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!

LCDM has met the enemy, and it is itself

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.

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

  1. ad hoc hypotheses
  2. modification of ostensive definitions
  3. doubting the reliability of the experimenter
  4. 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.


Tension in the Hubble constant

Tension in the Hubble constant

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.

Best fit values of the Hubble constant and the mass density from CMB satellite experiments (labeled). The blue lines demarcate the trench allowed in the space by the Planck data. Only the narrow space between the lines is allowed; the region above and below is excluded. The best fit values have simply marched along the floor of this trench over time.

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.

Ode to Vera

Ode to Vera

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.

Figure made by Vera Rubin from her measurements of the rotation curves of low surface brightness galaxies. Published in McGaugh, Rubin, & de Blok (2001).

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.

It was widely reported that Vera discovered dark matter or “proved” or “confirmed” its existence. I don’t think Vera would agree with this assessment, nor would many of her colleagues at DTM. I know this because we talked about it. A lot.

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.

“Early days.”

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.

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’?



So the always humorous, unabashedly nerdy xkcd recently published this comic:


This hits close to home for me, in many ways.

First, this is an every day experience for me. Hardly a day goes by that I don’t get an email, or worse, a phone call, from some wanna-be who has the next theory of everything. I try to be polite. I even read some of what I get sent. Mostly this is a waste of my time. News flash: at most, only one of you can be right. If the next Einstein is buried somewhere amongst these unsolicited, unrefereed, would-be theories, I wouldn’t know, because I do not have the time to sort through them all.

Second, it is true – it is a logical possibility that what we call dark matter is really just a proxy for a change in the law of gravity on galactic scales. It is also true that attempts to change the law of gravity on large scales do not work to explain the dark matter problem. (Attempts to do this to address the dark energy problem are a separate matter.)

Third, it is a logical fallacy. The implication of the structure of the statement is that the answer has to be dark matter. One could just as accurately turn the statement on its head and say “Yes, everybody has already had the idea, maybe it isn’t modified gravity – there’s just a lot of invisible mass on large scales!’ It sounds good but it doesn’t really fit the data.”

The trick is what data we’re talking about.

I have reviewed this problem many times (e.g., McGaugh & de Blok 1998, Sanders & McGaugh 2002, McGaugh 2006Famaey & McGaugh 2012, McGaugh 2015). Some of the data favor dark matter, some favor modified gravity. Which is preferable depends on how we weigh the different lines of evidence. If you think the situation is clear cut, you are not well informed of all the facts.* Most of the data that we cite to require dark matter are rather ambiguous and can usually be just as well interpreted in terms of modified gravity. The data that isn’t ambiguous points in opposite directions – see the review papers.

Note that I was careful above to say “galactic scales.” The scale that turns out to matter is not a size scale but an acceleration scale. Galaxies aren’t just big. The centripetal accelerations that hold stars in their orbits are incredibly low: about one part in 1011 of what we feel on the surface of the Earth. The only data that test gravity on this acceleration scale are the data that evince the missing mass problem. We only infer the need for dark matter at these very low accelerations. So while it is not possible to construct an empirically successful theory that modifies gravity on some large length scale, it remains a possibility that a modification can be made on an acceleration scale.

That the mass discrepancy problem occurs on an acceleration scale and not at some length scale has been known for many years. Failing to make the distinction between a length scale and an acceleration scale is fine for a comic strip. It is not OK for scientists working in the field. And yet I routinely encounter reasonable, intelligent scientists who are experts in some aspect of the dark matter problem but are unaware of this essential fact.

To end with another comic, the entire field is easily mocked:


The astute scientific reader will recognize that Mr. Breathed is conflating dark matter with dark energy. Before getting too dismissive, consider how you would go about explaining to him that our cosmic paradigm requires not just invisible mass to provide extra gravity, but also dark energy to act like antigravity. Do you really think that doubling down on ad hoc hypotheses makes for a strong case?

*Or worse, you may fall prey to cognitive dissonance and confirmation bias.

