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.

Neutrinos got mass!

In 1984, I heard Hans Bethe give a talk in which he suggested the dark matter might be neutrinos. This sounded outlandish – from what I had just been taught about the Standard Model, neutrinos were massless. Worse, I had been given the clear impression that it would screw everything up if they did have mass. This was the pervasive attitude, even though the solar neutrino problem was known at the time. This did not compute! so many of us were inclined to ignore it. But, I thought, in the unlikely event it turned out that neutrinos did have mass, surely that would be the answer to the dark matter problem.

Flash forward a few decades, and sure enough, neutrinos do have mass. Oscillations between flavors of neutrinos have been observed in both solar and atmospheric neutrinos. This implies non-zero mass eigenstates. We don’t yet know the absolute value of the neutrino mass, but the oscillations do constrain the separation between mass states (Δm_ν,21² = 7.53×10⁻⁵ eV² for solar neutrinos, and Δm_ν,31² = 2.44×10⁻³ eV² for atmospheric neutrinos).

Though the absolute values of the neutrino mass eigenstates are not yet known, there are upper limits. These don’t allow enough mass to explain the cosmological missing mass problem. The relic density of neutrinos is

Ω_νh² = ∑m_ν/(93.5 eV)

In order to make up the dark matter density (Ω ≈ 1/4), we need ∑m_ν ≈ 12 eV. The experimental upper limit on the electron neutrino mass is m_ν < 2 eV. There are three neutrino mass eigenstates, and the difference in mass between them is tiny, so ∑m_ν < 6 eV. Neutrinos could conceivably add up to more mass than baryons, but they cannot add up to be the dark matter.

In recent years, I have started to hear the assertion that we have already detected dark matter, with neutrinos given as the example. They are particles with mass that only interact with us through the weak nuclear force and gravity. In this respect, they are like WIMPs.

Here the equivalence ends. Neutrinos are Standard Model particles that have been known for decades. WIMPs are hypothetical particles that reside in a hypothetical supersymmetric sector beyond the Standard Model. Conflating the two to imply that WIMPs are just as natural as neutrinos is a false equivalency.

That said, massive neutrinos might be one of the few ways in which hierarchical cosmogony, as we currently understand it, is falsifiable. Whatever the dark matter is, we need it to be dynamically cold. This property is necessary for it to clump into dark matter halos that seed galaxy formation. Too much hot (relativistic) dark matter (neutrinos) suppresses structure formation. A nascent dark matter halo is nary a speed bump to a neutrino moving near the speed of light: if those fast neutrinos carry too much mass, they erase structure before it can form.

One of the great successes of ΛCDM is its explanation of structure formation: the growth of large scale structure from the small fluctuations in the density field at early times. This is usually quantified by the power spectrum – in the CMB at z > 1000 and from the spatial distribution of galaxies at z = 0. This all works well provided the dominant dark mass is dynamically cold, and there isn’t too much hot dark matter fighting it.

t16_galaxy_power_spectrum — The power spectrum from the CMB (low frequency/large scales) and the galaxy distribution (high frequency/”small” scales). Adapted from Whittle.

How much is too much? The power spectrum puts strong limits on the amount of hot dark matter that is tolerable. The upper limit is ∑m_ν < 0.12 eV. This is an order of magnitude stronger than direct experimental constraints.

Usually, it is assumed that the experimental limit will eventually come down to the structure formation limit. That does seem likely, but it is also conceivable that the neutrino mass has some intermediate value, say m_ν ≈ 1 eV. Such a result, were it to be obtained experimentally, would falsify the current CDM cosmogony.

Such a result seems unlikely, of course. Shooting for a narrow window such as the gap between the current cosmological and experimental limits is like drawing to an inside straight. It can happen, but it is unwise to bet the farm on it.

It should be noted that a circa 1 eV neutrino would have some desirable properties in an MONDian universe. MOND can form large scale structure, much like CDM, but it does so faster. This is good for clearing out the voids and getting structure in place early, but it tends to overproduce structure by z = 0. An admixture of neutrinos might help with that. A neutrino with an appreciable mass would also help with the residual mass discrepancy MOND suffers in clusters of galaxies.

