dark matter is stripped away. There is no reason to expect the MOND measure of external influence to apply (indeed, it should not), much less that it would correlate with indications of tidal disruption as seen above.

This seems to have been missed by more recent papers on the subject. Indeed, Fattahi et al. (2018) have reconstructed very much the chain of thought I describe above. The last sentence of their abstract states “In many cases, the resulting velocity dispersions are inconsistent with the predictions from Modified Newtonian Dynamics, a result that poses a possibly insurmountable challenge to that scenario.” This is exactly what I thought. (I have you now.) I was wrong.

Fattahi et al. are wrong for the same reasons I was wrong. They are applying equilibrium reasoning to a non-equilibrium situation. Ironically, the main point of the their paper is that many systems can’t be explained with dark matter, unless they are tidally stripped – i.e., the result of a non-equilibrium process. Oh, come on. If you invoke it in one dynamical theory, you might want to consider it in the other.

To quote the last sentence of our abstract from 2010, “We identify a test to distinguish between the ΛCDM and MOND based on the orbits of the dwarf satellites of the Milky Way and how stars are lost from them.” In ΛCDM, the sub-halos that contain dwarf satellites are expected to be on very eccentric orbits, with all the damage from tidal interactions with the host accruing during pericenter passage. In MOND, substantial damage may accrue along lower eccentricity orbits, leading to the expectation of more continuous disruption.

Gaia is measuring proper motions for stars all over the sky. Some of these stars are in the dwarf satellites. This has made it possible to estimate orbits for the dwarfs, e.g., work by Amina Helmi (et al!) and Josh Simon. So far, the results are definitely mixed. There are more dwarfs on low eccentricity orbits than I had expected in ΛCDM, but there are still plenty that are on high eccentricity orbits, especially among the ultrafaints. Which dwarfs have been tidally affected by interactions with their hosts is far from clear.

In short, reality is messy. It is going to take a long time to sort these matters out. These are early days.

Dwarf Satellite Galaxies and Low Surface Brightness Galaxies in the Field. I.

The Milky Way and its nearest giant neighbor Andromeda (M31) are surrounded by a swarm of dwarf satellite galaxies. Aside from relatively large beasties like the Large Magellanic Cloud or M32, the majority of these are the so-called dwarf spheroidals. There are several dozen examples known around each giant host, like the Fornax dwarf pictured above.

Dwarf Spheroidal (dSph) galaxies are ellipsoidal blobs devoid of gas that typically contain a million stars, give or take an order of magnitude. Unlike globular clusters, that may have a similar star count, dSphs are diffuse, with characteristic sizes of hundreds of parsecs (vs. a few pc for globulars). This makes them among the lowest surface brightness systems known.

This subject has a long history, and has become a major industry in recent years. In addition to the “classical” dwarfs that have been known for decades, there have also been many comparatively recent discoveries, often of what have come to be called “ultrafaint” dwarfs. These are basically dSphs with luminosities less than 100,000 suns, sometimes being comprised of as little as a few hundred stars. New discoveries are being made still, and there is reason to hope that the LSST will discover many more. Summed up, the known dwarf satellites are proverbial drops in the bucket compared to their giant hosts, which contain hundreds of billions of stars. Dwarfs could rain in for a Hubble time and not perturb the mass budget of the Milky Way.

Nevertheless, tiny dwarf Spheroidals are excellent tests of theories like CDM and MOND. Going back to the beginning, in the early ’80s, Milgrom was already engaged in a discussion about the predictions of his then-new theory (before it was even published) with colleagues at the IAS, where he had developed the idea during a sabbatical visit. They were understandably skeptical, preferring – as many still do – to believe that some unseen mass was the more conservative hypothesis. Dwarf spheroidals came up even then, as their very low surface brightness meant low acceleration in MOND. This in turn meant large mass discrepancies. If you could measure their dynamics, they would have large mass-to-light ratios. Larger than could be explained by stars conventionally, and larger than the discrepancies already observed in bright galaxies like Andromeda.

This prediction of Milgrom’s – there from the very beginning – is important because of how things change (or don’t). At that time, Scott Tremaine summed up the contrasting expectation of the conventional dark matter picture:

“There is no reason to expect that dwarfs will have more dark matter than bright galaxies.” *

This was certainly the picture I had in my head when I first became interested in low surface brightness (LSB) galaxies in the mid-80s. At that time I was ignorant of MOND; my interest was piqued by the argument of Disney that there could be a lot of as-yet undiscovered LSB galaxies out there, combined with my first observing experiences with the then-newfangled CCD cameras which seemed to have a proclivity for making clear otherwise hard-to-see LSB features. At the time, I was interested in finding LSB galaxies. My interest in what made them rotate came later.

The first indication, to my knowledge, that dSph galaxies might have large mass discrepancies was provided by Marc Aaronson in 1983. This tentative discovery was hugely important, but the velocity dispersion of Draco (one of the “classical” dwarfs) was based on only 3 stars, so was hardly definitive. Nevertheless, by the end of the ’90s, it was clear that large mass discrepancies were a defining characteristic of dSphs. Their conventionally computed M/L went up systematically as their luminosity declined. This was not what we had expected in the dark matter picture, but was, at least qualitatively, in agreement with MOND.

My own interests had focused more on LSB galaxies in the field than on dwarf satellites like Draco. Greg Bothun and Jim Schombert had identified enough of these to construct a long list of LSB galaxies that served as targets my for Ph.D. thesis. Unlike the pressure-supported ellipsoidal blobs of stars that are the dSphs, the field LSBs we studied were gas rich, rotationally supported disks – mostly late type galaxies (Sd, Sm, & Irregulars). Regardless of composition, gas or stars, low surface density means that MOND predicts low acceleration. This need not be true conventionally, as the dark matter can do whatever the heck it wants. Though I was blissfully unaware of it at the time, we had constructed the perfect sample for testing MOND.

