Of historical interest are a series of papers written in the mid-60s in collaboration with Margaret Burbidge. These show some early rotation curves. Many peter out around the turn-over of the rotation curve. With the benefit of hindsight, one can see what the data will do – extend more or less flat from the last measured points.
Here is an example from Burbidge et al. (1964). In this case, NGC 3521, they got a bit further than the turnover. You may judge for yourself how convincing the detection of flat rotation is.
As it happens, NGC 3521 is a near kinematic twin to the Milky Way. Here is the modern rotation curve from THINGS compared with an estimate of the Milky Way rotation curve.
Hopefully it is obvious why it helps to have extended data (usually from 21 cm data, as in the example from THINGS).
This reminds me of something Vera frequently said. Early Days. In many ways, we are far down the path of dark matter. But we still have no idea what it is, or even whether what we call dark matter now is merely a proxy for some more general concept.
Vera always appreciated this. In many ways, these are still Early Days.
There has been some hand-wringing of late about the tension between the value of the expansion rate of the universe – the famous Hubble constant, H0, measured directly from observed redshifts and distances, and that obtained by multi-parameter fits to the cosmic microwave background. Direct determinations consistently give values in the low to mid-70s, like Riess et al. (2016): H0 = 73.24 ± 1.74 km/s/Mpc while the latest CMB fit from Planck gives H0 = 67.8 ± 0.9 km/s/Mpc. These are formally discrepant at a modest level: enough to be annoying, but not enough to be conclusive.
The widespread presumption is that there is a subtle systematic error somewhere. Who is to blame depends on what you work on. People who work on the CMB and appreciate its phenomenal sensitivity to cosmic geometry generally presume the problem is with galaxy measurements. To people who work on local galaxies, the CMB value is a non-starter.
This subject has a long and sordid history which entire books have been written about. Many systematic errors have plagued the cosmic distance ladder. Hubble’s earliest (c. 1930) estimate of H0 = 500 km/s/Mpc was an order of magnitude off, and made the universe impossibly young by what was known to geologists at the time. Recalibration of the distance scale brought the number steadily down. There followed a long (1960s – 1990s) stand-off between H0 = 50 as advocated by Sandage and 100 as advocated by de Vaucouleurs. Obviously, there were some pernicious systematic errors lurking about. Given this history, it is easy to imagine that even today there persists some subtle systematic error in local galaxy distance measurements.
In the mid-90s, I realized that the Tully-Fisher method was effectively a first approximation – there should be more information in the full shape of the rotation curve. Playing around with this, I arrived at H0 = 72 ± 2. My work relied heavily on the work of Begeman, Broeils, & Sanders and in turn on the distances they had assumed. This was a much large systematic uncertainty. To firm up my estimate would require improved calibration of those distances quite beyond the scope of what I was willing to take on at that time, so I never published it.
In 2001, the HST Key Project on the Distance Scale – the primary motivation to build the Hubble Space Telescope – reported H0 = 72 ± 8. That uncertainty was still plagued by the same systematics that had befuddled me. Since that time, the errors have been beaten down. There have been many other estimates of increasing precision, mostly in the range 72 – 75. The serious-minded cosmologist always worries about some subtle remaining systematic error, but the issue seemed finally to be settled.
One weird consequence of this was that all my extensive notes on the distance scale no longer seemed essential to teaching graduate cosmology: all the arcane details that had occupied the field for decades suddenly seemed like boring minutia. That was OK – about that time, there finally started to be interesting data on the the cosmic microwave background. Explaining that neatly displaced the class time spent on the distance scale. No longer were the physics students stopping to ask, appalled, “what’s a distance modulus?”; now it was the astronomy students who were appalled to be confronted by the spherical harmonics they’d seen but not learned in quantum mechanics.
The first results from WMAP were entirely consistent with the results of the HST key project. This reinforced the feeling that the problem was solved. In the new century, we finally knew the value of the Hubble constant!
