Last time, we discussed the remarkable result that gravitational lensing extends the original remarkable result of flat rotation curves much farther out, as far as the data credibly probe. This corroborates and extends the result of Brouwer et al. They did a thorough job, but one thing they did not consider was Tully-Fisher. If the circular speed inferred from gravitational lensing remains constant, does this flat velocity fall on the same Tully-Fisher relation that is seen in kinematic data?

We set out to answer this question. Along the way, we did three new things: 1. Dr. Mistele derived an improved method for doing the lensing analysis, extending the radial range over which the data were credible. 2. He explored the criteria by which galaxies were judged to be isolated, finding a morphological type dependence on how far out one had to exclude. 3. We reanalyzed the stellar masses of the KiDS sample to be consistent with those we used when analyzing the kinematic data of SPARC galaxies. The first two are connected, as how far out we can trust the data depends on how well we can define a clean sample of isolated galaxies. The third resolved an apparent offset between early type galaxies (ETGs, aka ellipticals) and late type galaxies (LTGs, aka spirals) seen by Brouwer et al. That appears to be an artifact of stellar population modeling, as I suspected when I first discussed their result. We don’t need to do any fitting of the mass-to-light ratio; the the apparent offset between types disappears when we use use the same population models for both kinematic and lensing data.

I could write a lot about each of these, but most of it is the stuff of technical details that would be dull to many people. If you’re into that sort of thing, go and read the long science paper which is where such details reside. Here I just want to describe the Tully-Fisher result. Spoiler alert: it is the same as that from kinematics.

First off, I’m talking strictly about the Baryonic Tully-Fisher relation: the scaling between baryonic mass and the flat rotation speed. To address this, we bin the lensing data by mass. The mass of each bin is well defined by the average of the many thousands of galaxies within the bin. By far the dominant uncertainty is the systematic in stellar mass caused by stellar population modeling. We went through this with a fine tooth comb, and I’m confident we have an internally self-consistent result. That doesn’t preclude it being wrong in an absolute sense – such is the nature of astronomy – but we can at least make a straight comparison between kinematic and lensing data using the same best-effort stellar mass estimates.

For the velocity, we estimate the average effective rotation curve for each mass bin from the lensing data. We also split the data into morphological types to look for differences. The statistics go down when one divvies up the data like this, so the uncertainties go up, but there are enough KiDS galaxies to define four mass bins. Here are their inferred rotation curves:

Figure 1 from Mistele et al. (2024). Circular velocities implied by weak lensing for four baryonic mass bins (most to least massive from the top row to the bottom) for the whole sample (left column), for LTGs (middle column), and for ETGs (right column). The lowest ETG mass bin is not shown because it contains too few lenses. Instead we show results for lenses with spectroscopic redshifts from GAMA, without splitting by mass or type due to the small sample size (gray and white symbols). For comparison, we also show results for KiDS without splitting by mass or type (small yellow symbols). Open symbols at small radii indicate where lenses are not yet effective point masses. Light-colored symbols (not-outlined) at large radii indicate data points that may still be reliable but where the isolation criterion is less certain. The error bars show the statistical errors. Horizontal lines and the corresponding shaded regions indicate the inferred Vflat values and uncertainties that we use for the BTFR. The extent of the horizontal lines indicates the radial range we consider when calculating Vflat.

Note that the average over all KiDS data shown in the lower right bin is the data shown in the press release image in the previous post, but the x-axis is logarithmic here. The GAMA data in that bin provide an important cross-check, as these galaxies have spectroscopic redshifts. They give the same answer as the larger KiDS sample, which relies on photometric redshifts. We need the larger sample to consider finer bins in mass, which is the rest of the plot.

Another thing to note here is that all the data in all the bins are consistent with remaining flat. There are some hints of a turn down at very large radii, particularly for LTGs in the second and third row, but these are not statistically significant, and only happen where the data start to become untrustworthy. Where exactly that happens is a judgement call.