If experiments measure a neutrino mass in excess of the cosmological limit, it would be powerful motivation to consider MOND-like theories as a driver of structure formation. If instead the neutrino does prove to be tiny, ΛCDM will have survived another test. That wouldn’t falsify MOND (or really have any bearing on it), but it would remove one potential “out” for the galaxy cluster problem.

Tiny though they be, neutrinos got mass! And it matters!

LCDM has met the enemy, and it is itself

David Merritt recently published the article “Cosmology and convention” in Studies in History and Philosophy of Science. This article is remarkable in many respects. For starters, it is rare that a practicing scientist reads a paper on the philosophy of science, much less publishes one in a philosophy journal.

I was initially loathe to start reading this article, frankly for fear of boredom: me reading about cosmology and the philosophy of science is like coals to Newcastle. I could not have been more wrong. It is a genuine page turner that should be read by everyone interested in cosmology.

I have struggled for a long time with whether dark matter constitutes a falsifiable scientific hypothesis. It straddles the border: specific dark matter candidates (e.g., WIMPs) are confirmable – a laboratory detection is both possible and plausible – but the concept of dark matter can never be excluded. If we fail to find WIMPs in the range of mass-cross section parameters space where we expected them, we can change the prediction. This moving of the goal post has already happened repeatedly.

wimplimits2017 — The cross-section vs. mass parameter space for WIMPs. The original, “natural” weak interaction cross-section (10^-39) was excluded long ago, as were early attempts to map out the theoretically expected parameter space (upper pink region). Later predictions drifted to progressively lower cross-sections. These evaded experimental limits at the time, and confident predictions were made that the dark matter *would* be found. More recent data show otherwise: the gray region is excluded by PandaX (2016). [This plot was generated with the help of DMTools hosted at Brown.]

I do not find it encouraging that the goal posts keep moving. This raises the question, how far can we go? Arbitrarily low cross-sections can be extracted from theory if we work at it hard enough. How hard should we work? That is, what criteria do we set whereby we decide the WIMP hypothesis is mistaken?

There has to be some criterion by which we would consider the WIMP hypothesis to be falsified. Without such a criterion, it does not satisfy the strictest definition of a scientific hypothesis. If at some point we fail to find WIMPs and are dissatisfied with the theoretical fine-tuning required to keep them hidden, we are free to invent some other dark matter candidate. No WIMPs? Must be axions. Not axions? Would you believe light dark matter? [Worst. Name. Ever.] And so on, ad infinitum. The concept of dark matter is not falsifiable, even if specific dark matter candidates are subject to being made to seem very unlikely (e.g., brown dwarfs).

Faced with this situation, we can consult the philosophy science. Merritt discusses how many of the essential tenets of modern cosmology follow from what Popper would term “conventionalist stratagems” – ways to dodge serious consideration that a treasured theory is threatened. I find this a compelling terminology, as it formalizes an attitude I have witnessed among scientists, especially cosmologists, many times. It was put more colloquially by J.K. Galbraith:

“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.”

Boiled down (Keuth 2005), the conventionalist strategems Popper identifies are

ad hoc hypotheses
modification of ostensive definitions
doubting the reliability of the experimenter
doubting the acumen of the theorist

These are stratagems to be avoided according to Popper. At the least they are pitfalls to be aware of, but as Merritt discusses, modern cosmology has marched down exactly this path, doing each of these in turn.

The ad hoc hypotheses of ΛCDM are of course Λ and CDM. Faced with the observation of a metric that cannot be reconciled with the prior expectation of a decelerating expansion rate, we re-invoke Einstein’s greatest blunder, Λ. We even generalize the notion and give it a fancy new name, dark energy, which has the convenient property that it can fit any observed set of monotonic distance-redshift pairs. Faced with an excess of gravitational attraction over what can be explained by normal matter, we invoke non-baryonic dark matter: some novel form of mass that has no place in the standard model of particle physics, has yet to show any hint of itself in the laboratory, and cannot be decisively excluded by experiment.

