As soon as I wrote it, I realized that the title is much more general than anything that can be fit in a blog post. Bekenstein argued long ago that the missing mass problem should instead be called the acceleration discrepancy, because that’s what it is – a discrepancy that occurs in conventional dynamics at a particular acceleration scale. So in that sense, it is the entire history of dark matter. For that, I recommend the excellent book The Dark Matter Problem: A Historical Perspective by Bob Sanders.
Here I mean more specifically my own attempts to empirically constrain the relation between the mass discrepancy and acceleration. Milgrom introduced MOND in 1983, no doubt after a long period of development and refereeing. He anticipated essentially all of what I’m going to describe. But not everyone is eager to accept MOND as a new fundamental theory, and often suffer from a very human tendency to confuse fact and theory. So I have gone out of my way to demonstrate what is empirically true in the data – facts – irrespective of theoretical interpretation (MOND or otherwise).
What is empirically true, and now observationally established beyond a reasonable doubt, is that the mass discrepancy in rotating galaxies correlates with centripetal acceleration. The lower the acceleration, the more dark matter one appears to need. Or, as Bekenstein might have put it, the amplitude of the acceleration discrepancy grows as the acceleration itself declines.
Bob Sanders made the first empirical demonstration that I am aware of that the mass discrepancy correlates with acceleration. In a wide ranging and still relevant 1990 review, he showed that the amplitude of the mass discrepancy correlated with the acceleration at the last measured point of a rotation curve. It did not correlate with radius.
I was completely unaware of this when I became interested in the problem a few years later. I wound up reinventing the very same term – the mass discrepancy, which I defined as the ratio of dynamically measured mass to that visible in baryons: D = Mtot/Mbar. When there is no dark matter, Mtot = Mbar and D = 1.
My first demonstration of this effect was presented at a conference at Rutgers in 1998. This considered the mass discrepancy at every radius and every acceleration within all the galaxies that were available to me at that time. Though messy, as is often the case in extragalactic astronomy, the correlation was clear. Indeed, this was part of a broader review of galaxy formation; the title, abstract, and much of the substance remains relevant today.
I spent much of the following five years collecting more data, refining the analysis, and sweating the details of uncertainties and systematic instrumental effects. In 2004, I published an extended and improved version, now with over 5 dozen galaxies.
Here I’ve used a population synthesis model to estimate the mass-to-light ratio of the stars. This is the only unknown; everything else is measured. Note that the vast majority galaxies land on top of each other. There are a few that do not, as you can perceive in the parallel sets of points offset from the main body. But that happens in only a few cases, as expected – no population model is perfect. Indeed, this one was surprisingly good, as the vast majority of the individual galaxies are indistinguishable in the pile that defines the main relation.
I explored the how the estimation of the stellar mass-to-light ratio affected this mass discrepancy-acceleration relation in great detail in the 2004 paper. The details differ with the choice of estimator, but the bottom line was that the relation persisted for any plausible choice. The relation exists. It is an empirical fact.
At this juncture, further improvement was no longer limited by rotation curve data, which is what we had been working to expand through the early ’00s. Now it was the stellar mass. The measurement of stellar mass was based on optical measurements of the luminosity distribution of stars in galaxies. These are perfectly fine data, but it is hard to map the starlight that we measured to the stellar mass that we need for this relation. The population synthesis models were good, but they weren’t good enough to avoid the occasional outlier, as can be seen in the figure above.
One thing the models all agreed on (before they didn’t, then they did again) was that the near-infrared would provide a more robust way of mapping stellar mass than the optical bands we had been using up till then. This was the clear way forward, and perhaps the only hope for improving the data further. Fortunately, technology was keeping pace. Around this time, I became involved in helping the effort to develop the NEWFIRM near-infrared camera for the national observatories, and NASA had just launched the Spitzer space telescope. These were the right tools in the right place at the right time. Ultimately, the high accuracy of the deep images obtained from the dark of space by Spitzer at 3.6 microns were to prove most valuable.
