What should we expect for the radial acceleration relation?

In the previous post, I related some of the history of the Radial Acceleration Relation (henceforth RAR). Here I’ll discuss some of my efforts to understand it. I’ve spent more time trying to do this in terms of dark matter than pretty much anything else, but I have not published most of those efforts. As I related briefly in this review, that’s because most of the models I’ve considered are obviously wrong. Just because I have refrained from publishing explanations of the RAR that are manifestly incorrect has not precluded others from doing so.

A theory is only as good as its prior. If a theory makes a clear prediction, preferably ahead of time, then we can test it. If it has not done so ahead of time, that’s still OK, if we can work out what it would have predicted without being guided by the data. A good historical example of this is the explanation of the excess perihelion precession of Mercury provided by General Relativity. The anomaly had been known for decades, but the right answer falls out of the theory without input from the data. A more recent example is our prediction of the velocity dispersions of the dwarf satellites of Andromeda. Some cases were genuine a priori predictions, but even in the cases that weren’t, the prediction is what it is irrespective of the measurement.

Dark matter-based explanations of the RAR do not fall in either category. They have always chased the data and been informed by it. This has been going on for so long that new practitioners have entered field unaware of the extent to which the simulations they inherited had already been informed by the data. They legitimately seem to think that there has been no fine-tuning of the models because they weren’t personally present for every turn of the knob.

So let’s set the way-back machine. I became concerned about fine-tuning problems in the context of galaxy dynamics when I was trying to explain the Tully-Fisher relation of low surface brightness galaxies in the mid-1990s. This was before I was more than dimly aware that MOND existed, much less taken it seriously. Many of us were making earnest efforts to build proper galaxy formation theories at the time (e.g., Mo, McGaugh, & Bothun 1994, Dalcanton, Spergel, & Summers 1997; Mo, Mao, & White 1998 [MMW]; McGaugh & de Blok 1998), though of course these were themselves informed by observations to date. My own paper had started as an effort to exploit the new things we had discovered about low surface brightness galaxies to broaden our conventional theory of galaxy formation, but over the course of several years, turned into a falsification of some of the ideas I had considered in my 1992 thesis. Dalcanton’s model evolved from one that predicted a shift in Tully-Fisher (as mine had) to one that did not (after the data said no). It may never be possible to completely separate theoretical prediction from concurrent data, but it is possible to ask what a theory plausibly predicts. What is the LCDM prior for the RAR?

In order to do this, we need to predict both the baryon distribution (g_bar) and that of the dark matter (g_obs-g_bar). Unfortunately, nobody seems to really agree on what LCDM predicts for galaxies. There seems to be a general consensus that dark matter halos should start out with the NFW form, but opinions vary widely about whether and how this is modified during galaxy formation. The baryonic side of the issue is simply seen as a problem.

That there is no clear prediction is in itself a problem. I distinctly remember expressing my concerns to Martin Rees while I was still a postdoc. He said not to worry; galaxies were such non-linear entities that we shouldn’t be surprised by anything they do. This verbal invocation of a blanket dodge for any conceivable observation did not inspire confidence. Since then, I’ve heard that excuse repeated by others. I have lost count of the number of more serious, genuine, yet completely distinct LCDM predictions I have seen, heard, or made myself. Many dozens, at a minimum; perhaps hundreds at this point. Some seem like they might work but don’t while others don’t even cross the threshold of predicting both axes of the RAR. There is no coherent picture that adds up to an agreed set of falsifiable predictions. Individual models can be excluded, but not the underlying theory.