We didn’t accept these ad hoc add-ons easily or overnight. Persuasive astronomical evidence drove us there, but all these data really show is that something dire is wrong: General Relativity plus known standard model particles cannot explain the universe. Λ and CDM are more a first guess than a final answer. They’ve been around long enough that they have become familiar, almost beyond doubt. Nevertheless, they remain unproven ad hoc hypotheses.

The sentiment that is often asserted is that cosmology works so well that dark matter and dark energy must exist. But a more conservative statement would be that our present understanding of cosmology is correct if and only if these dark entities exist. The onus is on us to detect dark matter particles in the laboratory.

That’s just the first conventionalist stratagem. I could given many examples of violations of the other three, just from my own experience. That would make for a very long post indeed.

Instead, you should go read Merritt’s paper. There are too many things there to discuss, at least in a single post. You’re best going to the source. Be prepared for some cognitive dissonance.

Crater 2: the Bullet Cluster of LCDM

Recently I have been complaining about the low standards to which science has sunk. It has become normal to be surprised by an observation, express doubt about the data, blame the observers, slowly let it sink in, bicker and argue for a while, construct an unsatisfactory model that sort-of, kind-of explains the surprising data but not really, call it natural, then pretend like that’s what we expected all along. This has been going on for so long that younger scientists might be forgiven if they think this is how science is suppose to work. It is not.

At the root of the scientific method is hypothesis testing through prediction and subsequent observation. Ideally, the prediction comes before the experiment. The highest standard is a prediction made before the fact in ignorance of the ultimate result. This is incontrovertibly superior to post-hoc fits and hand-waving explanations: it is how we’re suppose to avoid playing favorites.

I predicted the velocity dispersion of Crater 2 in advance of the observation, for both ΛCDM and MOND. The prediction for MOND is reasonably straightforward. That for ΛCDM is fraught. There is no agreed method by which to do this, and it may be that the real prediction is that this sort of thing is not possible to predict.

The reason it is difficult to predict the velocity dispersions of specific, individual dwarf satellite galaxies in ΛCDM is that the stellar mass-halo mass relation must be strongly non-linear to reconcile the steep mass function of dark matter sub-halos with their small observed numbers. This is closely related to the M_*-M_halo relation found by abundance matching. The consequence is that the luminosity of dwarf satellites can change a lot for tiny changes in halo mass.

apj374168f11_lr — Fig. 11 from Tollerud et al. (2011, ApJ, 726, 108). The width of the bands illustrates the minimal scatter expected between dark halo and measurable properties. A dwarf of a given luminosity could reside in dark halos differing be two decades in mass, with a corresponding effect on the velocity dispersion.

Long story short, the nominal expectation for ΛCDM is a lot of scatter. Photometrically identical dwarfs can live in halos with very different velocity dispersions. The trend between mass, luminosity, and velocity dispersion is so weak that it might barely be perceptible. The photometric data should not be predictive of the velocity dispersion.

It is hard to get even a ballpark answer that doesn’t make reference to other measurements. Empirically, there is some correlation between size and velocity dispersion. This “predicts” σ = 17 km/s. That is not a true theoretical prediction; it is just the application of data to anticipate other data.

Abundance matching relations provide a highly uncertain estimate. The first time I tried to do this, I got unphysical answers (σ = 0.1 km/s, which is less than the stars alone would cause without dark matter – about 0.5 km/s). The application of abundance matching requires extrapolation of fits to data at high mass to very low mass. Extrapolating the M_*-M_halo relation over many decades in mass is very sensitive to the low mass slope of the fitted relation, so it depends on which one you pick.

Since my first pick did not work, lets go with the value suggested to me by James Bullock: σ = 11 km/s. That is the mid-value (the blue lines in the figure above); the true value could easily scatter higher or lower. Very hard to predict with any precision. But given the luminosity and size of Crater 2, we expect numbers like 11 or 17 km/s.