What is the Baryon Density, Anyway?

To continue… we had been discussing the baryon content of the universe, and the missing baryon problem. The problem exists because of a mismatch between the census of baryons locally and the density of baryons estimated from Big Bang Nucleosynthesis (BBN). How well do we know the latter? Either extremely well, or perhaps not so well, depending on which data we query.

At the outset let me say I do not doubt the basic BBN picture. BBN is clearly one of the great successes of early universe cosmology: it is pretty clear this is how the universe works. However, the absolute value we obtain for Ωb depends on the mutual agreement of independent measurements of the abundances of different isotopes. These agree well enough to establish the BBN paradigm, but not so well as to discount all debate about the exact value of Ωb – contrary to the impression one might get from certain segments of the literature.

BBN is thoroughly discussed elsewhere so I won’t belabor it here. In a nutshell, the primordial abundance of the isotopes of the light elements – especially deuterium, helium, and lithium relative to hydrogen – depends on the baryon density. Each isotope provides an independent constraint. This is perhaps the most (only?) over-constrained problem in cosmology.

It is instructive to look at the estimates of the baryon density over time. These are usually quoted as the baryon density multiplied by the square of the Hubble parameter normalized to 100 km/s/Mpc (Ωbh2). This is a hangover from the bad old days when we didn’t know H0 to a factor of 2.


The graph shows the baryon densities estimated by various methods by different people over the years. It starts with the compilation of Walker et al. (1991). By this time, BBN was already a mature subject, with an authoritative answer based on evidence from all the isotopes. Ωbh2 = 0.0125 ± 0.0025. It was Known, Khaleesi.

In the mid-90s there was a debate about the primordial deuterium abundance, largely between Hogan and Tytler. Deuterium (red tirangles) is a great isotope to measure for BBN because it is very sensitive to the baryon density, as it tends to get gobbled up into heavier isotopes like helium when there are lots of baryons around to react with. Moreover, one could measure it in the absorption along the line of sight to high redshift QSOs, presumably catching it before any nasty interstellar processing has polluted the primordial abundance. Unfortunately, Hogan found a high D/H (and hence a low baryon density) while Tytler found low D/H (hence high Ωbh2). This is a rare case when one side (Hogan) actually admitted error, and the standard density shot up to 0.019. At the time (1998) that seemed outrageously high, 2.6σ above the previous standard value. But we had bigger problems to wrap our head around (Λ) at the time, so this was accepted without much fuss.

The other elements (helium and lithium) preferred something in between at that time. Their uncertainties were large enough this didn’t seem a big deal. Helium in particular is notoriously hard to pin down. Not only is the measurement hard to make, but helium (unlike deuterium) is not particularly sensitive to the baryon density. You get about a quarter helium by mass out of BBN for any reasonable baryon density, so it is a great indicator that the basic picture is correct. But you really have to nail down the third decimal point to help distinguish between slightly lower or higher Ωb. So the new normal became Ωbh2 = 0.019 ± 0.001.

That was the summation of decades of work, but it wasn’t to last long. In 2000, cosmic microwave background (CMB) experiments like BOOMERanG and MAXIMA began to resolve the acoustic power spectrum. A funny thing emerged: the second peak was lower than expected. (At least by other people. I totally nailed this prediction.) In order to explain the low second peak conventionally (in the context of ΛCDM), one had to crank up the baryon density. This first point from the CMB (blue in plot above) was well above previously suspected levels.

Note the dotted lines in the figure. These denote the maximum baryon density (horizontal line) before the first relevant CMB data (vertical line). No isotope of any light element had ever suggested Ωbh2 > 0.02 prior to CMB constraints. Once those became available, this changed.

The change happened first to deuterium, which has not suggested Ωbh2 < 0.02 since the CMB said so. Helium was slower to respond, but it has also drifted slowly upward. Lithium has remained put. This is a serious problem that has not been satisfactorily resolved. The general presumption seems to be that this is a detail to be blamed on stellar rotation or some similarly obscure mechanism.

