Dwarf Galaxies on the Shoulders of Giants

Dwarf Galaxies on the Shoulders of Giants

The week of June 5, 2017, we held a workshop on dwarf galaxies and the dark matter problem. The workshop was attended by many leaders in the field – giants of dwarf galaxy research. It was held on the campus of Case Western Reserve University and supported by the John Templeton Foundation. It resulted in many fascinating discussions which I can’t possibly begin to share in full here, but I’ll say a few words.

Dwarf galaxies are among the most dark matter dominated objects in the universe. Or, stated more properly, they exhibit the largest mass discrepancies. This makes them great places to test theories of dark matter and modified gravity. By the end, we had come up with a few important tests for both ΛCDM and MOND. A few of these we managed to put on a white board. These are hardly a complete list, but provide a basis for discussion.

First, ΛCDM.

A few issues for ΛCDM identified during the workshop.

UFDs in field: Over the past few years, a number of extremely tiny dwarf galaxies have been identified as satellites of the Milky Way galaxy. These “ultrafaint dwarfs” are vaguely defined as being fainter than 100,000 solar luminosities, with the smallest examples having only a few hundred stars. This is absurdly small by galactic standards, having the stellar content of individual star clusters within the Milky Way. Indeed, it is not obvious to me that all of the ultrafaint dwarfs deserve to be recognized as dwarf galaxies, as some may merely be fragmentary portions of the Galactic stellar halo composed of stars coincident in phase space. Nevertheless, many may well be stellar systems external to the Milky Way that orbit it as dwarf satellites.

That multitudes of minuscule dark matter halos exist is a fundamental prediction of the ΛCDM cosmogony. These should often contain ultrafaint dwarf galaxies, and not only as satellites of giant galaxies like the Milky Way. Indeed, one expects to see many ultrafaints in the “field” beyond the orbital vicinity of the Milky Way where we have found them so far. These are predicted to exist in great numbers, and contain uniformly old stars. The “old stars” portion of the prediction stems from the reionization of the universe impeding star formation in the smallest dark matter halos. Upcoming surveys like LSST should provide a test of this prediction.

From an empirical perspective, I do expect that we will continue to discover galaxies of ever lower luminosity and surface brightness. In the field, I expect that these will be predominantly gas rich dwarfs like Leo P rather than gas-free, old stellar systems like the satellite ultrafaints. My expectation is an extrapolation of past experience, not a theory-specific prediction.

No Large Cores: Many of the simulators present at the workshop showed that if the energy released by supernovae was well directed, it could reshape the steep (‘cuspy’) interior density profiles of dark matter halos into something more like the shallow (‘cored’) interiors that are favored by data. I highlight the if because I remain skeptical that supernova energy couples as strongly as required and assumed (basically 100%). Even assuming favorable feedback, there seemed to be broad (in not unanimous) consensus among the simulators present that at sufficiently low masses, not enough stars would form to produce the requisite energy. Consequently, low mass halos should not have shallow cores, but instead retain their primordial density cusps. Hence clear measurement of a large core in a low mass dwarf galaxy (stellar mass < 1 million solar masses) would be a serious problem. Unfortunately, I’m not clear that we quantified “large,” but something more than a few hundred parsecs should qualify.

Radial Orbit for Crater 2: Several speakers highlighted the importance of the recently discovered dwarf satellite Crater 2. This object has a velocity dispersion that is unexpectedly low in ΛCDM, but was predicted by MOND. The “fix” in ΛCDM is to imagine that Crater 2 has suffered a large amount of tidal stripping by a close passage of the Milky Way. Hence it is predicted to be on a radial orbit (one that basically just plunges in and out). This can be tested by measuring the proper motion of its stars with Hubble Space Telescope, for which there exists a recently approved program.

DM Substructures: As noted above, there must exist numerous low mass dark matter halos in the cold dark matter cosmogony. These may be detected as substructure in the halos of larger galaxies by means of their gravitational lensing even if they do not contain dwarf galaxies. Basically, a lumpy dark matter halo bends light in subtly but detectably different ways from a smooth halo.

No Wide Binaries in UFDs: As a consequence of dynamical friction against the background dark matter, binary stars cannot remain at large separations over a Hubble time: their orbits should decay. In the absence of dark matter, this should not happen (it cannot if there is nowhere for the orbital energy to go, like into dark matter particles). Thus the detection of a population of widely separated binary stars would be problematic. Indeed, Pavel Kroupa argued that the apparent absence of strong dynamical friction already excludes particle dark matter as it is usually imagined.

Short dynamical times/common mergers: This is related to dynamical friction. In the hierarchical cosmogony of cold dark matter, mergers of halos (and the galaxies they contain) must be frequent and rapid. Dark matter halos are dynamically sticky, soaking up the orbital energy and angular momentum between colliding galaxies to allow them to stick and merge. Such mergers should go to completion on fairly short timescales (a mere few hundred million years).


A few distinctive predictions for MOND were also identified.