Over the past decade, the best-fit value of H0 from the CMB has done a slow walk away from the direct measurements in the local universe. It has gotten far enough to result in the present tension. The problem is that the CMB doesn’t measure the Hubble constant directly; it constrains a multi-dimensional parameters space that approximately projects to a constant of the product ΩmH03, as illustrated below.
Much of the progress in cosmology has been the steady reduction in the allowed range in the above parameter space. The CMB data now allow only a narrow trench. I worry that it may wink out entirely. Were that to happen, it would falsify our current model of cosmology.
For now the only thing that seems to be happening is that the χ2 for the CMB data is ever so slightly better for lower values of the Hubble constant. While the lines of the trench represent no-go zones – the data require cosmological parameters to fall between the lines – there isn’t much difference along the trench. It is like walking along the floor of the Grand Canyon: exiting by climbing up the cliffs is disfavored; meandering downstream is energetically favored.
That’s what it looks like to me. The CMB χ2 has meandered a bit down the trench. It is not obvious to me that the current Planck best-fit is all that preferable to that from WMAP3. I have asked a few experts what would be so terrible about imposing the local distance scale as a strong prior. Have yet to hear a good answer, so chime in if you know one. If we put the clamps on H0 it must come out somewhere else. Where? How terrible would it be?
This is not an idle question. If one can recover the local Hubble constant with only a small tweak to, say, the baryon density, then fine – we’ve already got a huge problem there with lithium that we’re largely ignoring – why argue about the Hubble constant if this tension can be resolved where there’s already a bigger problem? If instead, it requires something more radical, like a clear difference from the standard number of neutrinos, then OK, that’s interesting and potentially a big deal.
So what is it? What does it take to reconcile to Planck with local H0? Since this is an issue of geometry, I suspect it might be something like the best fit geometry of the universe becoming ever so slightly not-flat, at the 2σ level instead of 1σ.
While I have not come across a satisfactory explanation of what it would take to reconcile Planck with the local distance scale, I have seen many joint analyses of Planck plus lots of other data. They all seem consistent, so long as you ignore the high-L (L > 600) Planck data. It is only the high-L data that are driving the discrepancy (low L appear to be OK).
So I will say the obvious, for those who are too timid: it looks like the systematic error is most likely with the high-L data of Planck itself.
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.
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.
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.
One Law to rule them all, One Law to guide them, One Law to form them all and in the dark halo bind them.
Galaxies appear to obey a single universal effective force law.
Early indications of this have been around for some time. It has become particularly clear in our work using near-infrared surface photometry to trace the stellar mass distribution of late type galaxies (SPARC). It takes a while to wrap our heads around the implications.
The observed phenomenology constitutes a new law of nature. One Law to rule all galaxies.
So far, the ubiquitous effective force law had only been clearly demonstrated in rotating galaxies. Federico Lelli and Marcel Pawlowski went to great lengths to also include pressure supported galaxies, from giant ellipticals to dwarf spheroidals. They appear to follow the same effective force law as rotating galaxies.
This is not a fluke of a few special galaxies. It involves galaxies of all known morphological types spanning an enormous range in mass, size, and surface brightness. I have spent the last twenty years adding new data for all varieties of galaxy types to this relation in the expectation that it would break. Instead it has become stronger and clearer.
Understanding the observed relation is one of the pre-eminent challenges in modern physics. Once we exclude metaphysical nonsense like multiverses, it is arguably the most important unsolved problem. Why does this happen?
The usual ad hoc interpretation of rotation curves in terms of dark matter does nothing to anticipate the observed phenomenology, which is in fact quite troubling from this perspective as it requires excessive fine-tuning. This has been known (if widely ignored) for a while, but doesn’t preclude the more rabid advocates of dark matter from asserting that it all comes about naturally. Lets not mince words here: claims that the radial acceleration relation occurs naturally with dark matter are pure, unadulterated bullshit fueled by confirmation bias and cognitive dissonance. Perhaps dark matter is the root cause, but there is nothing natural about it.