Different communities work on each of these elements. Deuterium is the subject of high redshift astronomy, a field closely coupled to cosmology. Helium is the subject of nearby galaxies, a field aware of cosmology but less strongly tied to it. Lithium is measured in stars, a field that is not coupled to cosmology. Given the long history of confirmation bias in cosmology, it is hard not to be suspicious of the temporal variation in BBN baryon density estimates. The isotope most closely associated with cosmology, deuterium, quickly fell in line with the “right” result from the CMB. Helium has more gradually followed suit, while lithium continues to prefer lower baryon densities.

I do not doubt the sincerity of any particular measurement. But people talk. They have arguments about what is right and why. The communities that are closest are most likely to influence each other. Those further apart are less likely to be swayed. If we were suffering from confirmation bias, this is what it would look like.

The ΛCDM picture requires us to believe the CMB value, currently  Ωbh2 = 0.02230 ± 0.00023 (Planck 2015). You simply cannot fit the acoustic power spectrum with a number much different. Modern deuterium measurements are consistent with that, within the errors, so that has to be right, no? Lets just ignore lithium.

If instead we ignore the CMB and its associated baggage, this is not at all obvious. Perhaps the pre-CMB deuterium measurement is the one to trust. That is a bit higher than lithium, but consistent within the errors. Helium can go either way. So from a pure BBN (no CMB) perspective, maybe it is lithium and the other isotopes that are right and it is CMB fits that are misleading.

Where does this leave us with the missing baryons? The figure below shows the time evolution of the baryon density. The area is proportional to Ωbh2. This has grown over time, by an amount greater than the stated uncertainties (the circles show the change in area allowed). The baryon density has nearly doubled, being now ~4σ above the Known value of Walker.


As the baryon density has grown, the missing baryon problem has grown worse. If we still had the classical Walker baryon density, there would be no missing baryon problem at al.  Indeed, Shull’s inventory is a bit too large, though it is consistent within the errors. If we go up to the pre-CMB deuterium value, then there is a missing baryon problem. It is big enough to solve the cluster problem in MOND, but without a lot left over. If we insist on the CMB-fitted baryon density, then the missing baryon problem is severe, at a level where it is hard to figure where else they could be.

IF ΛCDM is the right picture, then I think a high baryon density is unavoidable. Accepting this, there must then be something wrong with lithium. There is no lack of papers motivated by this line of reasoning, though the most common approach seems to be to ignore lithium entirely. I’ve heard a lot of talks bragging about the excellent agreement between BBN and the CMB, but this  really only applies to post-CMB deuterium.

IF BBN, as originally posed, is correct so that lithium and the other pre-CMB measurements are not misleading, then it becomes impossible to fit the CMB with pure General Relativity. This is the case even if we spot it non-baryonic cold dark matter and dark energy. This situation might be considered a motivation to seek extensions of the theory.

Regardless of where the right answer ultimately lies, there is real tension between primordial lithium measurements and the ΛCDM interpretation of the CMB. Something is fishy in the state of the early universe.

Missing baryons in LCDM and MOND

Missing baryons in LCDM and MOND

People often ask for a straight up comparison between ΛCDM and MOND. This is rarely possible because the two theories are largely incommensurable. When one is eloquent the other is mute, and vice-versa.

It is possible to attempt a comparison about how bad the missing baryon problem is in each. In CDM, we expect a relation between dynamical mass and rotation speed of the form Mvir ∝ Vvir3. In MOND the equivalent relation has a different power law, Mb ∝ Vf4.

In CDM we speak of virial quantities – the total mass of everything, including dark matter, and the circular speed way out at the virial radius (typically far outside the luminous extent of a galaxy). In MOND, we use the observed baryonic mass (stars and gas) and the flat rotation speed. These are not the same, so strictly speaking, still incommensurable. But they provide a way to compare the baryonic mass with the total inferred mass.