Tangential Orbit for Crater 2: In contrast to ΛCDM, we expect that the `feeble giant’ Crater 2 could not survive a close encounter with the Milky Way. Even at its rather large distance of 120 kpc from the Milky Way, it is so feeble that it is not immune from the external field of its giant host. Consequently, we expect that Crater 2 must be on a more nearly circular orbit, and not on a radial orbit as suggested in ΛCDM. The orbit does not need to be perfectly circular of course, but is should be more tangential than radial.

This provides a nice test that distinguishes between the two theories. Either the orbit of Crater 2 is more radial or more tangential. Bear in mind that Crater 2 already constitutes a problem for ΛCDM. What we’re discussing here is how to close what is basically a loophole whereby we can excuse an otherwise unanticipated result in ΛCDM.

EFE: The External Field Effect is a unique prediction of MOND that breaks the strong equivalence principle. There is already clear if tentative evidence for the EFE in the dwarf satellite galaxies around Andromeda. There is no equivalent to the EFE in ΛCDM.

I believe the question mark was added on the white board to permit the logical if unlikely possibility that one could write a MOND theory with an undetectably small EFE.

Position of UFDs on RAR: We chose to avoid making the radial acceleration relation (RAR) a focus of the meeting – there was quite enough to talk about as it was – but it certainly came up. The ultrafaint dwarfs sit “too high” on the RAR, an apparent problem for MOND. Indeed, when I first worked on this subject with Joe Wolf, I initially thought this was a fatal problem for MOND.

My initial thought was wrong. This is not a problem for MOND. The RAR applies to systems in dynamical equilibrium. There is a criterion in MOND to check whether this essential condition may be satisfied. Basically all of the ultrafaints flunk this test. There is no reason to think they are in dynamical equilibrium, so no reason to expect that they should be exactly on the RAR.

Some advocates of ΛCDM seemed to think this was a fudge, a lame excuse morally equivalent to the fudges made in ΛCDM that its critics complain about. This is a false equivalency that reminds me of this cartoon:

I dare ya to step over this line!

The ultrafaints are a handful of the least-well measured galaxies on the RAR. Before we obsess about these, it is necessary to provide a satisfactory explanation for the more numerous, much better measured galaxies that establish the RAR in the first place. MOND does this. ΛCDM does not. Holding one theory to account for the least reliable of measurements before holding another to account for everything up to that point is like, well, like the cartoon… I could put an NGC number to each of the lines Bugs draws in the sand.

Long dynamical times/less common mergers: Unlike ΛCDM, dynamical friction should be relatively ineffective in MOND. It lacks the large halos of dark matter that act as invisible catchers’ mitts to make galaxies stick and merge. Personally, I do not think this is a great test, because we are a long way from understanding dynamical friction in MOND.

Non-evolution with redshift: If the Baryonic Tully-Fisher relation and the RAR are indeed the consequence of MOND, then their form is fixed by the theory. Consequently, their slope shouldn’t evolve with time. Conceivably their normalization might (e.g., the value of a0 could in principle evolve). Some recent data for high redshift galaxies place constraints on such evolution, but reports on these data are greatly exaggerated.

These are just a few of the topics discussed at the workshop, and all of those are only a few of the issues that matter to the bigger picture. While the workshop was great in every respect, perhaps the best thing was that it got people from different fields/camps/perspectives talking. That is progress.

I am grateful for progress, but I must confess that to me it feels excruciatingly slow. Models of galaxy formation in the context of ΛCDM have made credible steps forward in addressing some of the phenomenological issues that concern me. Yet they still seem to me to be very far from where they need to be. In particular, there seems to be no engagement with the fundamental question I have posed here before, and that I posed at the beginning of the workshop: Why does MOND get any predictions right?

Degenerating problemshift: a wedged paradigm in great tightness

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.

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.

“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!

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 (Δmν,212 = 7.53×10−5 eV2 for solar neutrinos, and Δmν,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

Ωνh2 = ∑mν/(93.5 eV)

In order to make up the dark matter density (Ω ≈ 1/4), we need ∑mν ≈ 12 eV. The experimental upper limit on the electron neutrino mass is mν < 2 eV. There are three neutrino mass eigenstates, and the difference in mass between them is tiny, so ∑mν < 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.

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 ∑mν < 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 mν ≈ 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!

Emergent Gravity hits a pothole

Emergent Gravity hits a pothole

We have in MOND a formula that has had repeated predictive successes. Many of these have been true a priori predictions, like the absolute nature of the baryonic Tully-Fisher relation, the large mass discrepancies evinced by low surface brightness galaxies, and the velocity dispersions of  many individual dwarf Spheroidal galaxies like Crater 2. I don’t see how these can be an accident. But what we lack is an underlying theoretical basis for the observed MONDian phenomenology: Why does this happen?

One apparently promising idea is the emergent gravity hypothesis by Erik Verlinde. Gravity is not a fundamental force so much as a consequence of microscopic entanglement. This manifests on scales comparable to the Hubble horizon, in particular, with an acceleration of order the speed of light times the Hubble expansion rate. This is  close to the acceleration scale of MOND.

An early test of emergent gravity was provided by weak gravitational lensing. It does remarkably well at predicting the observed lensing signal with no free parameters. This is promising – perhaps we finally have a physical basis for MOND. Indeed, as Milgrom points out, the equivalent success was already known in MOND.

