A unique prediction of MOND
One curious aspect of MOND as a theory is the External Field Effect (EFE). The modified force law depends on an absolute acceleration scale, with motion being amplified over the Newtonian expectation when the force per unit mass falls below the critical acceleration scale a0 = 1.2 x 10-10 m/s/s. Usually we consider a galaxy to be an island universe: it is a system so isolated that we need consider only its own gravity. This is an excellent approximation in most circumstances, but in principle all sources of gravity from all over the universe matter.
The EFE in dwarf satellite galaxies
An example of the EFE is provided by dwarf satellite galaxies – small galaxies orbiting a larger host. It can happen that the stars in such a dwarf feel a stronger acceleration towards the host than to each other – the external field exceeds the internal self-gravity of the dwarf . In this limit, they’re more a collection of stars in a common orbit around the larger host than they are a self-gravitating island universe.
A weird consequence of the EFE in MOND is that a dwarf galaxy orbiting a large host will behave differently than it would if it were isolated in the depths of intergalactic space. MOND obeys the Weak Equivalence Principle but does not obey local position invariance. That means it violates the Strong Equivalence Principle while remaining consistent with the Einstein Equivalence Principle, a subtle but important distinction about how gravity self-gravitates.
Nothing like this happens conventionally, with or without dark matter. Gravity is local; it doesn’t matter what the rest of the universe is doing. Larger systems don’t impact smaller ones except in the extreme of tidal disruption, where the null geodesics diverge within the lesser object because it is no longer small compared to the gradient in the gravitational field. An amusing, if extreme, example is spaghettification. The EFE in MOND is a much subtler effect: when near a host, there is an extra source of acceleration, so a dwarf satellite is not as deep in the MOND regime as the equivalent isolated dwarf. Consequently, there is less of a boost from MOND: stars move a little slower, and conventionally one would infer a bit less dark matter.
The importance of the EFE in dwarf satellite galaxies is well documented. It was essential to the a priori prediction of the velocity dispersion in Crater 2 (where MOND correctly anticipated a velocity dispersion of just 2 km/s where the conventional expectation with dark matter was more like 17 km/s) and to the correct prediction of that for NGC 1052-DF2 (13 rather than 20 km/s). Indeed, one can see the difference between isolated and EFE cases in matched pairs of dwarfs satellites of Andromeda. Andromeda has enough satellites that one can pick out otherwise indistinguishable dwarfs where one happens to be subject to the EFE while its twin is practically isolated. The speeds of stars in the dwarfs affected by the EFE are consistently lower, as predicted. For example, the relatively isolated dwarf satellite of Andromeda known as And XXVIII has a velocity dispersion of 5 km/s, while its near twin And XVII (which has very nearly the same luminosity and size) is affected by the EFE and consequently has a velocity dispersion of only 3 km/s.
The case of dwarf satellites is the most obvious place where the EFE occurs. In principle, it applies everywhere all the time. It is most obvious in dwarf satellites because the external field can be comparable to or even greater than the internal field. In principle, the EFE also matters even when smaller than the internal field, albeit only a little bit: the extra acceleration causes an object to be not quite as deep in the MOND regime.
The EFE from large scale structure
Even in the depths of intergalactic space, there is some non-zero acceleration due to everything else in the universe. This is very reminiscent of Mach’s Principle, which Einstein reputedly struggled hard to incorporate into General Relativity. I’m not going to solve that in a blog post, but note that MOND is much more in the spirit of Mach and Lorenz and Einstein than its detractors generally seem to presume.
Here I describe the apparent detection of the subtle effect of a small but non-zero background acceleration field. This is very different from the case of dwarf satellites where the EFE can exceed the internal field. It is just a small tweak to the dominant internal fields of very nearly isolated island universes. It’s like the lapping of waves on their shores: hardly relevant to the existence of the island, but a pleasant feature as you walk along the beach.
The universe has structure; there are places with lots of galaxies (groups, clusters, walls, sheets) and other places with very few (voids). This large scale structure should impose a low-level but non-zero acceleration field that should vary in amplitude from place to place and affect all galaxies in their outskirts. For this reason, we do not expect rotation curves to remain flat forever; even in MOND, there comes an over-under point where the gravity of everything else takes over from any individual object. A test particle at the see-saw balance point between the Milky Way and Andromeda may not know which galaxy to call daddy, but it sure knows they’re both there. The background acceleration field matters to such diverse subjects as groups of galaxies and Lyman alpha absorbers at high redshift.
As an historical aside, Lyman alpha absorbers at high redshift were initially found to deviate from MOND by many orders of magnitude. That was withoug the EFE. With the EFE, the discrepancy is much smaller, but persists. The amplitude of the EFE at high redshift is very uncertain. I expect it is higher in MOND than estimated because structure forms fast in MOND; this might suffice to solve the problem. Whether or not this is the case, it makes a good example of how a simple calculation can make MOND seem way off when it isn’t. If I had a dollar for every time I’ve seen that happen, I could fly first class.
I made an early estimate of the average intergalactic acceleration field, finding the typical environmental acceleration eenv to be about 2% of a0 (eenv ~ 2.6 x 10-12 m/s/s, see just before eq. 31). This is highly uncertain and should be location dependent, differing a lot from voids to richer environments. It is hard to find systems that probe much below 10% of a0, and the effect it would cause on the average (non-satellite) galaxy is rather subtle, so I have mostly neglected this background acceleration as, well, pretty negligible.
