There has already been one very quick attempt to match ΛCDM galaxy formation simulations to the radial acceleration relation (RAR). Another rapid preprint by the Durham group has appeared. It doesn’t do everything I ask for from simulations, but it does do a respectable number of them. So how does it do?

First, there is some eye-rolling language in the title and the abstract. Two words: natural (in the title) and accommodated (in the abstract). I can’t not address these before getting to the science.

Natural. As I have discussed repeatedly in this blog, and in the refereed literature, there is nothing natural about this. If it were so natural, we’d have been talking about it since Bob Sanders pointed this out in 1990, or since I quantified it better in 1998 and 2004. Instead, the modus operandi of much of the simulation community over the past couple of decades has been to pour scorn on the quality of rotation curve data because it did not look like their simulations. Now it is natural?


Accommodate. Accommodation is an important issue in the philosophy of science. I have no doubt that the simulators are clever enough to find a way to accommodate the data. That is why I have, for 20 years, been posing the question What would falsify ΛCDM? I have heard (or come up myself with) only a few good answers, and I fear the real answer is that it can’t be. It is so flexible, with so many freely adjustable parameters, that it can be made to accommodate pretty much anything. I’m more impressed by predictions that come ahead of time.

That’s one reason I want to see what the current generation of simulations say before entertaining those made with full knowledge of the RAR. At least these quick preprints are using existing simulations, so while not predictions in the strictest since, at least they haven’t been fine-tuned specifically to reproduce the RAR. Lots of other observations, yes, but not this particular one.

Ludlow et al. show a small number of model rotation curves that vary from wildly unrealistic (their NoAGN models peak at 500 km/s; no disk galaxy in the universe comes anywhere close to that… Vera Rubin once offered a prize for any that exceeded 300 km/s) to merely implausible (their StrongFB model is in the right ballpark, but has a very rapidly rising rotation curve). In all cases, their dark matter halos seem little affected by feedback, in contrast to the claims of other simulation groups. It will be interesting to follow the debate between simulators as to what we should really expect.

They do find a RAR-like correlation. Remarkably, the details don’t seem to depend much on the feedback scheme. This motivates some deeper consideration of the RAR.

The RAR plots observed centripetal acceleration, gobs, against that predicted by the observed distribution of baryons, gbar. We chose these coordinates because this seems to be the fundamental empirical correlation, and the two quantities are measured in completely independent ways: rotation curves vs. photometry. While measured independently, some correlation is guaranteed: physically, gobs includes gbar. Things only become weird when the correlation persists as gobs ≫ gbar.

The models are well fit by the functional form we found for the data, but with a different value of the fit parameter: g = 3 rather than 1.2 x 10-10 m s-2. That’s a factor of 2.5 off – a factor that is considered fatal for MOND in galaxy clusters. Is it OK here?

The uncertainty in the fit value is 1.20 ± 0.02. So formally, 3 is off by 90σ. However, the real dominant uncertainty is systematic: what is the true mean mass-to-light ratio at 3.6 microns? We estimated the systematic uncertainty to be ± 0.24 based on an extensive survey of plausible stellar population models. So 3 is only 7.5σ off.

The problem with systematic uncertainties is that they do not obey Gaussian statistics. So I decided to see what we might need to do to obtain g = 3 x 10-10 m s-2. This can be done if we take sufficient liberties with the mass-to-light ratio.

The radial acceleration relation as observed (open points fit by blue line) and modeled (red line). Filled points are the same data with the disk mass-to-light ratio reduced by a factor of two.

Indeed, we can get in the right ball park simply by reducing the assumed mass-to-light ratio of stellar disks by a factor of two. We don’t make the same factor of two adjustment to the bulge components, because the data don’t approach the 1:1 line at high accelerations if this is done. So rather than our fiducial model with M*/L(disk) = 0.5 M/L and M*/L(bulge) = 0.7 M/L (open points in plot), we have M*/L(disk) = 0.25 M/L and M*/L(bulge) = 0.7 M/L (filled points in plot). Lets pretend like we don’t know anything about stars and ignore the fact that this change corresponds to truncating the IMF of the stellar disk so that M dwarfs don’t exist in disks, but they do in bulges. We then find a tolerable match to the simulations (red line).

Amusingly, the data are now more linear than the functional form we assumed. If this is what we thought stars did, we wouldn’t have picked the functional form the simulations apparently reproduce. We would have drawn a straight line through the data – at least most of it.

That much isn’t too much of a problem for the models, though it is an interesting question whether they get the shape of the RAR right for the normalization they appear to demand. There is a serious problem though. That becomes apparent in the lowest acceleration points, which deviate strongly below the red line. (The formal error bars are smaller than the size of the points.)

It is easy to understand why this happens. As we go from high to low accelerations, we transition from bulge dominance to stellar disk dominance to gas dominance. Those last couple of bins are dominated by atomic gas, not stars. So it doesn’t matter what we adopt for the stellar mass-to-light ratio. That’s where the data sit: well off the simulated line.

Is this fatal for these models? As presented, yes. The simulations persist in predicting higher accelerations than observed. This has been the problem all along.

There are other issues. The scatter in the simulated RAR is impressively small. Much smaller than I expected. Smaller even than the observational scatter. But the latter is dominated by observational errors: the intrinsic relation is much tighter, consistent with a δ-function. The intrinsic scatter is what they should be comparing their results to. They either fail to understand, or conveniently choose to gloss over, the distinction between intrinsic scatter and that induced by random errors.