To give one example, let’s consider the specific model of MMW. I make this choice here for two reasons. One, it is a credible effort by serious workers and has become a touchstone in the field, to the point that a sizeable plurality of practitioners might recognize it as a plausible prior – i.e., the closest thing we can hope to get to a legitimate, testable prior. Two, I recently came across one of my many unpublished attempts to explain the RAR which happens to make use of it. Unix says that the last time I touched these files was nearly 22 years ago, in 2000. The postscript generated then is illegible now, so I have to update the plot:

At first glance, this might look OK. The trend is at least in the right direction. This is not a success so much as it is an inevitable consequence of the fact that the observed acceleration includes the contribution of the baryons. The area below the dashed line is excluded, as it is impossible to have g_obs < g_bar. Moreover, since g_obs = g_bar+g_DM, some correlation in this plane is inevitable. Quite a lot, if baryons dominate, as they always seem to do at high accelerations. Not that these models explain the high acceleration part of the RAR, but I’ll leave that detail for later. For now, note that this is a log-log plot. That the models miss the data a little to the eye translates to a large quantitative error. Individual model galaxies sometimes fall too high, sometimes too low: the model predicts considerably more scatter than is observed. The RAR is not predicted to be a narrow relation, but one with lots of scatter with large intrinsic deviations from the mean. That’s the natural prediction of MMW-type models.

I have explored many flavors of [L]CDM models. They generically predicts more scatter in the RAR than is observed. This is the natural expectation, and some fine-tuning has to be done to reduce the scatter to the observed level. The inevitable need for fine-tuning is why I became concerned for the dark matter paradigm, even before I became aware that MOND predicted exactly this. It is also why the observed RAR was considered to be against orthodoxy at the time: everybody’s prior was for a large scatter. It wasn’t just me.

In order to build a model, one has to make some assumptions. The obvious assumption to make, at the time, was a constant ratio of dark matter to baryons. Indeed, for many years, the working assumption was that this was about 10:1, maybe 20:1. This type of assumption is built into the models of MMW, who thought that they worked provided “(i) the masses of disks are a few percent of those of their haloes”. The (i) is there because it is literally their first point, and the assumption that everybody made. We were terrified of dropping this natural assumption, as the obvious danger is that it becomes a rolling fudge factor, assuming any value that is convenient for explaining any given observation.

Unfortunately, it had already become clear by this time from the data that a constant ratio of dark to luminous matter could not work. The earliest I said this on the record is 1996. [That was before LCDM had supplanted SCDM as the most favored cosmology. From that perspective, the low baryon fractions of galaxies seemed natural; it was clusters of galaxies that were weird.] I pointed out the likely failure of (i) to Mo when I first saw a draft of MMW (we had been office mates in Cambridge). I’ve written various papers about it since. The point here is that, from the perspective of the kinematic data, the ratio of dark to luminous mass has to vary. It cannot be a constant as we had all assumed. But it has to vary in a way that doesn’t introduce scatter into relations like the RAR or the Baryonic Tully-Fisher relation, so we have to fine-tune this rolling fudge factor so that it varies with mass but always obtains the same value at the same mass.

A constant ratio of dark to luminous mass wasn’t just a convenient assumption. There is good physical reason to expect that this should be the case. The baryons in galaxies have to cool and dissipate to form a galaxy in the center of a dark matter halo. This takes time, imposing an upper limit on galaxy mass. But the baryons in small galaxies have ample time to cool and condense, so one naively expects that they should all do so. That would have been natural. It would also lead to a steeply increasing luminosity function, which is not observed, leading to the over-cooling and missing satellite problems.

Reconciling the observed and predicted mass functions is one of the reasons we invoke feedback. The energy produced by the stars that form in the first gas to condense are an energy source that feeds back into the surrounding gas. This can, in principle, reheat the remaining gas or expel it entirely, thereby precluding it from condensing and forming more stars as in the naive expectation. In principle. In practice, we don’t know how this works, or even if the energy provided by star formation couples to the surrounding gas in a way that does what we need it to do. Simulations do not have the resolution to follow feedback in detail, so instead make some assumptions (“subgrid physics”) about how this might happen, and tune the assumed prescription to fit some aspect of the data. Once this is done, it is possible to make legitimate predictions about other aspects of the data, provided they are unrelated. But we still don’t know if that’s how feedback works, and in no way is it natural. Rather, it is a deus ex machina that we invoke to save us from a glaring problem without really knowing how it works or even if it does. This is basically just theoretical hand-waving in the computational age.

People have been invoking feedback as a panacea for all ills in galaxy formation theory for so long that it has become familiar. Once something becomes familiar, everybody knows it. Since everybody knows that feedback has to play some role, it starts to seem like it was always expected. This is easily confused with being natural.