Having studied the properties of our sample of LSB galaxies, I developed strong ideas about their formation and evolution. Everything we had learned – their blue colors, large gas fractions, and low star formation rates – suggested that they evolved slowly compared to higher surface brightness galaxies. Star formation gradually sputtered along, having a hard time gathering enough material to make stars in their low density interstellar media. Perhaps they even formed late, an idea I took a shining to in the early ’90s. This made two predictions: field LSB galaxies should be less strongly clustered than bright galaxies, and should spin slower at a given mass.

The first prediction follows because the collapse time of dark matter halos correlates with their larger scale environment. Dense things collapse first and tend to live in dense environments. If LSBs were low surface density because they collapsed late, it followed that they should live in less dense environments.

I didn’t know how to test this prediction. Fortunately, fellow postdoc and office mate in Cambridge at the time, Houjun Mo, did. It came true. The LSB galaxies I had been studying were clustered like other galaxies, but not as strongly. This was exactly what I expected, and I thought sure we were on to something. All that remained was to confirm the second prediction.

At the time, we did not have a clear idea of what dark matter halos should be like. NFW halos were still in the future. So it seemed reasonable that late forming halos should have lower densities (lower concentrations in the modern terminology). More importantly, the sum of dark and luminous density was certainly less. Dynamics follow from the distribution of mass as Velocity² ∝ Mass/Radius. For a given mass, low surface brightness galaxies had a larger radius, by construction. Even if the dark matter didn’t play along, the reduction in the concentration of the luminous mass should lower the rotation velocity.

Indeed, the standard explanation of the Tully-Fisher relation was just this. Aaronson, Huchra, & Mould had argued that galaxies obeyed the Tully-Fisher relation because they all had essentially the same surface brightness (Freeman’s law) thereby taking variation in the radius out of the equation: galaxies of the same mass all had the same radius. (If you are a young astronomer who has never heard of Freeman’s law, you’re welcome.) With our LSB galaxies, we had a sample that, by definition, violated Freeman’s law. They had large radii for a given mass. Consequently, they should have lower rotation velocities.

Up to that point, I had not taken much interest in rotation curves. In contrast, colleagues at the University of Groningen were all about rotation curves. Working with Thijs van der Hulst, Erwin de Blok, and Martin Zwaan, we set out to quantify where LSB galaxies fell in relation to the Tully-Fisher relation. I confidently predicted that they would shift off of it – an expectation shared by many at the time. They did not.

BTFSBallwlinessmall — The Tully-Fisher relation: disk mass vs. flat rotation speed (circa 1996). Galaxies are binned by surface brightness with the highest surface brightness galaxies marked red and the lowest blue. The lines show the expected shift following the argument of Aaronson et al. Contrary to this expectation, galaxies of all surface brightnesses follow the same Tully-Fisher relation.

I was flummoxed. My prediction was wrong. That of Aaronson et al. was wrong. Poking about the literature, everyone who had made a clear prediction in the conventional context was wrong. It made no sense.

I spent months banging my head against the wall. One quick and easy solution was to blame the dark matter. Maybe the rotation velocity was set entirely by the dark matter, and the distribution of luminous mass didn’t come into it. Surely that’s what the flat rotation velocity was telling us? All about the dark matter halo?

Problem is, we measure the velocity where the luminous mass still matters. In galaxies like the Milky Way, it matters quite a lot. It does not work to imagine that the flat rotation velocity is set by some property of the dark matter halo alone. What matters to what we measure is the combination of luminous and dark mass. The luminous mass is important in high surface brightness galaxies, and progressively less so in lower surface brightness galaxies. That should leave some kind of mark on the Tully-Fisher relation, but it doesn’t.

CRVfresid — Residuals from the Tully-Fisher relation as a function of size at a given mass. Compact galaxies are to the left, diffuse ones to the right. The red dashed line is what Newton predicts: more compact galaxies should rotate faster at a given mass. Fundamental physics? Tully-Fisher don’t care. Tully-Fisher don’t give a sh*t.

I worked long and hard to understand this in terms of dark matter. Every time I thought I had found the solution, I realized that it was a tautology. Somewhere along the line, I had made an assumption that guaranteed that I got the answer I wanted. It was a hopeless fine-tuning problem. The only way to satisfy the data was to have the dark matter contribution scale up as that of the luminous mass scaled down. The more stretched out the light, the more compact the dark – in exact balance to maintain zero shift in Tully-Fisher.

This made no sense at all. Over twenty years on, I have yet to hear a satisfactory conventional explanation. Most workers seem to assert, in effect, that “dark matter does it” and move along. Perhaps they are wise to do so.

repomanfoxharris — Working on the thing can drive you mad.

As I was struggling with this issue, I happened to hear a talk by Milgrom. I almost didn’t go. “Modified gravity” was in the title, and I remember thinking, “why waste my time listening to that nonsense?” Nevertheless, against my better judgement, I went. Not knowing that anyone in the audience worked on either LSB galaxies or Tully-Fisher, Milgrom proceeded to derive the MOND prediction:

“The asymptotic circular velocity is determined only by the total mass of the galaxy: V_f⁴ = a₀GM.”

In a few lines, he derived rather trivially what I had been struggling to understand for months. The lack of surface brightness dependence in Tully-Fisher was entirely natural in MOND. It falls right out of the modified force law, and had been explicitly predicted over a decade before I struggled with the problem.

I scraped my jaw off the floor, determined to examine this crazy theory more closely. By the time I got back to my office, cognitive dissonance had already started to set it. Couldn’t be true. I had more pressing projects to complete, so I didn’t think about it again for many moons.

When I did, I decided I should start by reading the original MOND papers. I was delighted to find a long list of predictions, many of them specifically to do with surface brightness. We had just collected fresh data on LSB galaxies, which provided a new window on the low acceleration regime. I had the data to finally falsify this stupid theory.