This changed recently thanks to Kyu-Hyun Chae and Harry Desmond. We met at a conference in Bonn a year ago September. (Remember travel? I used to complain about how much travel work involved. Now I miss it – especially as experience demonstrates that some things really do require in-person interaction.) Kyu thought we should be able to tease out the EFE from SPARC data in a statistical way, and Harry offered to make a map of the environmental acceleration based on the locations of known galaxies. This is a distinct improvement over the crude average of my ancient first estimate as it specifies the EFE that ought to occur at the location of each individual galaxy. The results of this collaboration were recently published open-access in the Astrophysical Journal.
This did not come easily. I think I mentioned that the predicted effect is subtle. We’re no longer talking about the effect of a big host on a tiny dwarf up close to it. We’re talking about the background of everything on giant galaxies. Space is incomprehensibly vast, so every galaxy is far, far away, and the expected effect is small. So my first reaction was “Sure. Great idea. No way can we do this with current data.” I am please to report that I was wrong: with lots of hard work, perseverance, and the power of Bayesian statistics, we have obtained a positive detection of the EFE.
One reason for my initial skepticism was the importance of data quality. The rotation curves in SPARC are a heterogeneous lot, being the accumulated work of an entire community of radio astronomers over the course of several decades. Some galaxies are bright and their data stupendous, others… not so much. Having started myself working on low surface brightness galaxies – the least stupendous of them all – and having spent much of the past quarter century working long and hard to improve the data, I tend to be rather skeptical of what can be accomplished.
An example of a galaxy with good data is NGC 5055 (aka M63, aka the Sunflower galaxy, pictured atop as viewed by the Hubble Space Telescope). NGC 5055 happens to reside in a relatively high acceleration environment for a spiral, with eenv ~ 9% of a0. For comparison, the acceleration at the last measured point of its rotation curve is about 15% of a0. So they’re within a factor of two, which is pretty much the strongest effect in the whole sample. This additional bit of acceleration means NGC 5055 is not quite as deep in the MOND regime as it would be all by its lonesome, with the net effect that the rotation curve is predicted to decline a little bit faster than it would in the isolated case, as you can see in the figure below. See that? Or is it too subtle? I think I mentioned the effect was pretty subtle.
That this case works well is encouraging. I like to start with a good case: if you can’t see what you’re looking for in the best of the data, stop. But I still didn’t hold out much hope for the rest of the sample. Then Kyu showed that the most isolated galaxies – those subject to the lowest environmental accelerations – showed no effect. That sounds boring, but null results are important. It could happen that the environmental acceleration was a promiscuous free parameter that appears to improve a fit without really adding any value. That it declined to do that in cases where it shouldn’t was intriguing. The galaxies in the most extreme environments show an effect when they should, but don’t when they shouldn’t.
Statistical detection of the EFE
Statistics become useful for interpreting the entirety of the large sample of galaxies. Because of the variability in data quality, we knew some cases would go astray. But we only need to know if the fit for any galaxy is improved relative to the case where the EFE is neglected, so each case sets its own standard. This relative measure is more robust than analyses that require an assessment of the absolute fit quality. All we’re really asking the data is whether the presence of an EFE helps. To my initial and ongoing amazement, it does.
The figure above shows the amplitude of the EFE that best fits each rotation curve along the x-axis. The median is 5% of a0. This is non-zero at 4.7σ, and our detection of the EFE is comparable in quality to that of the Baryon Acoustic Oscillation or the accelerated expansion of the universe when these were first accepted. Of course, these were widely anticipated effects, while the EFE is expected only in MOND. Personally, I think it is a mistake to obsess over the number of σ, which is not as robust as people like to think. I am more impressed that the peak of the color map (the darkest color in the data density map above) is positive definite and clearly non-zero.
Taken together, the data prefer a small but clearly non-zero EFE. That’s a statistical statement for the whole sample. Of course, the amplitude (e) of the EFE inferred for individual galaxies is uncertain, and is occasionally negative. This is unphysical: it shouldn’t happen. Nevertheless, it is statistically expected given the amount of uncertainty in the data: for error bars this size, some of the data should spill over to e < 0.
I didn’t initially think we could detect the EFE in this way because I expected that the error bars would wash out the effect. That is, I expected the colored blob above would be smeared out enough that the peak would encompass zero. That’s not what happened, me of little faith. I am also encouraged that the distribution skews positive: the error bars scatter points in both direction, and wind up positive more often than negative. That’s an indication that they started from an underlying distribution centered on e > 0, not e = 0.
The y-axis in the figure above is the estimate of the environmental acceleration based on the 2M++ galaxy catalog. This is entirely independent of the best fit e from rotation curves. It is the expected EFE from the distribution of mass that we know about. The median environmental EFE found in this way is 3% of a0. This is pretty close to the 2% I estimated over 20 years ago. Given the uncertainties, it is quite compatible with the median of 5% found from the rotation curve fits.
In an ideal world where all quantities are perfectly known, there would be a correlation between the external field inferred from the best fit to the rotation curves and that of the environment predicted by large scale structure. We are nowhere near to that ideal. I can conceive of improving both measurements, but I find it hard to imagine getting to the point where we can see a correlation between e and eenv. The data quality required on both fronts would be stunning.
Then again, I never thought we could get this far, so I am game to give it a go.