The natural explanation of a single effective force law is that it is caused by a truly universal force law.
So far, the theory that comes closest to explaining these data is MOND. Milgrom, understandably enough, argues that these data require MOND. He has a valid point. It is a good argument, but does it suffice to overcome the other problems MOND faces? These are not as great as widely portrayed, but they aren’t exactly negligible, either. I tried to look at the problem from both perspectives in this review for the Canadian Journal of Physics. [Being able to see things from both sides is an essential skill if one is to be objective, an important value in science that seems disturbingly absent in its modern practice.]
MOND anticipates an asymptotic slope of 1/2 at low acceleration (gobs ~ gbar1/2). In the figure above, the data for the faintest (“ultrafaint”) dwarf spheroidals show a flattening in the empirical law at low accelerations that is not predicted by MOND. Perhaps the underlying force law is subtly different from pure MOND? On the other hand, weak lensingobservations show that the MOND slope extrapolates well to much lower accelerations.
It is possible that the data for ultrafaint dwarfs are in some cases misleading. Are these objects in dynamical equilibrium (a prerequisite for analysis)? Are they even dwarf galaxies? Some of the ultrafaints are not clearly distinct objects in the sense of dSph satellites like Crater 2: it is not clear that all of them deserve the status of “dwarf galaxy.” Some are little more than a handful of stars that occupy a similar cell in phase space – perhaps they are fragmentary structures in the Galactic stellar halo? Or the rump end of dissolving satellites? This is anticipated to occur in both ΛCDM and MOND. If so, their velocity dispersions probably tell us more about their disruption history than their gravitational potential, in which case their location in the plot is misleading.
Detailed questions like these are the subject of much current research. For now, lets take a step back and appreciate the data for what they say, irrespective of the underlying theoretical reason for it. We’re looking at a new law of nature! How cool is that?
Vera Rubin passed away a few weeks ago. This was not surprising: she had lived a long, positive, and fruitful life, but had faced the usual health problems of those of us who make it to the upper 80s. Though news of her death was not surprising, it was deeply saddening. It affected me more than I had anticipated, even armed with the intellectual awareness that the inevitable must be approaching. It saddens me again now trying to write this, which must inevitably be an inadequate tribute.
In the days after Vera Rubin passed away, I received a number of inquiries from the press asking me to comment on her life and work for their various programs. I did not respond. I guess I understand the need to recognize and remark on the passing of a great scientist and human being, and I’m glad the press did in fact acknowledge her many accomplishments. But I wondered if, by responding, I would be providing a tribute to Vera, or merely feeding the needs of the never-ending hyperactive news cycle. Both, I guess. At any rate, I did not feel it was my place to comment. It did not seem right to air my voice where hers would never be heard again.
I knew Vera reasonably well, but there are plenty who knew her better and were her colleagues over a longer period of time. Also, at the back of my mind, I was a tiny bit afraid that no matter what I said, someone would read into it some sort of personal scientific agenda. My reticence did not preclude other scientists who knew her considerably less well from doing exactly that. Perhaps it is unavoidable: to speak of others, one must still use one’s own voice, and that inevitably is colored by our own perspective. I mention this because many of the things recently written about Vera do not do justice to her scientific opinions as I know them from conversations with her. This is important, because Vera was all about the science.
One thing I distinctly remembering her saying to me, and I’m sure she repeated this advice to many other junior scientists, was that you had to do science because you had a need to Know. It was not something to be done for awards or professional advancement; you could not expect any sort of acknowledgement and would likely be disappointed if you did. You had to do it because you wanted to find out how things work, to have even a brief moment when you felt like you understood some tiny fraction of the wonders of the universe.
Despite this attitude, Vera was very well rewarded for her science. It came late in her career – she did devote a lot of energy to raising a large family; she and her husband Bob Rubin were true life partners in the ideal sense of the term: family came first, and they always supported each other. It was deeply saddening when Bob passed, and another blow to science when their daughter Judy passed away all too early. We all die, sometimes sooner rather than later, but few of us take it well.