The measured velocity dispersion is σ = 2.7 ± 0.3 km/s.

This is incredibly low. Shockingly so, considering the enormous size of the system (1 kpc half light radius). The NFW halos predicted by ΛCDM don’t do that.

To illustrate how far off this is, I have adopted this figure from Boylan-Kolchin et al. (2012).

mbkplusdwarfswcraterii — Fig. 1 of MNRAS, 422, 1203 illustrating the “too big to fail” problem: observed dwarfs have lower velocity dispersions than sub-halos that must exist and should host similar or even more luminous dwarfs that apparently do not exist. I have had to extend the range of the original graph to lower velocities in order to include Crater 2.

Basically, NFW halos, including the sub-halos imagined to host dwarf satellite galaxies, have rotation curves that rise rapidly and stay high in proportion to the cube root of the halo mass. This property makes it very challenging to explain a low velocity at a large radius: exactly the properties observed in Crater 2.

Lets not fail to appreciate how extremely wrong this is. The original version of the graph above stopped at 5 km/s. It didn’t extend to lower values because they were absurd. There was no reason to imagine that this would be possible. Indeed, the point of their paper was that the observed dwarf velocity dispersions were already too low. To get to lower velocity, you need an absurdly low mass sub-halo – around 10⁷ M_☉. In contrast, the usual inference of masses for sub-halos containing dwarfs of similar luminosity is around 10⁹ M_☉to 10¹⁰ M_☉. So the low observed velocity dispersion – especially at such a large radius – seems nigh on impossible.

More generally, there is no way in ΛCDM to predict the velocity dispersions of particular individual dwarfs. There is too much intrinsic scatter in the highly non-linear relation between luminosity and halo mass. Given the photometry, all we can say is “somewhere in this ballpark.” Making an object-specific prediction is impossible.

Except that it is possible. I did it. In advance.

The predicted velocity dispersion is σ = 2.1 +0.9/-0.6 km/s.

I’m an equal opportunity scientist. In addition to ΛCDM, I also considered MOND. The successful prediction is that of MOND. (The quoted uncertainty reflects the uncertainty in the stellar mass-to-light ratio.) The difference is that MOND makes a specific prediction for every individual object. And it comes true. Again.

MOND is a funny theory. The amplitude of the mass discrepancy it induces depends on how low the acceleration of a system is. If Crater 2 were off by itself in the middle of intergalactic space, MOND would predict it should have a velocity dispersion of about 4 km/s.

But Crater 2 is not isolated. It is close enough to the Milky Way that there is an additional, external acceleration imposed by the Milky Way. The net result is that the acceleration isn’t quite as low as it would be were Crater 2 al by its lonesome. Consequently, the predicted velocity dispersion is a measly 2 km/s. As observed.

In MOND, this is called the External Field Effect (EFE). Theoretically, the EFE is rather disturbing, as it breaks the Strong Equivalence Principle. In particular, Local Position Invariance in gravitational experiments is violated: the velocity dispersion of a dwarf satellite depends on whether it is isolated from its host or not. Weak equivalence (the universality of free fall) and the Einstein Equivalence Principle (which excludes gravitational experiments) may still hold.

We identified several pairs of photometrically identical dwarfs around Andromeda. Some are subject to the EFE while others are not. We see the predicted effect of the EFE: isolated dwarfs have higher velocity dispersions than their twins afflicted by the EFE.

If it is just a matter of sub-halo mass, the current location of the dwarf should not matter. The velocity dispersion certainly should not depend on the bizarre MOND criterion for whether a dwarf is affected by the EFE or not. It isn’t a simple distance-dependency. It depends on the ratio of internal to external acceleration. A relatively dense dwarf might still behave as an isolated system close to its host, while a really diffuse one might be affected by the EFE even when very remote.