Let’s take a closer look, with a comparison to radio data:

Figure 2 from Mistele et al. (2024). The circular velocities from weak lensing (circles) compared with those from gas kinematics (diamonds). The individual galaxies illustrated here have among the most extended 21 cm rotation curves in their mass bins; the lensing data continue to much larger radii still. The error bars show the statistical error, while the gray band indicates the systematic uncertainty in the radial accelerations. Symbol colors are as in Figure 1. Open symbols at large radii indicate where lenses are not sufficiently isolated. The solid green lines indicate the circular velocities of NFW halos and baryonic point masses appropriate for each mass bin. Green crosses indicate each NFW halo’s virial radius. The light green band adds a qualitative estimate of a two-halo term contribution to the NFW halo, which may become important at large radii in case our isolation criterion is imperfect there.

Again we see that the lensing data, averaged over many galaxies, extend much further out than the rotation curve of any one individual. The x-axis is again logarithmic, so the lensing data go way further out. They trace to 1 Mpc, which is crazy far beyond the observed ends of the most extended individual galaxies. A more conservative limit is the 300 kpc estimated by Brouwer et al. Surely we can go further than that, but how much further remains a judgement call.

What should we expect? The green lines show the rotation curve we’d expect for galaxy in an NFW halo with parameters specified by the stellar mass-halo mass relation of Kravtsov et al. (2018). Not all such relations agree well with kinematic data; this is the case that agrees most closely. We have intentionally cherry-picked the relation that makes LCDM look best. And it does look good up to a point, for example in the top two mass bins out to the virial radius of the halo (tick marks). Beyond that, not so much, and not at all for the two lower mass bins. The data extend far enough out that we should see the predicted decline. We do not.

The green line only represents the expected halo of the primary galaxy. When one gets so far out, one has to worry about all the other stuff out there. We’ve selected galaxies to be isolated, so there isn’t much that is luminous. But we can only exclude down to some sensitivity limit, there might be lots of tiny dwarf galaxies whose mass adds up and starts to affect the result. And of course there can be completely invisible dark matter. The green band attempts to account for this extra stuff in the so-called 2-halo term. This is hard to do, but we’ve made our best estimate based on the LCDM power spectrum. I’m sure the 2-halo term can be adjusted, but the shape is wrong. It will take some fine-tuning to get an effectively flat rotation curve out of the 1-halo+2 halo terms. They don’t naturally do that.

Something that is easy to do is define a flat value of the rotation speed. That’s just the average over the lensing data. We exclude the points at R < 50 kpc, as the assumption of a spherical mass that we make in the lensing analysis isn’t really valid at those comparatively small scales. We tried averaging over a bunch of different ranges, all of which gave pretty much the same answer. For illustration, we show two cases: a conservative one that only uses the data at R < 300 kpc, and another that goes out to 1 Mpc. Having measured Vflat over these ranges, we can plot Tully-Fisher:

Figure 3 from Mistele et al. (2024). The baryonic Tully–Fisher relation implied by weak lensing for the entire sample (yellow symbols, left column) and for ETGs and LTGs separately (red and blue symbols, right column). The Vflat values are weighted averages of the Vc values shown in Figure 1 for 50 kpc < R < 300 kpc (first row) and 50 kpc < R < 1000 kpc (second row). Vertical error bars represent a 0.1 dex systematic uncertainty on M*/L. For comparison, we also show the best fit to the kinematic data from Lelli et al. (2019; solid gray line) and the corresponding binned kinematic data (white diamonds).

Lo and behold, we find the same Baryonic Tully-Fisher relation from lensing data as we find with kinematics. This does not surprise me, but it didn’t have to be true. It shouldn’t be true in LCDM: if we can measure out to the virial radius, we should see some indication of a decline in velocity. We have and we don’t.

We also see no statistically significant separation between ETGs and LTGs. This is important, as a theory like MOND predicts that there should be no morphology dependence: only the baryonic mass matters. Brouwer et al. did see an indication of such a split, but it was small compared to the uncertainty in stellar population models. We don’t see it when we use our own stellar mass estimates. This is particularly true in the more conservative (300 kpc) case. There is a hint of a segregation when we average out to 1000 kpc, but the statistics say this isn’t significant. Since the lowest mass bin is most affected, I suspect this is a hint that the isolation criterion is failing first for the smallest galaxies. That makes sense, as the sensitivity limit on interlopers makes the lowest mass bin most susceptible to having its signal inappropriately boosted. It also makes sense that ETGs would be affected first, as ETGs are known to be more clustered than LTGs. It is really hard to define an isolated sample of ETGs, as discussed at length by Mistele et al.