This plot shows the detected baryon fraction as a function of mass. The top panel is identical to last time. In ΛCDM we see most of the baryons in the most massive systems, but progressively less in smaller systems. In MOND the situation is reversed. The check-sum is complete in galaxies, but falls short in clusters of galaxies. (Note that the error bars have been divided by an extra power of velocity in the lower panel, which amplifies their appearance.) The reader may judge for himself which of these problems is more serious.

Critics of MOND frequently cite the bullet cluster as having falsified MOND. Period. No room for debate. See the linked press release from NASA: dark_matter_proven.

OK, what kind of dark matter? As discussed previously, we need at least two kinds of dark matter in ΛCDM: non-baryonic cold dark matter (some entirely novel particle) and dark baryons (normal matter not yet detected). Unfortunately, “dark matter” is a rather generic, catch-all term that allows these two distinct problems to be easily confused. We see the need for unseen mass in objects like the bullet cluster, and make the natural leap to conclude that we are seeing the non-baryonic cold dark matter that we expect in cosmology. There it is, case closed.

This is an example of a logical fallacy. There is nothing about the missing mass problem suffered by MOND in clusters that demands the unseen mass be non-baryonic. Indeed, even in ΛCDM we suffer some missing baryon problem on top of the need for non-baryonic cold dark matter. In both theories, there is a missing baryon problem in clusters. In both cases, this missing baryon problem is more severe at small radii, suggestive of a connection with the also-persistent cooling flow problem. Basically, the X-ray emitting gas observed in the inner 200 kpc or so of clusters have time to cool, so it ought to be condensing into – what? Stars? MACHOs? Something normal but as yet unseen.

It is not obvious that cooling flows can solve MOND’s problem in clusters. The problem is both serious and persistent. It was first pointed out in 1988 by The & White, and is discussed in this 2002 Annual Review. A factor or two (or even a bit more) of the expected baryons in clusters are missing (the red portion of the plot above). Note, however, that this problem was known long before the bullet cluster was discovered. From this perspective, it would have been very strange had the bullet cluster not shown the same discrepancy as every other cluster in the sky.

I do not know if the missing mass in clusters is baryonic. I am at a loss to suggest a plausible form that the missing baryons might be lurking in. Certainly others have tried. But lets take a step back and as if it is plausible.

As seen above, we have a missing baryon problem in both theories. It just manifests in different places. Advocates of ΛCDM do not, by and large, seem to consider the baryon discrepancy in galaxies to be a problem. The baryons were blown out, or are there but just not detected yet. No Problem. I’m not as lenient, but if we are to extend that grace to ΛCDM, why not also to MOND?

Recall that Shull et al. found that about 30% of baryons remain undetected in the local universe. In order to solve the problem MOND suffers in clusters, we need a mass in baryons about equal to the ICM wedge in this pie chart:


Note that the missing wedge is much larger than the ICM wedge. There are more than enough baryons out there to solve this problem. Indeed, it hardly makes a dent in the global missing baryon problem. Those baryons “must” be somewhere, so why not some in clusters of galaxies?

The short answer is cognitive dissonance. If one comes to the problem sure of the answer, then one sees in the data what one expects to see. MOND fits rotation curves? That’s just a fluke: it bounces off the wall of cognitive dissonance without serious consideration. MOND needs dark matter in clusters? Well of course – we knew that it had to be wrong in the first place.

I understand this perspective exceedingly well. It is where I started from myself. But the answer I wanted is not the conclusion that a more balanced evaluation of the evidence leads one to. The challenge is not in the evidence – it is to give an unorthodox idea a chance in the first place.

Missing Baryons

A long standing problem in cosmology is that we do not have a full accounting of all the baryons that we believe to exist. Big Bang Nucleosynthesis (BBN) teaches us that the mass density in normal matter is Ωb ≈ 5%. One can put a more precise number on it, but that’s close enough for our purposes here.

Ordinary matter fails to account for the closure density by over an order of magnitude. To make matters worse, if we attempt an accounting of where these baryons are, we again fall short. As well as the dynamical missing mass problem, we also have a missing baryon problem.