Professionally, Vera was all about the science. Work was like breathing. Something you just did; doing it was its own reward. Vera always seemed to take great joy in it. Success, in terms of awards, came late, but it did come, and in many prestigious forms – membership in the National Academy of Sciences, the Gold Medal of the Royal Astronomical Society, and the National Medal of Science, to name a few of her well-deserved honors. Much has been made of the fact that this list does not include a Nobel Prize, but I never heard Vera express disappointment about that, or even aspiration to it. Quite the contrary, she, like most modest people, didn’t seem to consider it to be appropriate. I think part of the reason for this was that she self-identified as an astronomer, not as a physicist (as some publications mis-report). That distinction is worthy of an entire post so I’ll leave it for now.
Astronomer though she was, her work certainly had an outsized impact on physics. I have written before as to why she was deserving of a Nobel Prize, if for slightly different reasons than others give. But I do not dread that she died in any way disappointed by the lack of a Nobel Prize. It was not her nature to fret about such things.
Nevertheless, Vera was an obvious scientist to recognize with a Nobel Prize. No knowledgeable scientist would have disputed her as a choice. And yet the history of the physics Nobel prize is incredibly lacking in female laureates (see definition 4). Only two women have been recognized in the entire history of the award: Marie Curie (1903) and Maria Goeppert-Mayer (1963). She was an obvious woman to have honored in this way. It is hard to avoid the conclusion that the awarding of the prize is inherently sexist. Based on two data points, it has become more sexist over time, as there is a longer gap between now and the last award to a woman (63 years) than between the two awards (60 years).
Why should gender play any role in the search for knowledge? Or the recognition of discoveries made in that search? And yet women scientists face antiquated attitudes and absurd barriers all the time. Not just in the past. Now.
Vera was always a strong advocate of women in science. She has been an inspiration to many. A Nobel prize awarded to Vera Rubin would have been great for her, yes, but the greater tragedy of this missed opportunity is what it would have meant to all the women who are scientists now and who will be in the future.
Well, those are meta-issues raised by Vera’s passing. I don’t think it is inappropriate, because these were issues dear to her heart. I know the world is a better place for her efforts. But I hadn’t intended to go off on meta-tangents. Vera was a very real, warm, positive human being. So I what I had meant to do was recollect a few personal anecdotes. These seem so inadequate: brief snippets in a long and expansive life. Worse, they are my memories, so I can’t see how to avoid making it at least somewhat about me when it should be entirely about her. Still. Here are a few of the memories I have of her.
I first met Vera in 1985 on Kitt Peak. In retrospect I can’t imagine a more appropriate setting. But at the time it was only my second observing run, and I had no clue as to what was normal or particularly who Vera Rubin was. She was just another astronomer at the dinner table before a night of observing.
A very curious astronomer. She kindly asked what I was working on, and followed up with a series of perceptive questions. She really wanted to know. Others have remarked on her ability to make junior people feel important, and she could indeed do that. But I don’t think she tried, in particular. She was just genuinely curious.
At the time, I was a senior about to graduate from MIT. I had to beg permission to take some finals late so I could attend this observing run. My advisor, X-ray astronomer George Whipple Clark, kindly bragged about how I had actually got my thesis in on time (most students took advantage of a default one-week grace period) in order to travel to Kitt Peak. Vera, ever curious, asked about my thesis, what galaxies were involved, how the data were obtained… all had been from a run the semester before. As this became clear, Vera got this bemused look and asked “What kind of thesis can be written from a single observing run?” “A senior thesis!” I volunteered: undergraduate observers were rare on the mountain in those days; up till that point I think she had assumed I was a grad student.