The weak lensing signal for galaxies stacked in several mass bins from Brouwer et al. (2016). The straight line predicted by emergent gravity (EG) is a good fit to the data with no free parameters to adjust.

Weak lensing occurs deep in the MOND regime, at very low accelerations far from the lensing galaxy. In that regard, the results of Brouwer et al. can be seen as an extension of the radial acceleration relation to much lower accelerations. In this limit, it is fair to treat galaxies as point masses – hence the similarity of the solid and dashed lines in the figure above.

For rotation curves, it is not fair to approximate galaxies as point masses. Rotation curves are observed in the midst of the stellar and gaseous mass distribution. One must take account of the detailed distribution of baryons to treat the problem properly. This is something MOND is very successful at.

Emergent gravity converges to the same limit as MOND in the point mass case, which holds for any mass distribution once you get far enough away. It is not identical for finite mass distributions. When one solves the equation of emergent gravity for a finite mass distribution, you get one term that looks like MOND, which gives the success noted above. But you also get an additional term that depends on the gradient of the mass distribution, dM/dr.

The additional term that emerges for extended mass distributions in emergent gravity lead to different predictions than MOND. This is good, in that it makes the theories distinguishable. This is bad, in that MOND already provides good fits to rotation curves. Additional terms are likely to mess that up.

And so it does. Two independent studies recently come to this conclusion: one including myself (Lelli et al. 2017) and another by Hees et al. (2017). The studies differ in their approach. We show that the additional term in emergent gravity leads to the prediction of a larger mass discrepancy than in MOND, driving one to abnormally low stellar mass-to-light ratios and degrading the empirical radial acceleration relation. Hees et al. make detailed rotation curve fits, showing that the dM/dr term over-amplifies bumps & wiggles in the rotation curve. It has always been intriguing that MOND gets these right: this is a non-trivial success to reproduce.

Rotation curves predicted for various exponential disks (top row) by Newton (dotted line), MOND (red dashed line), and emergent gravity (solid line). Note that emergent gravity predicts more of a hump at the peak of the rotation curve, leading to the hook in the corresponding radial acceleration relation (lower panels). From Lelli et al. (2017).

The situation looks bad for emergent gravity. One caveat is that at present we only have solutions for emergent gravity in the case of a spherical cow. Conceivably a better treatment of the geometry would change the result, but it won’t eliminate the dM/dr term. So this seems unlikely to help with the fundamental problem: this term needs not to exist.

Perhaps emergent gravity is a clue to what ultimately is going on – a single step in the right direction. Or perhaps the similarity to MOND is misleading. For now, the search for a satisfactory explanation for the observed phenomenology continues.

Crater 2: the Bullet Cluster of LCDM

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*-Mhalo relation found by abundance matching. The consequence is that the luminosity of dwarf satellites can change a lot for tiny changes in halo mass.

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*-Mhalo 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).

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 Mto 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).

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 Mcould 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.

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?

Crater 2: prediction verified.

Crater 2: prediction verified.

The arXiv brought an early Xmas gift in the form of a measurement of the velocity dispersion of Crater 2. Crater 2 is an extremely diffuse dwarf satellite of the Milky Way. Upon its discovery, I realized there was an opportunity to predict its velocity dispersion based on the reported photometry. The fact that it is very large (half light radius a bit over 1 kpc) and relatively far from the Milky Way (120 kpc) make it a unique and critical case. I will expand on that in another post, or you could read the paper. But for now:

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

This prediction appeared in press in advance of the measurement (ApJ, 832, L8). The uncertainty reflects the uncertainty in the mass-to-light ratio.

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

as reported by Caldwell et al.

Isn’t that how science is suppose to work? Make the prediction first? Not just scramble to explain it after the fact?

Pulp Science

Pulp Science


Vincent: Want to talk about MOND?

Jules: No man, I don’t consider MOND.

Vincent: Are you biased?

Jules: Nah, I ain’t biased, I just don’t dig MOND, that’s all.

Vincent: Why not?

Jules: MOND is an ugly theory. I don’t consider ugly theories.

Vincent: MOND makes predictions that come true. Fits galaxy data gooood.

Jules: Hey, MOND may fit every galaxy in the universe, but I’d never know ’cause I wouldn’t consider the ugly theory. MOND has no generally covariant extension. That’s an ugly theory. I ain’t considering nothin’ that ain’t got a proper cosmology.

Vincent: How about ΛCDM? ΛCDM has lots of small scale problems.

Jules: I don’t care about small scale problems.

Vincent: Yeah, but do you consider ΛCDM to be an ugly theory?

Jules: I wouldn’t go so far as to call ΛCDM ugly, but it’s definitely fine-tuned. But, ΛCDM’s got the CMB. The CMB goes a long way.

Vincent: Ah, so by that rationale, if a theory of modified dynamics fit the CMB, it would cease to be an ugly theory. Is that true?

Jules: Well, we’d have to be talkin’ about one charming eff’n theory of modified dynamics. I mean, it’d have to be ten times more charmin’ than MOND, you know what I’m sayin’?