For a long time, this was also an order of magnitude problem. The stars and gas we could most readily see added up to < 1%, well short of even 5%. More recent work has shown that many, but not all, of the missing baryons are in the intergalactic medium (IGM).  The IGM is incredibly diffuse – a better vacuum than we can make in the laboratory by many orders of magnitude – but it is also very, very, very, well, very big. So all that nothing does add up to a bit of something.


A thorough accounting has been made by Shull et al. (2012). A little over half of detected baryons reside in the IGM, in either the Lyman alpha forest (Ly a in the pie chart above) or in the so-called warm-hot intergalactic medium (WHIM). There are also issues of double-counting, which Shull has taken care to avoid.

Gravitationally bound objects like galaxies and clusters of galaxies contain a minority of the baryons. Stars and cold (HI) gas in galaxies are small wedges of the pie, hence the large problem we initially had. Gas in the vicinity of galaxies (CGM) and in the intracluster medium of clusters of galaxies (ICM) also matter. Indeed, in the most massive clusters, the ICM outweighs all the stars in the galaxies there. This situation reverses as we look at lower mass groups. Rich clusters dominated by the ICM are rare; objects like our own Local Group are more typical. There’s no lack of circum-galactic gas (CGM), but it does not obviously outweigh the stars around L* galaxies.

There are of course uncertainties, so one can bicker and argue about the relative size of each slice of the pie. Even so, it remains hard to make their sum add up to 5% of the closure density. It appears that ~30% of the baryons that we believe to exist from BBN are still unaccounted for in the local universe.

The pie diagram only illustrates the integrated totals. For a long time I have been concerned about the baryon budget in individual objects. In essence, each dark matter halo should start with a cosmically equal share of baryons and dark matter. Yet in most objects, the ratio of baryons to total mass falls well short of the cosmic baryon fraction.

The value of the cosmic baryon fraction is well constrained by a variety of data, especially the cosmic microwave background. The number we persistently get is

fb = Ωbm = 0.17

or maybe 0.16, depending on which CMB analysis you consult.  But it isn’t 0.14 nor 0.10 nor 0.01. For sticklers, note that this the fraction of the total gravitating mass in baryons, not the ratio of baryons to dark matter: Ωm includes both. For numerologists, note that within the small formal uncertainties, 1/fb = 2π.

This was known long before the CMB experiments provided constraints that mattered. Indeed, one of the key findings that led us to repudiate standard SCDM in favor of ΛCDM was the recognition that clusters of galaxies had too many baryons for their dynamical mass. We could measure the baryon fraction in clusters. If we believe that these are big enough chunks of the universe to be representative of the whole, and we also believe BBN, then we are forced to conclude that Ωm ≈ 0.3.

Why stop with clusters? One can do this accounting in every gravitationally bound object. The null hypothesis is that every object should be composed of the universal composition, roughly 1 part baryons for every 5 parts dark matter. This almost works in rich clusters of galaxies. It fails in small clusters and groups of galaxies, and gets worse as you examine progressively smaller systems. So: not only are we missing baryons in the cosmic sum, there are some missing in each individual object.


The figure shows the ratio of detected baryons to those expected in individual systems. I show the data I compiled in McGaugh et al. (2010), omitting the tiniest dwarfs for which the baryon content becomes imperceptible on a linear scale. By detected baryons I mean all those seen to exist in the form of stars or gas in each system (Mb = M*+Mg), such that

fd = Mb/(fbMvir)

where Mvir is the total mass of each object. This `virial’ mass is a rather uncertain quantity, but in this plot it can only slide the data up and down a little bit. The take-away is that not a single, gravitationally bound object appears to contain its fair share of cosmic baryons. There is a missing baryon problem not just globally, but in each and every object.

This halo-by-halo missing baryon problem is least severe in the most massive systems, rich clusters. Indeed, the baryon fraction of clusters is a rising function of radius, so a case could be made that the observations simply don’t reach far enough out to encompass a fair total. This point has been debated at great length in the literature, and I have little to add to it, except to observe that rich clusters are perhaps like horseshoes – close enough.