When Crater 2 was first discovered, I ground through the math and tweeted the prediction. I didn’t want to write a paper for just one object. However, I eventually did so because I realized that Crater 2 is important as an extreme example of a dwarf so diffuse that it is affected by the EFE despite being very remote (120 kpc from the Milky Way). This is not easy to reproduce any other way. Indeed, MOND with the EFE is the only way that I am aware of whereby it is possible to predict, in advance, the velocity dispersion of this particular dwarf.

If I put my ΛCDM hat back on, it gives me pause that any method can make this prediction. As discussed above, this shouldn’t be possible. There is too much intrinsic scatter in the halo mass-luminosity relation.

If we cook up an explanation for the radial acceleration relation, we still can’t make this prediction. The RAR fit we obtained empirically predicts 4 km/s. This is indistinguishable from MOND for isolated objects. But the RAR itself is just an empirical law – it provides no reason to expect deviations, nor how to predict them. MOND does both, does it right, and has done so before, repeatedly. In contrast, the acceleration of Crater 2 is below the minimum allowed in ΛCDM according to Navarro et al.

For these reasons I consider Crater 2 to be the bullet cluster of ΛCDM. Just as the bullet cluster seems like a straight-up contradiction to MOND, so too does Crater 2 for ΛCDM. It is something ΛCDM really can’t do. The difference is that you can just look at the bullet cluster. With Crater 2 you actually have to understand MOND as well as ΛCDM, and think it through.

So what can we do to save ΛCDM?

Whatever it takes, per usual.

One possibility is that Crater II may represent the “bright” tip of the extremely low surface brightness “stealth” fossils predicted by Bovill & Ricotti. Their predictions are encouraging for getting the size and surface brightness in the right ballpark. But I see no reason in this context to expect such a low velocity dispersion. They anticipate dispersions consistent with the ΛCDM discussion above, and correspondingly high mass-to-light ratios that are greater than observed for Crater 2 (M/L ≈ 10⁴ rather than ~50).

A plausible suggestion I heard was from James Bullock. While noting that reionization should preclude the existence of galaxies in halos below 5 km/s, as we need for Crater 2, he suggested that tidal stripping could reduce an initially larger sub-halo to this point. I am dubious about this, as my impression from the simulations of Penarrubia was that the outer regions of the sub-halo were stripped first while leaving the inner regions (where the NFW cusp predicts high velocity dispersions) largely intact until near complete dissolution. In this context, it is important to bear in mind that the low velocity dispersion of Crater 2 is observed at large radii (1 kpc, not tens of pc). Still, I can imagine ways in which this might be made to work in this particular case, depending on its orbit. Tony Sohn has an HST program to measure the proper motion; this should constrain whether the object has ever passed close enough to the center of the Milky Way to have been tidally disrupted.

Josh Bland-Hawthorn pointed out to me that he made simulations that suggest a halo with a mass as low as 10⁷ M_☉could make stars before reionization and retain them. This contradicts much of the conventional wisdom outlined above because they find a much lower (and in my opinion, more realistic) feedback efficiency for supernova feedback than assumed in most other simulations. If this is correct (as it may well be!) then it might explain Crater 2, but it would wreck all the feedback-based explanations given for all sorts of other things in ΛCDM, like the missing satellite problem and the cusp-core problem. We can’t have it both ways.

maxresdefault — Without super-efficient supernova feedback, the Local Group would be filled with a million billion ultrafaint dwarf galaxies!

I’m sure people will come up with other clever ideas. These will inevitably be ad hoc suggestions cooked up in response to a previously inconceivable situation. This will ring hollow to me until we explain why MOND can predict anything right at all.

In the case of Crater 2, it isn’t just a matter of retrospectively explaining the radial acceleration relation. One also has to explain why exceptions to the RAR occur following the very specific, bizarre, and unique EFE formulation of MOND. If I could do that, I would have done so a long time ago.

No matter what we come up with, the best we can hope to do is a post facto explanation of something that MOND predicted correctly in advance. Can that be satisfactory?

Triton Station

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

Category: LCDM