Or so I thought. As I went through the list of predictions, my assumption that MOND had to be wrong was challenged by each item. It was barely an afternoon’s work: check, check, check. Everything I had struggled for months to understand in terms of dark matter tumbled straight out of MOND.

I was faced with a choice. I knew this would be an unpopular result. I could walk away and simply pretend I had never run across it. That’s certainly how it had been up until then: I had been blissfully unaware of MOND and its perniciously successful predictions. No need to admit otherwise.

Had I realized just how unpopular it would prove to be, maybe that would have been the wiser course. But even contemplating such a course felt criminal. I was put in mind of Paul Gerhardt’s admonition for intellectual honesty:

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

Ignoring what I had learned seemed tantamount to just that. So many predictions coming true couldn’t be an accident. There was a deep clue here; ignoring it wasn’t going to bring us closer to the truth. Actively denying it would be an act of wanton vandalism against the scientific method.

Still, I tried. I looked long and hard for reasons not to report what I had found. Surely there must be some reason this could not be so?

Indeed, the literature provided many papers that claimed to falsify MOND. To my shock, few withstood critical examination. Commonly a straw man representing MOND was falsified, not MOND itself. At a deeper level, it was implicitly assumed that any problem for MOND was an automatic victory for dark matter. This did not obviously follow, so I started re-doing the analyses for both dark matter and MOND. More often than not, I found either that the problems for MOND were greatly exaggerated, or that the genuinely problematic cases were a problem for both theories. Dark matter has more flexibility to explain outliers, but outliers happen in astronomy. All too often the temptation was to refuse to see the forest for a few trees.

The first MOND analysis of the classical dwarf spheroidals provides a good example. Completed only a few years before I encountered the problem, these were low surface brightness systems that were deep in the MOND regime. These were gas poor, pressure supported dSph galaxies, unlike my gas rich, rotating LSB galaxies, but the critical feature was low surface brightness. This was the most directly comparable result. Better yet, the study had been made by two brilliant scientists (Ortwin Gerhard & David Spergel) whom I admire enormously. Surely this work would explain how my result was a mere curiosity.

Indeed, reading their abstract, it was clear that MOND did not work for the dwarf spheroidals. Whew: LSB systems where it doesn’t work. All I had to do was figure out why, so I read the paper.

As I read beyond the abstract, the answer became less and less clear. The results were all over the map. Two dwarfs (Sculptor and Carina) seemed unobjectionable in MOND. Two dwarfs (Draco and Ursa Minor) had mass-to-light ratios that were too high for stars, even in MOND. That is, there still appeared to be a need for dark matter even after MOND had been applied. One the flip side, Fornax had a mass-to-light ratio that was too low for the old stellar populations assumed to dominate dwarf spheroidals. Results all over the map are par for the course in astronomy, especially for a pioneering attempt like this. What were the uncertainties?

Milgrom wrote a rebuttal. By then, there were measured velocity dispersions for two more dwarfs. Of these seven dwarfs, he found that

“within just the quoted errors on the velocity dispersions and the luminosities, the MOND M/L values for all seven dwarfs are perfectly consistent with stellar values, with no need for dark matter.”

Well, he would say that, wouldn’t he? I determined to repeat the analysis and error propagation.

MdB98bFig8_dSph — Mass-to-light ratios determined with MOND for eight dwarf spheroidals (named, as published in McGaugh & de Blok 1998). The various symbols refer to different determinations. Mine are the solid circles. The dashed lines show the plausible range for stellar populations.

The net result: they were both right. M/L was still too high for Draco and Ursa Minor, and still too low for Fornax. But this was only significant at the 2σ level, if that – hardly enough to condemn a theory. Carina, Leo I, Leo II, Sculptor, and Sextans all had fairly reasonable mass-to-light ratios. The voting is different now. Instead of going 2 for 5 as Gerhard & Spergel found, MOND was now 5 for 8. One could choose to obsess about the outliers, or one could choose to see a more positive pattern. Either a positive or a negative spin could be put on this result. But it was clearly more positive than the first attempt had indicated.

The mass estimator in MOND scales as the fourth power of velocity (or velocity dispersion in the case of isolated dSphs), so the too-high M*/L of Draco and Ursa Minor didn’t disturb me too much. A small overestimation of the velocity dispersion would lead to a large overestimation of the mass-to-light ratio. Just about every systematic uncertainty one can think of pushes in this direction, so it would be surprising if such an overestimate didn’t happen once in a while.

Given this, I was more concerned about the low M*/L of Fornax. That was weird.

Up until that point (1998), we had been assuming that the stars in dSphs were all old, like those in globular clusters. That corresponds to a high M*/L, maybe 3 in solar units in the V-band. Shortly after this time, people started to look closely at the stars in the classical dwarfs with the Hubble. Low and behold, the stars in Fornax were surprisingly young. That means a low M*/L, 1 or less. In retrospect, MOND was trying to tell us that: it returned a low M*/L for Fornax because the stars there are young. So what was taken to be a failing of the theory was actually a predictive success.

Hmm.

And Gee. This is a long post. There is a lot more to tell, but enough for now.

*I have a long memory, but it is not perfect. I doubt I have the exact wording right, but this does accurately capture the sentiment from the early ’80s when I was an undergraduate at MIT and Scott Tremaine was on the faculty there.

The next cosmic frontier: 21cm absorption at high redshift

There are two basic approaches to cosmology: start at redshift zero and work outwards in space, or start at the beginning of time and work forward. The latter approach is generally favored by theorists, as much of the physics of the early universe follows a “clean” thermal progression, cooling adiabatically as it expands. The former approach is more typical of observers who start with what we know locally and work outwards in the great tradition of Hubble, Sandage, Tully, and the entire community of extragalactic observers that established the paradigm of the expanding universe and measured its scale. This work had established our current concordance cosmology, ΛCDM, by the mid-90s.*

Both approaches have taught us an enormous amount. Working forward in time, we understand the nucleosynthesis of the light elements in the first few minutes, followed after a few hundred thousand years by the epoch of recombination when the universe transitioned from an ionized plasma to a neutral gas, bequeathing us the cosmic microwave background (CMB) at the phenomenally high redshift of z=1090. Working outwards in redshift, large surveys like Sloan have provided a detailed map of the “local” cosmos, and narrower but much deeper surveys provide a good picture out to z = 1 (when the universe was half its current size, and roughly half its current age) and beyond, with the most distant objects now known above redshift 7, and maybe even at z > 11. JWST will provide a good view of the earliest (z ~ 10?) galaxies when it launches.