Irrespective of whether we consider the most massive clusters to be close enough to the cosmic baryon fraction or not, no other system comes close to close enough. There is already a clear discrepancy among smaller clusters, and an apparent trend with mass. This trend continues smoothly and continuously over many decades in baryonic mass through groups, then individual L* galaxies, and on to the tiniest dwarfs.

A respectively massive galaxy like the Milky Way has many tens of billions of solar masses in form of stars, and another ten billion or so in the form of cold gas. Yet this huge mass represents only a 1/4 or so of the baryons that should reside in the halo of the Milky Way. As we look at progressively smaller galaxies, the detected baryon fraction decreases further. For a galaxy with a mere few hundred million stars, fd ≈ 6%. It drops below 1% for M* < 107 solar masses.

That’s a lot of missing baryons. In the case of the Milky Way, all those stars and cold gas are within a radius of 20 kpc. The dark matter halo extends out to at least 150 kpc. So there is plenty of space in which the missing baryons might lurk in some tenuous form. But they have to remain pretty well hidden. Joel Bregman has spent a fair amount of his career searching for such baryonic reservoirs. While there is certainly some material out there, it does not appear to add up to be enough.

It is still harder to see this working in smaller galaxies. The discrepancy that is a factor of a few in the Milky Way grows to an order of magnitude and more in dwarfs. A common hypothesis is that these baryons do indeed lurk there, probably in a tenuous, hot gas. If so, direct searches have yet to see them. Another common idea is that the baryons get expelled entirely from the small potential wells of dwarf galaxy dark matter halos, driven by winds powered by supernovae. It that were the case, I’d expect to see a break at a critical mass where the potential well was or wasn’t deep enough to prevent this. If there is any indication of this, it is at still lower mass than shown above, and begs the question as to where those baryons are now.

So we don’t have a single missing mass problem in cosmology. We have at least two. One is the need for non-baryonic dark matter. The other is the need for unseen normal matter: dark baryons. This latter problem has at least two flavors. One is that the global sum of baryons comes up short. The other is that each and every individual gravitationally bound object comes up short in the number of baryons it should have.

An obvious question is whether accounting for the missing baryons in individual objects helps with the global problem. The wedges in the pie chart represent what is seen, not what goes unseen. Or do they? The CGM is the hot gas around galaxies, the favored hiding place for the object-by-object missing baryon problem.

Never mind the potential for double counting. Lets amp up the stars wedge by the unseen baryons indicated in red in the figure above. Just take for granted, for the moment, that these baryons are there in some form, associated in the proper ratio. We can then reevaluate the integrated sum and… still come up well short.

Low mass galaxies appear to have lots of missing baryons. But they are low mass. Even when we boost their mass in this way, they still contribute little to the integral.

This is a serious problem. Is it hopeless? No. Is it easily solved? No. At a minimum, it means we have at least two flavors of dark matter: non-baryonic [cosmic] dark matter, and dark baryons.

Does this confuse things immensely? Oh my yes.

What is theory?

What is theory?

OK, I’m not even going to try to answer that one. But I am going to do some comparison exploration.

A complaint often leveled against MOND is that it is not a theory. Or not a complete theory. Or somehow not a proper one. Sometimes people confuse MOND with the empirical observations that display MONDian phenomenology.

I would say that MOND is a hypothesis, as is dark matter. We observe a discrepancy between the motions observed in extragalactic systems and what is predicted by application of the known law of gravity to the mass visible in ordinary baryonic matter. Either we need more mass (dark matter) or need to change the force law (modify dynamical laws, i.e., gravity). MOND is just one example of the latter type of hypothesis.

Put this way, dark matter is the more conservative hypothesis. It doesn’t require any change to well established, fundamental theory. There’s just more mass there than we see.

