Finally the team I met in Tenerife have formally published this result as a Letter. This is not as detailed as a full article (which I presume is forthcoming), but all the major points are addressed. In fact it's even more interesting than I was expecting.
There are two main ways to explain a deviation from the standard TFR. The first is that the rotation speed could have been incorrectly estimated. This was my main concern when I saw the original data, which comes from the ALFALFA survey. Arecibo has extremely high sensitivity but doesn't have the resolution to map exactly how the gas is moving. It can place a lower limit on how fast the gas in a galaxy is moving along our line of sight, but that's not the same as measuring how it's rotating. In principle, the gas could actually be rotating much more quickly than the basic estimate implies. And because of selection effects, it's possible that the sample was unexpectedly biased towards detecting galaxies which looked like they were rotating more slowly than they actually are.
This was a real possibility, but not a very likely one as even the first paper had some observations taken with less sensitive but higher resolution telescopes. The authors here do the same for three more galaxies, giving them a sample of six. Once you can resolve a galaxy, you can determine its rotation much more accurately : you can correct for the effect of viewing it at an angle, and you can see if its motions display the pattern characteristic of ordered rotation or if it's doing something more chaotic. All six galaxies here show ordered rotation (though it would be nice to see the full maps), and the viewing angle definitely isn't enough to explain the deviation. This effectively demolishes the argument that the sample is biased.
The second way to explain the offset is that the total mass (here meaning the combination of gas and stellar mass) is incorrect. The problem with telescopes that give you nice resolution is that they, for technical reasons, give you crappy sensitivity. But in this case that can't be a problem, since they measure about the same amount of gas as for the earlier, higher sensitivity observations. The only other way the mass could be wrong is if the galaxies were much closer than expected. But that simply isn't possible for these objects - it would require them to have extremely high peculiar velocities, and since they're all pretty isolated objects this is just not credible.
So the galaxies really do seem to deviate. They've got rather high gas masses, around a billion times the mass of the Sun - not far off how much there is in a typical massive spiral. Whereas the Milky Way rotates at 220 km/s, these guys rotate at just 30 km/s or so. They don't have high velocity dispersions so there can't be much in the way of stellar winds. They seem to be in equilibrium, since they have nice ordered motions, so they can't have experienced some recent event that could have disturbed their kinematics. They're also isolated : at least 350 kpc from the nearest galaxy with a mean distance of 1 Mpc. So there's not much chance of them being produced in tidal encounters between galaxies. They really do seem to be primordial galaxies that rotate very, very slowly.
There's been a lot of hoo-hah about galaxies lacking dark matter recently. The consensus seems to be that those objects have incorrect distance estimates. But that's not credible for these objects (although surely people will investigate this too), so it really does seem that such weird galaxies do exist after all : their position on the TFR is exactly consistent with them completely lacking dark matter. They also seem to have no missing baryons.
What does all this mean ? Potentially, lots. No galaxy formation theory I've ever heard of predicts the existence of objects like this, so on the one hand, this is a major and unexpected development. The main question is why objects with similar gas masses apparently have radically different levels of star formation activity. Why do some gas-rich objects, the normal galaxies we're familiar with,convert most of their gas into stars, while others, like these, barely form any at all ?
On the other hand, the good news for the standard model is that these dark matter deficient galaxies could be strong evidence that dark matter really does exist. That sounds ironic, but in modified gravity theories the appearance of dark matter (i.e. high rotation speeds) arises solely from the gas and stars. Thus, any object of the same mass and distribution should show the same fast rotation. That these objects don't is likely to prove a big headache. It's easy in the dark matter model to claim that some galaxies simply don't have dark matter, whereas it's very difficult for modified gravity theories to say that identical objects should rotate differently. And the low scatter in the TFR has long been a puzzle, so this may finally solve this too.
Of course, watch this space. Hopefully this will be the start of the next interesting controversy !
EDIT : There's a second, quite similar paper here. I'm not giving it it's own post for several reasons : it doesn't cite the original Leisman paper, it uses an outdated version of the ALFALFA catalogue, some sentences are garbled, and one graph doesn't have proper axes labels (and it's unclear to which journal - if any - it's been submitted). It also uses only the ALFALFA data and doesn't have resolved observations. Still, the population of galaxies it describes is different to those presented in the other papers and their deviation from the TFR, and I can find no reason to dispute its main result.
Off the baryonic Tully-Fisher relation: a population of baryon-dominated ultra-diffuse galaxies
We study the gas kinematics traced by the 21-cm emission of a sample of six HI$-$rich low surface brightness galaxies classified as ultra-diffuse galaxies (UDGs). Using the 3D kinematic modelling code $\mathrm{^{3D}}$Barolo we derive robust circular velocities, revealing a startling feature: HI$-$rich UDGs are clear outliers from the baryonic Tully-Fisher relation, with circular velocities much lower than galaxies with similar baryonic mass.
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