The lensing data corroborate previous kinematic work. Rotation curves are flat. The amplitude of the flat rotation speed correlates with baryonic mass as Mb ∝ Vf4. The radial acceleration relation extends to very low accelerations. These are all predictions of MOND. Moreover they are unique predictions: predictions made a priori by MOND and only by MOND. Dark matter models so far provide no satisfactory explanation*.

That hasn’t prevented people from overlooking these basic facts in order to get to the apparent if statistically meaningless difference between ETGs and LTGs. Nevermind the successes! The slight offset between ETGs and LTGs falsify MOND! Seriously: other scientists have already made this argument to me while completely eliding the successes of MOND. It’s a case of refusing to see the forest for a tree that’s a little away from the others.

I think I said something about how this would happen when I first wrote about Brouwer et al‘s lensing result. Ah yes, here it is:

MOND predicted this behavior well in advance of the observation, so one would have to bend over backwards, rub one’s belly, and simultaneously punch oneself in the face to portray this as anything short of a fantastic success of MOND.

I say that because I’m sure people will line up to punch themselves in the face in exactly this fashion.

And so it has come to pass. Sometimes human behavior is as predictable as galaxy dynamics.

*There are many claims to explain limited portions of these results, but none are satisfactory. There is no LCDM model that matches the entire dynamic range of the radial acceleration relation. See, for example, Fig. 5 of Brouwer et al. (reproduced below), which shows the MICE and BAHAMAS simulations. Neither extend into the regime that is well-constrained by kinematic data; there is no reason to think they would successfully do so and good reason to think otherwise. MICE comes nowhere close to this regime and has no baryonic physics that would allow it do even address this question. BAHAMAS comes close but appears to turn away from the kinematic data before it gets there. We’ve built our own LCDM models; they don’t work either. We can make them come close, but only over a limited dynamic range, not over the full span of the data. It isn’t good enough to only explain a limited range of the data. One has to explain the full range, and the only theory that does that so far is MOND.

Fig. 5 from Brouwer et al. showing the radial acceleration relation inferred from the MICE (red band) and BAHAMAS (orange band) simulations. Not also that in our assessment of stellar masses, the lower acceleration points translate a bit to the right.
Stacy McGaugh is an astrophysicist and cosmologist who studies galaxies, dark matter, and theories of modified gravity. He is an expert on low surface brightness galaxies, a class of objects in which the stars are spread thin compared to bright galaxies like our own Milky Way. He demonstrated that these dim galaxies appear to be dark matter dominated, providing unique tests of theories of galaxy formation and modified gravity. Professor McGaugh is currently the chair of the Department of Astronomy at Case Western Reserve University in Cleveland, Ohio, and director of the Warner and Swasey Observatory. Previously he was a member of the faculty at the University of Maryland, having also held research fellowships at Rutgers, the Department of Terrestrial Magnetism of the Carnegie Institution of Washington, and the Institute of Astronomy at the University of Cambridge after earning his Ph.D. from the University of Michigan.

28 thoughts on “Tully-Fisher from gravitational lensing”

  1. More consistent empirical evidence showing that dark matter is an artificial construct that nobody has ever observed, but very convenient and maleable to keep the current framework “free” of obvious contradictions.

    It seems that the dark matter coffin needs a very large number of nails to be completed before burying it, for more and more people it already has enough nails.

    Thans.

  2. Hi Stacy,

    Thank you for this post on “Tully-Fisher relation”.

    Please can you clarify where the mass estimates come from? Presumably, with the weak gravitational lensing data you should end up with estimates for the total mass (baryonic + dark matter) as well as estimates for the circular speed. In which case your Tully-Fisher plot would be for total mass not baryonic mass. I’m sure I’ve missed something somewhere. Thanks.

    1. The mass we’re talking about here is explicitly the baryonic mass: the sum of stars and gas. It is an inventory of what we can see; no dark matter.