But what is it? Dark matter as so far stated is not a valid scientific hypothesis. It is a concept – there is unseen stuff out there. To turn it into science, we need to hypothesize a specific candidate.

An example of a dark matter candidate that most people would agree has been falsified at this point is brown dwarfs. These are very faint, sub-stellar objects – failed stars if you like, things not quite massive enough to ignite nuclear fusion in their cores to shine as stars. In the early days of dark matter, it was quite reasonable to believe there could be an enormous amount of mass in the sum of these objects. Indeed, the mass spectrum of stars as then known (via the Salpeter IMF) diverged when extrapolated to the low masses of brown dwarfs. It appeared that there had to be lots of them, and their integrated mass could easily add up to lots and lots – potentially enough to be the dark matter.

The hypothesis of brown dwarf-like dark matter, dubbed MACHOs (MAssive Compact Halo Objects), was tested by a series of microlensing experiments. Remarkably, if you stare at the stars in the Large Magellanic Cloud long enough, you should occasionally witness a MACHO pass in front of one of them. You don’t see the MACHO directly, but you can see an enhancement to the brightness of the background star due to the gravitational lensing effect of the MACHO.

Long story short: microlensing events are observed, but not nearly enough are seen for the dark matter halo of the Milky Way to be composed of brown dwarf MACHOs. Nowadays we have a better handle on the stellar mass spectrum. Lots of brown dwarfs are indeed known, but nothing like the numbers necessary to compose the dark matter.

Many of us, including me, never gave MACHOs much of a chance. In order to add up to the total mass density we need in dark matter cosmologically, we need an amount 5 or 6 times as great as the density allowed in baryons by Big Bang Nucleosynthesis. So MACHO dark matter would break some pretty fundamental theory after all.

The most popular hypothesis, then and now, is some form of non-baryonic dark matter. Most prominent among these are WIMPs (Weakly Interacting Massive Particles). This is a valid, specific hypothesis that can be tested in the laboratory. Indeed, it has been. If the WIMP hypothesis were correct, we really should have detected them by now. It only persists because it is very flexible: we can keep adjusting the interaction cross-section to keep them invisible.

It would be a long post to revisit all the ways in which the WIMP hypothesis has repeatedly disappointed. Here I’d like to point out merely that WIMPs are hypothetical particles that exist in a hypothetical supersymmetric sector. There are compelling theoretical arguments in favor of supersymmetry, but so far it too has repeatedly disappointed. Anybody else remember how the decay of the Bs meson was suppose to be the Golden Test of supersymmetry? No? Nobody seems to talk about it anymore because it flunked badly. So supersymmetry itself is in dire shape. No supersymmetry, no WIMPs.

Like WIMPs, supersymmetry can be made more complicated to avoid falsification. This allows it to persist, but it is not the sign of a healthy theory. Still, everybody seems to agree that it is a theory, and most people seem to think it is a good one.

Unlike MACHOs, WIMPs do require a fundamentally new theory. Supersymmetry is not a part of the highly successful Standard Model of particle physics. It is a hypothetical extension thereof. So they aren’t really as conservative as just saying there is some unseen mass. There have to be invisible particles that reside in an entirely novel and itself hypothetical dark sector. That they have never been detected in the laboratory, and so far we have zero laboratory evidence to support the existence of the supersymmetric sector in which they reside, despite enormous (and expensive) effort (e.g., the LHC), might strike some as cause for concern.

So why do WIMPs persist? Time lag and training. If you are an astronomer, you don’t really care what the dark matter particle is, just that it is there. You are unlikely to keep close tabs on the tribulations of dark matter detection experiments. If you are an astroparticle physicist, dark matter particles are your bread and butter. We all know the Standard Model is incomplete; surely the dark matter problem is just a sign of that. Suggesting that the problem might instead be with gravity is to admit that the entire field is an oxymoron. Yes, we need new physics. But that would be the wrong kind of new physics!