I could rant about the difficulty of making predictions with feedback afflicted models, but never mind the details. Let’s find some aspect of the data that is independent of the kinematics that we can use to specify the dark to luminous mass ratio. The most obvious candidate is abundance matching, in which the number density of observed galaxies is matched to the predicted number density of dark matter halos. We don’t have to believe feedback-based explanations to apply this, we merely have to accept that there is some mechanism to make the dark to luminous mass ratio variable. Whatever it is that makes this happen had better predict the right thing for both the mass function and the kinematics.

When it comes to the RAR, the application of abundance matching to assign halo masses to observed galaxies works out much better than the natural assumption of a constant ratio. This was first pointed out by Di Cintio & Lelli (2016), which inspired me to consider appropriately modified models. All I had to do was update the relation between stellar and halo mass from a constant ratio to a variable specified by abundance matching. This gives rather better results:

This looks considerably better! The predicted scatter is much lower. How is this accomplished?

Abundance matching results in a non-linear relation bewteen stellar mass and halo mass. For the RAR, the scatter is reduced by narrowing the dynamic range of halo masses relative to the observed stellar masses. There is less variation in g_DM. Empirically, this is what needs to happen – to a crude first approximation, the data are roughly consistent with all galaxies living in the same halo – i.e., no variation in halo mass with stellar mass. This was already known before abundance matching became rife; both the kinematic data and the mass function push us in this direction. There’s nothing natural about any of this; it’s just what we need to do to accommodate the data.

Still, it is tempting to say that we’ve succeeded in explaining the RAR. Indeed, some people have built the same kind of models to claim exactly this. While matters are clearly better, really we’re just less far off. By reducing the dynamic range in halo masses that are occupied by galaxies, the partial contribution of g_DM to the g_obs axis is compressed, and model lines perforce fall closer together. There’s less to distinguish an L* galaxy from a dwarf galaxy in this plane.

Nevertheless, there’s still too much scatter in the models. Harry Desmond made a specific study of this, finding that abundance matching “significantly overpredicts the scatter in the relation and its normalisation at low acceleration”, which is exactly what I’ve been saying. The offset in the normalization at low acceleration is obvious from inspection in the figure above: the models overshoot the low acceleration data. This led Navarro et al. to argue that there was a second acceleration scale, “an effective minimum acceleration probed by kinematic tracers in isolated galaxies” a little above 10^-11 m/s/s. The models do indeed do this, over a modest range in g_bar, and there is some evidence for it in some data. This does not persist in the more reliable data; those shown above are dominated by atomic gas so there isn’t even the systematic uncertainty of the stellar mass-to-light ratio to save us.

The astute observer will notice some pink model lines that fall well above the RAR in the plot above. These are for the most massive galaxies, those with luminosities in excess of L*. Below the knee in the Schechter function, there is a small range of halo masses for a given range of stellar masses. Above the knee, this situation is reversed. Consequently, the nonlinearity of abundance matching works against us instead of for us, and the scatter explodes. One can suppress this with an apt choice of abundance matching relation, but we shouldn’t get to pick and choose which relation we use. It can be made to work only because there remains enough uncertainty in abundance matching to select the “right” one. There is nothing natural about any this.

There are also these little hooks, the kinks at the high acceleration end of the models. I’ve mostly suppressed them here (as did Navarro et al.) but they’re there in the models if one plots to small enough radii. This is the signature of the cusp-core problem in the RAR plane. The hooks occur because the exponential disk model has a maximum acceleration at a finite radius that is a little under one scale length; this marks the maximum value that such a model can reach in g_bar. In contrast, the acceleration g_DM of an NFW halo continues to increase all the way to zero radius. Consequently, the predicted g_obs continues to increase even after g_bar has peaked and starts to decline again. This leads to little hook-shaped loops at the high acceleration end of the models in the RAR plane.

These hooks were going to be the segue to discuss more sophisticated models built by Pengfei Li, but that’s going to be a whole ‘nother post because these are quite enough words for now. So, until next time, don’t invest in bitcoins, Russian oil, or LCDM models that claim to explain the RAR.