This is wonderful progress, but there is a gap from 10 < z < 1000. Not only is it hard to observe objects so distant that z > 10, but at some point they shouldn’t exist. It takes time to form stars and galaxies and the supermassive black holes that fuel quasars, especially when starting from the smooth initial condition seen in the CMB. So how do we probe redshifts z > 10?

It turns out that the universe provides a way. As photons from the CMB traverse the neutral intergalactic medium, they are subject to being absorbed by hydrogen atoms – particularly by the 21cm spin-flip transition. Long anticipated, this signal has recently been detected by the EDGES experiment. I find it amazing that the atomic physics of the early universe allows for this window of observation, and that clever scientists have figured out a way to detect this subtle signal.

So what is going on? First, a mental picture. In the image below, an observer at the left looks out to progressively higher redshift towards the right. The history of the universe unfolds from right to left.

cosmicdarkagesillustration — An observer’s view of the history of the universe. Nearby, at low redshift, we see mostly empty space sprinkled with galaxies. At some high redshift (z ~ 20?), the first stars formed, flooding the previously dark universe with UV photons that reionize the gas of the intergalactic medium. The backdrop of the CMB provides the ultimate limit to electromagnetic observations as it marks the boundary (at z = 1090) between a mostly transparent and completely opaque universe.

Pritchard & Loeb give a thorough and lucid account of the expected sequence of events. As the early universe expands, it cools. Initially, the thermal photon bath that we now observe as the CMB has enough energy to keep atoms ionized. The mean free path that a photon can travel before interacting with a charged particle in this early plasma is very short: the early universe is opaque like the interior of a thick cloud. At z = 1090, the temperature drops to the point that photons can no longer break protons and electrons apart. This epoch of recombination marks the transition from an opaque plasma to a transparent universe of neutral hydrogen and helium gas. The path length of photons becomes very long; those that we see as the CMB have traversed the length of the cosmos mostly unperturbed.

Immediately after recombination follows the dark ages. Sources of light have yet to appear. There is just neutral gas expanding into the future. This gas is mostly but not completely transparent. As CMB photons propagate through it, they are subject to absorption by the spin-flip transition of hydrogen, a subtle but, in principle, detectable effect: one should see redshifted absorption across the dark ages.

After some time – perhaps a few hundred million years? – the gas has had enough time to clump up enough to start to form the first structures. This first population of stars ends the dark ages and ushers in cosmic dawn. The photons they release into the vast intergalactic medium (IGM) of neutral gas interacts with it and heats it up, ultimately reionizing the entire universe. After this time the IGM is again a plasma, but one so thin (thanks to the expansion of the universe) that it remains transparent. Galaxies assemble and begin the long evolution characterized by the billions of years lived by the stars the contain.

This progression leads to the expectation of 21cm absorption twice: once during the dark ages, and again at cosmic dawn. There are three temperatures we need to keep track of to see how this happens: the radiation temperature T_γ, the kinetic temperature of the gas, T_k, and the spin temperature, T_S. The radiation temperature is that of the CMB, and scales as (1+z). The gas temperature is what you normally think of as a temperature, and scales approximately as (1+z)². The spin temperature describes the occupation of the quantum levels involved in the 21cm hyperfine transition. If that makes no sense to you, don’t worry: all that matters is that absorption can occur when the spin temperature is less than the radiation temperature. In general, it is bounded by T_k < T_S < T_γ.

The radiation temperature and gas temperature both cool as the universe expands. Initially, the gas remains coupled to the radiation, and these temperatures remain identical until decoupling around z ~ 200. After this, the gas cools faster than the radiation. The radiation temperature is extraordinarily well measured by CMB observations, and is simply T_γ = (2.725 K)(1+z). The gas temperature is more complicated, requiring the numerical solution of the Saha equation for a hydrogen-helium gas. Clever people have written codes to do this, like the widely-used RECFAST. In this way, one can build a table of how both temperatures depend on redshift in any cosmology one cares to specify.

This may sound complicated if it is the first time you’ve encountered it, but the physics is wonderfully simple. It’s just the thermal physics of the expanding universe, and the atomic physics of a simple gas composed of hydrogen and helium in known amounts. Different cosmologies specify different expansion histories, but these have only a modest (and calculable) effect on the gas temperature.

Wonderfully, the atomic physics of the 21cm transition is such that it couples to both the radiation and gas temperatures in a way that matters in the early universe. It didn’t have to be that way – most transitions don’t. Perhaps this is fodder for people who worry that the physics of our universe is fine-tuned.

There are two ways in which the spin temperature couples to that of the gas. During the dark ages, the coupling is governed simply by atomic collisions. By cosmic dawn collisions have become rare, but the appearance of the first stars provides UV radiation that drives the Wouthuysen–Field effect. Consequently, we expect to see two absorption troughs: one around z ~ 20 at cosmic dawn, and another at still higher redshift (z ~ 100) during the dark ages.

Observation of this signal has the potential to revolutionize cosmology like detailed observations of the CMB did. The CMB is a snapshot of the universe during the narrow window of recombination at z = 1090. In principle, one can make the same sort of observation with the 21cm line, but at each and every redshift where absorption occurs: z = 16, 17, 18, 19 during cosmic dawn and again at z = 50, 100, 150 during the dark ages, with whatever frequency resolution you can muster. It will be like having the CMB over and over and over again, each redshift providing a snapshot of the universe at a different slice in time.