The MOND hypothesis is an example of the wrong kind of new physics. No new particles; rather, new dynamics. The idea is to tweak the force law below a critical acceleration scale (of order 1 Å/s/s). Intriguingly, this can be interpreted as either a modification of gravity (which gets stronger) or of inertia (which gets less, so particles become easier to push around).

From such a hypothesis, one must construct a proper theory – whatever that is. One thing is for sure – the motivation is the opposite of supersymmetry. Supersymmetry is motivated by theory. It is a Good Idea that therefore ought to be true, even if it appears that Nature declined to implement it. MOND has no compelling theoretical motivation or basis. (Who ordered that?) Rather, it is empirically motivated. It started by seeking a possible explanation for a particular observation: the apparent flatness of spiral galaxy rotation curves. In this regard, it could be considered an effective theory, though it does have strong implications for what the underlying cause is.

The original (1983) MOND formula did not conserve energy or momentum. That’s not a property of a healthy theory. Some people seem to think it is still stuck there.

The first step towards building a proper theory was taken by Bekenstein and Milgrom in 1984 with AQUAL. They introduced an aquadratic Lagrangian that led to a modified Poisson equation, a form of modified gravity. Being derived form a Lagrangian, it automatically satisfies the conservation laws.

Since then, a variety of MOND theories have been posited. By this, I mean distinct theories that lead to the hypothesized behavior at low acceleration. These may be modifications of either gravity or inertia, and can lead to subtly different higher order predictions.

So far most MOND theories are extensions of Newtonian dynamics. MOND always contains Newton in the high acceleration limit, just as General Relativity contains Newton in the appropriate limit. The trick is to write a theory that does both. That’s the theoretical Holy Grail.

The following Venn diagram might help:


Both MOND and General Relativity encompass Newtonian dynamics. However, they do not contain each other. Since General Relativity came first, I think when people say MOND is not a theory they usually mean that it doesn’t capture all the previous theory that it needs to. We know General Relativity is correct – so far as we have tested it – so it doesn’t suffice to write down a theory that is merely an extension of Newton. We need a theory that does both – the Holy Grail.

Of course I agree that we want to have it all. I also think it is appropriate to take one step at a time. If Newtonian dynamics is in itself a valid theory – and I think it is – the so too is MOND, as it contains all of Newton in the appropriate limit. MOND is an incomplete theory, but it is certainly a theory.

For many years, an argument against MOND was that Bekenstein had sought the Holy Grail long and hard without success. Bekenstein was really smart, implying that if he couldn’t do it, it couldn’t be done.  In 2004, Bekenstein published TeVeS (for Tensor-Vector-Scalar), the first example of a theory that contained both General Relativity and MOND without obviously having some dreadful failing, like ghosts. The argument then became that TeVeS was inelegant.

It is not clear that TeVeS is the correct generalized version of General Relativity. Indeed, it is not the only such theory possible. Hence the question mark in the Venn diagram. If we falsify TeVeS, it doesn’t falsify the MOND hypothesis – just that particular realization thereof.

What theory the question mark in the Venn diagram represents is what we should be trying to figure out. Unfortunately, most scientists interested in the subject are not trained nor equipped to do this sort of work, and for the most part are conditioned to be actively hostile to the project. That’s the wrong kind of new physics!

I find this a strange attitude. We all know that, as yet, there is no widely accepted theory of quantum theory. In this regard, General Relativity is itself incomplete. It is a noble endeavor to seek a quantum theory of gravity. How can we be sure that there is no intermediate step? Perhaps some of the difficulty in getting there stems from playing with an incomplete deck. I sometimes wonder if some string theorist has already come up with the correct theory but discarded it because it predicted this crazy low acceleration behavior he didn’t know might actually be desirable.

Whatever the final theory may be, be it dark matter based or a modification of dynamics, it must explain the empirical phenomena we observe. An enormous amount of galaxy phenomenology can be put down to one simple fact: galaxies behave as if MOND is the effective force law. We can write down a single formula that describes the dynamics of hundreds of measured galaxies and has had tremendous predictive success. If you don’t find that compelling, your physical intuition needs a check up.