I encountered Vera occasionally over the following years, but only in passing. In 1995, she offered me a Carnegie fellowship at DTM. This was a reprieve in a tight job market. As it happened, we were both visiting the Kapteyn Institute, and Renzo Sancisi had invited us both to dinner, so she took the opportunity to explain that their initial hire had moved on to a faculty position so the fellowship was open again. She managed to do this without making me feel like an also-ran. I had recently become interested in MOND, and here was the queen of dark matter offering me a job I desperately needed. It seemed right to warn her, so I did: would she have a problem with a postdoc who worked on MOND? She was visibly shocked, but only for an instant. “Of course not,” she said. “As a Carnegie Fellow, you can work on whatever you want.”
Vera was very supportive throughout my time at DTM, and afterwards. We had many positive scientific interactions, but we didn’t really work together then. I tried to get her interested in the rotation curves of low surface brightness galaxies, but she had a full plate. It wasn’t until a couple of years after I left DTM that we started collaborating.
Vera loved to measure. The reason I chose the picture featured at top is that it shows her doing what she loved. By the time we collaborated, she had moved on to using a computer to measure line positions for velocities. But that is what she loved to do. She did all the measurements for the rotation curves we measured, like the ones shown above. As the junior person, I had expected to do all that work, but she wanted to do it. Then she handed it on to me to write up, with no expectation of credit. It was like she was working for me as a postdoc. Vera Rubin was an awesome postdoc!
She also loved to observe. Mostly that was a typically positive, fruitful experience. But she did have an intense edge that rarely peaked out. One night on Las Campanas, the telescope broke. This is not unusual, and we took it in stride. For a half hour or so. Then Vera started calmly but assertively asking the staff why we were not yet back up and working. Something was very wrong, and it involved calling in extra technicians who led us into the mechanical bowels of the du Pont telescope, replete with steel cables and unidentifiable steam-punk looking artifacts. Vera watched them like a hawk. She never said a negative word. But she silently, intently watched them. Tension mounted; time slowed to a crawl till it seemed that I could feel like a hard rain the impact of every photon that we weren’t collecting. She wanted those photons. Never said a negative word, but I’m sure the staff felt a wall of pressure that I was keenly aware of merely standing in its proximity. Perhaps like a field mouse under a raptor’s scrutiny.
Vera was not normally like that, but every good observer has in her that urgency to get on sky. This was the only time I saw it come out. Other typical instrumental guffaws she bore in stride. This one took too long. But it did get fixed, and we were back on sky, and it was as if there had never been a problem in the world.
Ultimately, Vera loved the science. She was one of the most intrinsically curious souls I ever met. She wanted to know, to find out what was going on up there. But she was also content with what the universe chose to share, reveling in the little discoveries as much as the big ones. Why does the Hα emission extend so far out in UGC 2885? What is the kinematic major axis of DDO 154, anyway? Let’s put the slit in a few different positions and work it out. She kept a cheat sheet taped on her desk for how the rotation curve changed if the position angle were missed – which never happened, because she prepared so carefully for observing runs. She was both thorough and extremely good at what she did.
Vera was very positive about the discoveries of others. Like all good astronomers, she had a good BS detector. But she very rarely said a negative word. Rarely, not never. She was not a fan of Chandrasekhar, who was the editor of the ApJ when she submitted her dissertation paper there. Her advisor, Gamow, had posed the question to her, is there a length scale in the sky? Her answer would, in the modern parlance, be called the correlation length of galaxies. Chandrasekhar declined to consider publishing this work, explaining in a letter that he had a student working on the topic, and she should wait for the right answer. The clear implication was that this was a man’s job, and the work of a woman was not to be trusted. Ultimately her work was published in the proceedings of the National Academy, of which Gamow was a member. He had predicted that this is how Chandrasekhar would behave, afterwards sending her a postcard saying only “Told you so.”
On another occasion, in the mid-90s when “standard” CDM meant SCDM with Ωm = 1, not ΛCDM, she confided to me in hushed tones that the dark matter had to be baryonic. Other eminent dynamicists have said the same thing to me at times, always in the same hushed tones, lest the cosmologists overhear. As well they might. To my ears this was an absurdity, and I know well the derision it would bring. What about Big Bang Nucleosynthesis? This was the only time I recall hearing Vera scoff. “If I told the theorists today that I could prove Ωm = 1, tomorrow they would explain that away.”