The information density available from the 21cm signal is in principle quite large. Before we can make use of any of this information, we have to detect it first. Therein lies the rub. This is an incredibly weak signal – we have to be able to detect that the CMB is a little dimmer than it would have been – and we have to do it in the face of much stronger foreground signals from the interstellar medium of our Galaxy and from man-made radio interference here on Earth. Fortunately, though much brighter than the signal we seek, these foregrounds have a different frequency dependence, so it should be possible to sort out, in principle.

Saying a thing can be done and doing it are two different things. This is already a long post, so I will refrain from raving about the technical challenges. Lets just say it’s Real Hard.

Many experimentalists take that as a challenge, and there are a good number of groups working hard to detect the cosmic 21cm signal. EDGES appears to have done it, reporting the detection of the signal at cosmic dawn in February. Here some weasel words are necessary, as the foreground subtraction is a huge challenge, and we always hope to see independent confirmation of a new signal like this. Those words of caution noted, I have to add that I’ve had the chance to read up on their methods, and I’m really impressed. Unlike the BICEP claim to detect primordial gravitational waves that proved to be bogus after being rushed to press release before refereering, the EDGES team have done all manner of conceivable cross-checks on their instrumentation and analysis. Nor did they rush to publish, despite the importance of the result. In short, I get exactly the opposite vibe from BICEP, whose foreground subtraction was obviously wrong as soon as I laid eyes on the science paper. If EDGES proves to be wrong, it isn’t for want of doing things right. In the meantime, I think we’re obliged to take their result seriously, and not just hope it goes away (which seems to be the first reaction to the impossible).

Here is what EDGES saw at cosmic dawn:

nature25792-f2 — Fig. 2 from the EDGES detection paper. The dip, detected repeatedly in different instrumental configurations, shows a decrease in brightness temperature at radio frequencies, as expected from the 21cm absorbing some of the radiation from the CMB.

The unbelievable aspect of the EDGES observation is that it is too strong. Feeble as this signal is (a telescope brightness decrement of half a degree Kelvin), after subtracting foregrounds a thousand times stronger, it is twice as much as is possible in ΛCDM.

I made a quick evaluation of this, and saw that the observed signal could be achieved if the baryon fraction of the universe was high – basically, if cold dark matter did not exist. I have now had the time to make a more careful calculation, and publish some further predictions. The basic result from before stands: the absorption should be stronger without dark matter than with it.

The reason for this is simple. A universe full of dark matter decelerates rapidly at early times, before the acceleration of the cosmological constant kicks in. Without dark matter, the expansion more nearly coasts. Consequently, the universe is relatively larger from 10 < z < 1000, and the CMB photons have to traverse a larger path length to get here. They have to go about twice as far through the same density of hydrogen absorbers. It’s like putting on a second pair of sunglasses.

Quantitatively, the predicted absorption, both with dark matter and without, looks like:

predict21cmsignal — The predicted 21cm absorption with dark matter (red broken line) and without (blue line). Also shown (in grey) is the signal observed by EDGES.

The predicted absorption is consistent with the EDGES observation, within the errors, if there is no dark matter. More importantly, ΛCDM is not consistent with the data, at greater than 95% confidence. At cosmic dawn, I show the maximum possible signal. It could be weaker, depending on the spectra of the UV radiation emitted by the first stars. But it can’t be stronger. Taken at face value, the EDGES result is impossible in ΛCDM. If the observation is corroborated by independent experiments, ΛCDM as we know it will be falsified.

There have already been many papers trying to avoid this obvious conclusion. If we insist on retaining ΛCDM, the only way to modulate the strength of the signal is to alter the ratio of the radiation temperature to the gas temperature. Either we make the radiation “hotter,” or we make the gas cooler. If we allow ourselves this freedom, we can fit any arbitrary signal strength. This is ad hoc in the way that gives ad hoc a bad name.

We do not have this freedom – not really. The radiation temperature is measured in the CMB with great accuracy. Altering this would mess up the genuine success of ΛCDM in fitting the CMB. One could postulate an additional source, something that appears after recombination but before cosmic dawn to emit enough radio power throughout the cosmos to add to the radio brightness that is being absorbed. There is zero reason to expect such sources (what part of `cosmic dawn’ was ambiguous?) and no good way to make them at the right time. If they are primordial (as people love to imagine but are loathe to provide viable models for) then they’re also present at recombination: anything powerful enough to have the necessary effect will likely screw up the CMB.

Instead of magically increasing the radiation temperature, we might decrease the gas temperature. This seems no more plausible. The evolution of the gas temperature is a straightforward numerical calculation that has been checked by several independent codes. It has to be right at the time of recombination, or again, we mess up the CMB. The suggestions that I have heard seem mostly to invoke interactions between the gas and dark matter that offload some of the thermal energy of the gas into the invisible sink of the dark matter. Given how shy dark matter has been about interacting with normal matter in the laboratory, it seems pretty rich to imagine that it is eager to do so at high redshift. Even advocates of this scenario recognize its many difficulties.

For those who are interested, I cite a number of the scientific papers that attempt these explanations in my new paper. They all seem like earnest attempts to come to terms with what is apparently impossible. Many of these ideas also strike me as a form of magical thinking that stems from ΛCDM groupthink. After all, ΛCDM is so well established, any unexpected signal must be a sign of exciting new physics (on top of the new physics of dark matter and dark energy) rather than an underlying problem with ΛCDM itself.

The more natural interpretation is that the expansion history of the universe deviates from that predicted by ΛCDM. Simply taking away the dark matter gives a result consistent with the data. Though it did not occur to me to make this specific prediction a priori for an experiment that did not yet exist, all the necessary calculations had been done 15 years ago.

Using the same model, I make a genuine a priori prediction for the dark ages. For the specific NoCDM model I built in 2004, the 21cm absorption in the dark ages should again be about twice as strong as expected in ΛCDM. This seems fairly generic, but I know the model is not complete, so I wouldn’t be upset if it were not bang on.