Jim Schombert and I spent much of the following decade observing in the near-infrared. Many other observers were doing this as well, filling the Spitzer archive with useful data while we concentrated on our own list of low surface brightness galaxies. This paragraph cannot suffice to convey the long term effort and enormity of this program. But by the mid-teens, we had accumulated data for hundreds of galaxies, including all those for which we also had rotation curves and HI observations. The latter had been obtained over the course of decades by an entire independent community of radio observers, and represent an integrated effort that dwarfs our own.
On top of the observational effort, Jim had been busy building updated stellar population models. We have a sophisticated understanding of how stars work, but things can get complicated when you put billions of them together. Nevertheless, Jim’s work – and that of a number of independent workers – indicated that the relation between Spitzer’s 3.6 micron luminosity measurements and stellar mass should be remarkably simple – basically just a constant conversion factor for nearly all star forming galaxies like those in our sample.
Things came together when Federico Lelli joined Case Western as a postdoc in 2014. He had completed his Ph.D. in the rich tradition of radio astronomy, and was the perfect person to move the project forward. After a couple more years of effort, curating the rotation curve data and building mass models from the Spitzer data, we were in the position to build the relation for over a dozen dozen galaxies. With all the hard work done, making the plot was a matter of running a pre-prepared computer script.
Federico ran his script. The plot appeared on his screen. In a stunned voice, he called me into his office. We had expected an improvement with the Spitzer data – hence the decade of work – but we had also expected there to be a few outliers. There weren’t. Any.
All. the. galaxies. fell. right. on. top. of. each. other.
This plot differs from those above because we had decided to plot the measured acceleration against that predicted by the observed baryons so that the two axes would be independent. The discrepancy, defined as the ratio, depended on both. D is essentially the ratio of the y-axis to the x-axis of this last plot, dividing out the unity slope where D = 1.
This was one of the most satisfactory moments of my long career, in which I have been fortunate to have had many satisfactory moments. It is right up there with the eureka moment I had that finally broke the long-standing loggerhead about the role of selection effects in Freeman’s Law. (Young astronomers – never heard of Freeman’s Law? You’re welcome.) Or the epiphany that, gee, maybe what we’re calling dark matter could be a proxy for something deeper. It was also gratifying that it was quickly recognized as such, with many of the colleagues I first presented it to saying it was the highlight of the conference where it was first unveiled.
Regardless of the ultimate interpretation of the radial acceleration relation, it clearly exists in the data for rotating galaxies. The discrepancy appears at a characteristic acceleration scale, g† = 1.2 x 10-10 m/s/s. That number is in the data. Why? is a deeply profound question.
It isn’t just that the acceleration scale is somehow fundamental. The amplitude of the discrepancy depends systematically on the acceleration. Above the critical scale, all is well: no need for dark matter. Below it, the amplitude of the discrepancy – the amount of dark matter we infer – increases systematically. The lower the acceleration, the more dark matter one infers.
The relation for rotating galaxies has no detectable scatter – it is a near-perfect relation. Whether this persists, and holds for other systems, is the interesting outstanding question. It appears, for example, that dwarf spheroidal galaxies may follow a slightly different relation. However, the emphasis here is on slighlty. Very few of these data pass the same quality criteria that the SPARC data plotted above do. It’s like comparing mud pies with diamonds.
Whether the scatter in the radial acceleration relation is zero or merely very tiny is important. That’s the difference between a new fundamental force law (like MOND) and a merely spectacular galaxy scaling relation. For this reason, it seems to be controversial. It shouldn’t be: I was surprised at how tight the relation was myself. But I don’t get to report that there is lots of scatter when there isn’t. To do so would be profoundly unscientific, regardless of the wants of the crowd.
Of course, science is hard. If you don’t do everything right, from the measurements to the mass models to the stellar populations, you’ll find some scatter where perhaps there isn’t any. There are so many creative ways to screw up that I’m sure people will continue to find them. Myself, I prefer to look forward: I see no need to continuously re-establish what has been repeatedly demonstrated in the history briefly outlined above.