It is worth noting that some of the same authors make this same mistake – and it is a straight up mistake – in discussing the scatter in the baryonic Tully-Fisher relation. The assertion there is “the scatter in the simulated BTF is smaller than observed”. But the observed scatter is dominated by observational errors, which we have taken great care to assess. Once this is done, there is practically no room left over for intrinsic scatter, which is what the models display. This is important, as it completely inverts the stated interpretation. Rather than having less scatter than observed, the simulations exhibit more scatter than allowed.

Can these problems be fixed? No doubt. See the comments on accommodation above.

12 thoughts on “Another quick-trick simulation result

  1. About intrinsic vs extrinsic scatter: in Fig 3. (lower panel) of your RAR paper, what was your procedure to assess the extrinsic scatter (the two solid red lines)?

    In particular, within the extrinsic scatter, could you please elaborate on how one goes about estimating observational errors, i.e. where the numbers in Table 1 come from? Only the mass-to-light line in this table is discussed in the maint text, as far as I can tell.


    1. This is just error propagation. The errors on distances, velocities, etc. are described in the SPARC data paper and we’ll go over the propagation into the RAR in more detail in a paper due out on the arXiv on Monday.
      The one thing I’d like most to improve is the distances to dwarf galaxies. This is a straightforward Hubble project. It isn’t even a huge HST project, but so far the TAC hasn’t seen the importance of it.
      One subtlety worth noting: because of the varying slope of the relation, the uncertainty in the mass-to-light ratio matters more at high accelerations than at low. In contrast, the data get progressively less accurate at low accelerations. These two effects counteract each other, so the expected scatter (the red lines to which you refer) don’t flare too much at either end of the relation.


  2. Stacy,
    thank you for this post. I entirely agree that the community is behaving unnaturally by trying to accommodate the word natural into contrived models. Methinks this is damaging science at its core.

    Concerning rigorous science: Congratulations on providing an extremely important contribution with your and your collaborator’s paper and to this fundamental field. The data very convincingly demonstrate that effective gravitation departs from Newton’s law in the classical very weak field limit. Stunning is that the data beautifully confirm the prediction by Milgrom in 1983! Indeed, the data can be reproduced neither with cold nor with warm dark matter models, as this detailed analysis by Xufen Wu and myself has shown: .
    In-line with this is Mike Disney et al.’s important observation that disk galaxies are all far too similar to be a result of the dark-matter-driven merging history, where each galaxy has its own unknowable and stochastic merging tree:

    But allow me to disagree with you on one point: you write that the dark matter models cannot be falsified. I know this comes from a frustratingly long history of dealing with such issues, but it is not correct, and one must differentiate between the scientifically-robust procedure and the sociologically-driven unscientific attempts to negate this by parts of the community (we had this in history before).

    Concerning the former: there are two explicit and very hard tests which already rule out the existence of dark matter. The one is that the dark matter halos imply dynamical friction of any galaxy with its own dark matter halo entering a larger host dark matter halo. The observational evidence does not show this process. The other test is that the spatial distribution in rotating disk-like systems of satellite galaxies of our Milky Way, of Andromeda, of the M81 group and the Cen A group are entirely inconsistent with the expected spheroidal distribution, even taking into account infall from dark matter and baryonic cosmological filaments. This has been conclusively shown by the recent ground-breaking work by Marcel Pawlowsky and Rodrigo Ibata, and was already evident in 2005 as published here:…431..517K .
    For those interested for a discussion of these and other tests:

    Pavel Kroupa

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  3. To quote one response I got to the query I posed: Tad Pryor (1997): “CDM has been falsified many times.”
    If we hold CDM as a theory to the same standards as other theories (e.g., MOND) then yes, it has been falsified repeatedly, having many problems in addition to the ones you note that are as severe for it as the bullet cluster is widely perceived to be for MOND. Nevertheless, we seem to repeatedly accommodate these problems, or choose to ignore them.
    I was thinking of a more general, weaker standard. Dark matter as a concept cannot be falsified. If it isn’t WIMPs, it’s Axions. If it isn’t Axions, we are free to make up something else. And then something else again, ad infinitum. We can make up no end of hypothetical particles to serve the role of non-baryonic dark matter. It is in this sense that I mean that dark matter cannot be falsified. At best we can exclude specific hypotheses, like WIMPs of mass ~100 GeV with cross-sections > 1E-45. Even for WIMPs we are free to move the goal posts, and have repeatedly done so. I call this the express elevator to hell:


  4. One would need to postulate a type of “dark matter” which does not provide dynamical friction, and this contradicts this speculative dark matter having mass, which however it must have to fulfill the cosmological constraints. Thus the falsification via dynamical friction alone is terminal.


  5. I must say, the words by Roger Cotes on the preface to Newton’s Principia, second edition, probably written by Newton himself, sound eerily relevant… When describing exactly how some people completely go down the wrong path in science:

    “But when they take a liberty of imagining at pleasure unknown figures and magnitudes, and uncertain situations and motion of the parts; and moreover of supposing occult fluids, freely pervading the pores of bodies, endued with an all-performing subtilty, and agitated, with occult motions; they now run out into dreams and chimera’s, and neglect the true constitution of things; which certainly is not to be expected from fallacious conjectures, when we can scarce reach it by the most certain observations. Those who fetch from by hypotheses the foundation on which they build their speculations, may form indeed an ingenious romance, but a romance it will still be.”

    Presenting DM as a proven scientific fact (rather than as an ad hoc hypothesis of diminishing plausibility) is not only patently false, it is doing science a disservice.


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