I would be upset if ΛCDM were not bang on. The only thing that drives the signal in the dark ages is atomic scattering. We understand this really well. ΛCDM is now so well constrained by Planck that, if right, the 21cm absorption during the dark ages must follow the red line in the inset in the figure. The amount of uncertainty is not much greater than the thickness of the line. If ΛCDM fails this test, it would be a clear falsification, and a sign that we need to try something completely different.

Unfortunately, detecting the 21cm absorption signal during the dark ages is even harder than it is at cosmic dawn. At these redshifts (z ~ 100), the 21cm line (1420 MHz on your radio dial) is shifted beyond the ionospheric cutoff of the Earth’s atmosphere at 30 MHz. Frequencies this low cannot be observed from the ground. Worse, we have made the Earth itself a bright foreground contaminant of radio frequency interference.

Undeterred, there are multiple proposals to measure this signal by placing an antenna in space – in particular, on the far side of the moon, so that the moon shades the instrument from terrestrial radio interference. This is a great idea. The mere detection of the 21cm signal from the dark ages would be an accomplishment on par with the original detection of the CMB. It appears that it might also provide a decisive new way of testing our cosmological model.

There are further tests involving the shape of the 21cm signal, its power spectrum (analogous to the power spectrum of the CMB), how structure grows in the early ages of the universe, and how massive the neutrino is. But that’s enough for now.

Most likely beer. Or a cosmo. That’d be appropriate. I make a good pomegranate cosmo.

*Note that a variety of astronomical observations had established the concordance cosmology before Type Ia supernovae detected cosmic acceleration and well-resolved observations of the CMB found a flat cosmic geometry.

The dwarf galaxy NGC1052-DF2

A recently discovered dwarf galaxy designated NGC1052-DF2 has been in the news lately. Apparently a satellite of the giant elliptical NGC 1052, DF2 (as I’ll call it from here on out) is remarkable for having a surprisingly low velocity dispersion for a galaxy of its type. These results were reported in Nature last week by van Dokkum et al., and have caused a bit of a stir.

It is common for giant galaxies to have some dwarf satellite galaxies. As can be seen from the image published by van Dokkum et al., there are a number of galaxies in the neighborhood of NGC 1052. Whether these are associated physically into a group of galaxies or are chance projections on the sky depends on the distance to each galaxy.

NGC 1052 is listed by the NASA Extragalactic Database (NED) as having a recession velocity of 1510 km/s and a distance of 20.6 Mpc. The next nearest big beastie is NGC 1042, at 1371 km/s. The difference of 139 km/s is not much different from 115 km/s, which is the velocity that Andromeda is heading towards the Milky Way, so one could imagine that this is a group similar to the Local Group. Except that NED says the distance to NGC 1042 is 7.8 Mpc, so apparently it is a foreground object seen in projection.

Van Dokkum et al. assume DF2 and NGC 1052 are both about 20 Mpc distant. They offer two independent estimates of the distance, one consistent with the distance to NGC 1052 and the other more consistent with the distance to NGC 1042. Rather than wring our hands over this, I will trust their judgement and simply note, as they do, that the nearer distance would change many of their conclusions. The redshift is 1803 km/s, larger than either of the giants. It could still be a satellite of NGC 1052, as ~300 km/s is not unreasonable for an orbital velocity.

So why the big fuss? Unlike most galaxies in the universe, DF2 appears not to require dark matter. This is inferred from the measured velocity dispersion of ten globular clusters, which is 8.4 km/s. That’s fast to you and me, but rather sluggish on the scale of galaxies. Spread over a few kiloparsecs, that adds up to a dynamical mass about equal to what we expect for the stars, leaving little room for the otherwise ubiquitous dark matter.

This is important. If the universe is composed of dark matter, it should on occasion be possible to segregate the dark from the light. Tidal interactions between galaxies can in principle do this, so a galaxy devoid of dark matter would be good evidence that this happened. It would also be evidence against a modified gravity interpretation of the missing mass problem, because the force law is always on: you can’t strip it from the luminous matter the way you can dark matter. So ironically, the occasional galaxy lacking dark matter would constitute evidence that dark matter does indeed exist!

DF2 appears to be such a case. But how weird is it? Morphologically, it resembles the dwarf spheroidal satellite galaxies of the Local Group. I have a handy compilation of those (from Lelli et al.), so we can compute the mass-to-light ratio for all of these beasties in the same fashion, shown in the figure below. It is customary to refer quantities to the radius that contains half of the total light, which is 2.2 kpc for DF2.

dwarfMLdyn — The dynamical mass-to-light ratio for Local Group dwarf Spheroidal galaxies measured within their half-light radii, as a function of luminosity (left) and average surface brightness within the half-light radius (right). DF2 is the blue cross with low M/L. The other blue cross is Crater 2, a satellite of the Milky Way discovered after the compilation of Local Group dwarfs was made. The dotted line shows M/L = 2, which is a good guess for the stellar mass-to-light ratio. That DF2 sits on this line implies that stars are the only mass that’s there.

Perhaps the most obvious respect in which DF2 is a bit unusual relative to the dwarfs of the Local Group is that it is big and bright. Most nearby dwarfs have half light radii well below 1 kpc. After DF2, the next most luminous dwarfs is Fornax, which is a factor of 5 lower in luminosity.

DF2 is called an ultradiffuse galaxy (UDG), which is apparently newspeak for low surface brightness (LSB) galaxy. I’ve been working on LSB galaxies my entire career. While DF2 is indeed low surface brightness – the stars are spread thin – I wouldn’t call it ultra diffuse. It is actually one of the higher surface brightness objects of this type. Crater 2 and And XIX (the leftmost points in the right panel) are ultradiffuse.