I was unconvinced. But it made clear to me that I put a lot of faith in Big Bang Nucleosynthesis, and this need not be true for all intelligent scientists. Vera – and the others I allude to, who still live so I won’t name – had good reasons for her assertion. She had already recognized that there was a connection between the baryon distribution and the dynamics of galaxies, and that this made a lot more sense if the dark and luminous component were closely related – for example, if the dark matter – or at least some important fraction of it in galaxies – were itself baryonic. Even if we believe in Big Bang Nucleosynthesis, we’re still missing a lot of baryons.
The proper interpretation of this evidence is still debated today. What I learned from this was to be more open to the possibility that things I thought I knew for sure might turn out to be wrong. After all, that pretty much sums up the history of cosmology.
To my mind, what Vera discovered is both more specific and more profound than the dark matter paradigm it helped to create. What she discovered observationally is that rotation curves are very nearly flat, and continue to be so to indefinitely large radius. Over and over again, for every galaxy in the sky. It is a law of nature for galaxies, akin to Kepler’s laws for planets. Dark matter is an inference, a subsidiary result. It is just one possible interpretation, a subset of amazing and seemingly unlikely possibilities opened up by her discovery.
The discovery itself is amazing enough without conflating it with dark matter or MOND or any other flavor of interpretation of which the reader might be fond. Like many great discoveries, it has many parents. I would give a lot of credit to Albert Bosma, but there are also others who had early results, like Mort Roberts and Seth Shostak. But it was Vera whose persistence overcame the knee-jerk conservatism of cosmologists like Sandage, who she said dismissed her early flat rotation curve of M31 (obtained in collaboration with Roberts) as “the effect of looking at a bright galaxy.” “What does that even mean?” she asked me rhetorically. She also recalled Jim Gunn gasping “But… that would mean most of the mass is dark!” Indeed. It takes time to wrap our heads around these things. She obtained rotation curve after rotation curve in excess of a hundred to ensure we realized we had to do so.
Vera realized the interpretation was never as settled as the data. Her attitude (and that of many of us, including myself) is nicely summarized by her exchange with Tohline at the end of her 1982 talk at IAU 100. One starts with the most conservative – or at least, least outrageous – possibility, which at that time was a mere factor of two in hidden mass, which could easily have been baryonic. Yet much more more recently, at the last conference I attended with her (in 2009), she reminded the audience (to some visible consternation) that it was still “early days” for dark matter, and we should not be surprised to be surprised – up to, and including, how gravity works.
At this juncture, I expect some readers will accuse me of what I warned about above: using this for my own agenda. I have found it is impossible to avoid having an agenda imputed to me by people who don’t like what they imagine my agenda to be, whether they imagine right or not – usually not. But I can’t not say these things if I want to set the record straight – these were Vera’s words. She remained concerned all along that it might be gravity to blame rather than dark matter. Not convinced, nor even giving either the benefit of the doubt. There was, and remains, so much to figure out.
I suppose, in the telling, it is often more interesting to relate matters of conflict and disagreement than feelings of goodwill. In that regards, some of the above anecdotes are atypical: Vera was a very positive person. It just isn’t compelling to relate episodes like her gushing praise for Rodrigo Ibata’s discovery of the Sagittarius dwarf satellite galaxy. I probably only remember that myself because I had, like Rodrigo, encountered considerable difficulty in convincing some at Cambridge that there could be lots of undiscovered low surface brightness galaxies out there, even in the Local Group. Some of these same people now seem to take for granted that there are a lot more in the Local Group than I find plausible.
I have been fortunate in my life to have known many talented scientists. I have met many people from many nations, most of them warm, wonderful human beings. Vera was the best of the best, both as a scientist and as a human being. The world is a better place for having had her in it, for a time.
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.
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.
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.
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.
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?
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