Astronomers love vague terminology, and as a result often reinvent terms that already exist. Dwarf, LSB, UDG, have all been used interchangeably and with considerable slop. I was sufficiently put out by this that I tried to define some categories is the mid-90s. This didn’t catch on, but by my definition, DF2 is VLSB – very LSB, but only by a little – it is much closer to regular LSB than to extremely (ELSB). Crater 2 and And XIX, now they’re ELSB, being more diffuse than DF2 by 2 orders of magnitude.

SBdefinitiontable — Surface brightness categories from McGaugh (1996).

Whatever you call it, DF2 is low surface brightness, and LSB galaxies are always dark matter dominated. Always, at least among disk galaxies: here is the analogous figure for galaxies that rotate:

MLdynDisk — Dynamical mass-to-light ratios for rotationally supported disk galaxies, analogous to the plot above for pressure supported disks. The lower the surface brightness, the higher the mass discrepancy. The correlation with luminosity is secondary, as a result of the correlation between luminosity and surface brightness. From McGaugh (2014).

Pressure supported dwarfs generally evince large mass discrepancies as well. So in this regard, DF2 is indeed very unusual. So what gives?

Perhaps DF2 formed that way, without dark matter. This is anathema to everything we know about galaxy formation in ΛCDM cosmology. Dark halos have to form first, with baryons following.

Perhaps DF2 suffered one or more tidal interactions with NGC 1052. Sub-halos in simulations are often seen to be on highly radial orbits; perhaps DF2 has had its dark matter halo stripped away by repeated close passages. Since the stars reside deep in the center of the subhalo, they’re the last thing to be stripped away. So perhaps we’ve caught this one at that special time when the dark matter has been removed but the stars still remain.

This is improbable, but ought to happen once in a while. The bigger problem I see is that one cannot simply remove the dark matter halo like yanking a tablecloth and leaving the plates. The stars must respond to the change in the gravitational potential; they too must diffuse away. That might be a good way to make the galaxy diffuse, ultimately perhaps even ultradiffuse, but the observed motions are then not representative of an equilibrium situation. This is critical to the mass estimate, which must perforce assume an equilibrium in which the gravitational potential well of the galaxy is balanced against the kinetic motion of its contents. Yank away the dark matter halo, and the assumption underlying the mass estimate gets yanked with it. While such a situation may arise, it makes it very difficult to interpret the velocities: all tests are off. This is doubly true in MOND, in which dwarfs are even more susceptible to disruption.

Then there are the data themselves. Blaming the data should be avoided, but it does happen once in a while that some observation is misleading. In this case, I am made queasy by the fact that the velocity dispersion is estimated from only ten tracers. I’ve seen plenty of cases where the velocity dispersion changes in important ways when more data are obtained, even starting from more than 10 tracers. Andromeda II comes to mind as an example. Indeed, several people have pointed out that if we did the same exercise with Fornax, using its globular clusters as the velocity tracers, we’d get a similar answer to what we find in DF2. But we also have measurements of many hundreds of stars in Fornax, so we know that answer is wrong. Perhaps the same thing is happening with DF2? The fact that DF2 is an outlier from everything else we know empirically suggests caution.

Throwing caution and fact-checking to the wind, many people have been predictably eager to cite DF2 as a falsification of MOND. Van Dokkum et al. point out the the velocity dispersion predicted for this object by MOND is 20 km/s, more than a factor of two above their measured value. They make the MOND prediction for the case of an isolated object. DF2 is not isolated, so one must consider the external field effect (EFE).

The criterion by which to judge isolation in MOND is whether the acceleration due to the mutual self-gravity of the stars is less than the acceleration from an external source, in this case the host NGC 1052. Following the method outlined by McGaugh & Milgrom, and based on the stellar mass (adopting M/L=2 as both we and van Dokkum assume), I estimate an internal acceleration of DF2 to be g_in = 0.15 a₀. Here a₀ is the critical acceleration scale in MOND, 1.2 x 10^-10 m/s/s. Using this number and treating DF2 as isolated, I get the same 20 km/s van Dokkum et al. estimate.

Estimating the external field is more challenging. It depends on the mass of NGC 1052, and the separation between it and DF2. The projected separation at the assumed distance is 80 kpc. That is well within the range that the EFE is commonly observed to matter in the Local Group. It could be a bit further granted some distance along the line of sight, but if this becomes too large then the distance by association with NGC 1052 has to be questioned, and all bets are off. The mass of NGC 1052 is also rather uncertain, or at least I have heard wildly different values quoted in discussions about this object. Here I adopt 10¹¹ M_☉ as estimated by SLUGGS. To get the acceleration, I estimate the asymptotic rotation velocity we’d expect in MOND, V⁴ = a₀GM. This gives 200 km/s, which is conservative relative to the ~300 km/s quoted by van Dokkum et al. At a distance of 80 kpc, the corresponding external acceleration g_ex = 0.14 a₀. This is very uncertain, but taken at face value is indistinguishable from the internal acceleration. Consequently, it cannot be ignored: the calculation published by van Dokkum et al. is not the correct prediction for MOND.

The velocity dispersion estimator in MOND differs when g_ex < g_in and g_ex > g_in (see equations 2 and 3 of McGaugh & Milgrom). Strictly speaking, these apply in the limits where one or the other field dominates. When they are comparable, the math gets more involved (see equation 59 of Famaey & McGaugh). The input data are too uncertain to warrant an elaborate calculation for a blog, so I note simply that the amplitude of the mass discrepancy in MOND depends on how deep in the MOND regime a system is. That is, how far below the critical acceleration scale it is. The lower the acceleration, the larger the discrepancy. This is why LSB galaxies appear to be dark matter dominated; their low surface densities result in low accelerations.

For DF2, the absolute magnitude of the acceleration is approximately doubled by the presence of the external field. It is not as deep in the MOND regime as assumed in the isolated case, so the mass discrepancy is smaller, decreasing the MOND-predicted velocity dispersion by roughly the square root of 2. For a factor of 2 range in the stellar mass-to-light ratio (as in McGaugh & Milgrom), this crude MOND prediction becomes

σ = 14 ± 4 km/s.

Like any erstwhile theorist, I reserve the right to modify this prediction granted more elaborate calculations, or new input data, especially given the uncertainties in the distance and mass of the host. Indeed, we should consider the possibility of tidal disruption, which can happen in MOND more readily than with dark matter. Indeed, at one point I came very close to declaring MOND dead because the velocity dispersions of the ultrafaint dwarf galaxies were off, only realizing late in the day that MOND actually predicts that these things should be getting tidally disrupted (as is also expected, albeit somewhat differently, in ΛCDM), so that the velocity dispersions might not reflect the equilibrium expectation.

In DF2, the external field almost certainly matters. Barring wild errors of the sort discussed or unforeseen, I find it hard to envision the MONDian velocity dispersion falling outside the range 10 – 18 km/s. This is not as high as the 20 km/s predicted by van Dokkum et al. for an isolated object, nor as small as they measure for DF2 (8.4 km/s). They quote a 90% confidence upper limit of 10 km/s, which is marginally consistent with the lower end of the prediction (corresponding to M/L = 1). So we cannot exclude MOND based on these data.

That said, the agreement is marginal. Still, 90% is not very high confidence by scientific standards. Based on experience with such data, this likely overstates how well we know the velocity dispersion of DF2. Put another way, I am 90% confident that when better data are obtained, the measured velocity dispersion will increase above the 10 km/s threshold.

More generally, experience has taught me three things:

In matters of particle physics, do not bet against the Standard Model.
In matters cosmological, do not bet against ΛCDM.
In matters of galaxy dynamics, do not bet against MOND.

The astute reader will realize that these three assertions are mutually exclusive. The dark matter of ΛCDM is a bet that there are new particles beyond the Standard Model. MOND is a bet that what we call dark matter is really the manifestation of physics beyond General Relativity, on which cosmology is based. Which is all to say, there is still some interesting physics to be discovered.

Degenerating problemshift: a wedged paradigm in great tightness

Reading Merritt’s paper on the philosophy of cosmology, I was struck by a particular quote from Lakatos:

A research programme is said to be progressing as long as its theoretical growth anticipates its empirical growth, that is as long as it keeps predicting novel facts with some success (“progressive problemshift”); it is stagnating if its theoretical growth lags behind its empirical growth, that is as long as it gives only post-hoc explanations either of chance discoveries or of facts anticipated by, and discovered in, a rival programme (“degenerating problemshift”) (Lakatos, 1971, pp. 104–105).

The recent history of modern cosmology is rife with post-hoc explanations of unanticipated facts. The cusp-core problem and the missing satellites problem are prominent examples. These are explained after the fact by invoking feedback, a vague catch-all that many people agree solves these problems even though none of them agree on how it actually works.

FeedbackCartoonSilkMamon — Cartoon of the feedback explanation for the difference between the galaxy luminosity function (blue line) and the halo mass function (red line). From Silk & Mamon (2012).

There are plenty of other problems. To name just a few: satellite planes (unanticipated correlations in phase space), the emptiness of voids, and the early formation of structure (see section 4 of Famaey & McGaugh for a longer list and section 6 of Silk & Mamon for a positive spin on our list). Each problem is dealt with in a piecemeal fashion, often by invoking solutions that contradict each other while buggering the principle of parsimony.

It goes like this. A new observation is made that does not align with the concordance cosmology. Hands are wrung. Debate is had. Serious concern is expressed. A solution is put forward. Sometimes it is reasonable, sometimes it is not. In either case it is rapidly accepted so long as it saves the paradigm and prevents the need for serious thought. (“Oh, feedback does that.”) The observation is no longer considered a problem through familiarity and exhaustion of patience with the debate, regardless of how [un]satisfactory the proffered solution is. The details of the solution are generally forgotten (if ever learned). When the next problem appears the process repeats, with the new solution often contradicting the now-forgotten solution to the previous problem.

This has been going on for so long that many junior scientists now seem to think this is how science is suppose to work. It is all they’ve experienced. And despite our claims to be interested in fundamental issues, most of us are impatient with re-examining issues that were thought to be settled. All it takes is one bold assertion that everything is OK, and the problem is perceived to be solved whether it actually is or not.

8631e895433bc3d1fa87e3d857fc7500 — *“Is there any more?”*

That is the process we apply to little problems. The Big Problems remain the post hoc elements of dark matter and dark energy. These are things we made up to explain unanticipated phenomena. That we need to invoke them immediately casts the paradigm into what Lakatos called degenerating problemshift. Once we’re there, it is hard to see how to get out, given our propensity to overindulge in the honey that is the infinity of free parameters in dark matter models.

Note that there is another aspect to what Lakatos said about facts anticipated by, and discovered in, a rival programme. Two examples spring immediately to mind: the Baryonic Tully-Fisher Relation and the Radial Acceleration Relation. These are predictions of MOND that were unanticipated in the conventional dark matter picture. Perhaps we can come up with post hoc explanations for them, but that is exactly what Lakatos would describe as degenerating problemshift. The rival programme beat us to it.

In my experience, this is a good description of what is going on. The field of dark matter has stagnated. Experimenters look harder and harder for the same thing, repeating the same experiments in hope of a different result. Theorists turn knobs on elaborate models, gifting themselves new free parameters every time they get stuck.

On the flip side, MOND keeps predicting novel facts with some success, so it remains in the stage of progressive problemshift. Unfortunately, MOND remains incomplete as a theory, and doesn’t address many basic issues in cosmology. This is a different kind of unsatisfactory.

In the mean time, I’m still waiting to hear a satisfactory answer to the question I’ve been posing for over two decades now. Why does MOND get any predictions right? It has had many a priori predictions come true. Why does this happen? It shouldn’t. Ever.

Triton Station

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

Category: LCDM