      The dynamical mass would include dark matter. That is connected to the velocity through M(R) = R*V^2/G. Since V = constant here, the inferred dynamical mass increases linearly without bound.

      1. “In the standard lambda-CDM model of cosmology, the mass–energy content of the universe is 5% ordinary matter, 26.8% dark matter, and 68.2% a form of energy”.

        Something is deeply wrong with the standard lambda-CDM model of cosmology, but that is really irrelevant when it continues to be a cash cow, the gate keepers control funding and you have to be a “team member” to get that funding or that promotion, still the dark matter thing is already a living dead.

  3. Just asking, although I suppose it depends on the specific transition from newtonian to MOND regime: what’s the fuss about Cassini? They claim like 10 sigma, but I doubt it – actually dunno if we can know saturn’s mass that accurately.

    1. I don’t know and don’t care. As a phenomenon in the very high acceleration regime, the orbit of Cassini is incapable of testing any fundamental prediction of MOND. A similar fuss was made about pulsars a decade ago; people were testing TeVeS (which did fail) and equating that with all conceivable MOND theories. It’s a classic overstatement.

      That said, without looking, I expect it has something to do with whether or not there is evidence for a perturbation from the external field of the Milky Way. Some forms of MOND (those that modify the Poisson equation) predict such a thing, so I presume this is a signal that isn’t seen. But not all possibilities for MOND require such a signal, e.g., modified intertia.

      Oddly, this situation is analogous to that for dark matter: a positive signal would be clear corroborative evidence, but a negative signal just tells you what you haven’t found. So it’s like not seeing WIMP in a DM detection experiment – it excludes that particular kind of WIMP, but it doesn’t falsify the idea of DM.

      I am not paying close attention to the solar system and wide binary results because there are a lot of straight-up contradictory claims. I don’t know which to believe. What I do know is that phenomena in the low acceleration regime where MOND does make fundamental predictions show the predicted behavior and that this is not satisfactorily explained by dark matter. That’s what I care about, and what requires explanation in any final theory. All we know so far is what doesn’t work.

      1. In the search for a hypothetical Planet 9, would one expect MOND to have a measurable effect?

  4. I’d like to ask about the time variation of a0 (on the scale of billions of years). What’s the situation now?

    1. I am not aware of any credible evidence that suggests that a0 varies with time.

      The best evidence against time variation is probably that the Baryonic Tully-Fisher relation remains fixed, within fairly modest uncertainties, since z ~ 2.5, which was 11 billion years ago.

  5. Has any work been done on the effect of dynamical friction? As I understand it, the gravitational drag of stars and gas upon each other would increase towards the galactic centre, as a function of their density in the disc, and slow rotation velocities correspondingly – and vice versa in the other direction. Beyond a certain point, the drag would be negligible.

    Regarding the anomaly of the Milky Way’s RC, I have just come across a paper by MH Chan arguing that the MW data do not support a universal radial acceleration relation. This would suggest that the anomaly is real.

    1. Oh my yes. Pavel Kroupa asserts that dynamical friction falsifies dark matter, arguing that there must be a lot more than is observed.

      The paper by Chan that you cite is comically wrong. Not even worth debunking.

  6. Yes, but the concept of dynamical friction is not confined to interaction with dark matter. I was wondering how the decelerating effect might apply to the ordinary matter in galaxies, which thins away from the centre. Presumably some of the rotation curve profiles can be attributed to there being less matter, specifically gas, and therefore less friction further out. Is there literature addressing this particular aspect?

      1. Well, that’s weird. Don’t know where WordPress deep-sixed that reply. But yes, of course – dynamical friction happens between stars as well as dark matter: it’s just gravity; it doesn’t matter what they’re made of. But there is a lot less dynamical friction if there is only the stars rather than stars plus ten (or more) times as much dark matter. So it is a matter of magnitude. For the rotation curve itself, dynamical friction is negligible. It does matter for things like bars in the centers of galaxies, for which the observed pattern speeds exclude a dense (cuspy) halo but not DM distributions with low density cores.

        1. Yes, thank you, but I was explicitly asking about friction within the gas component. Is there not a distinction to be made between friction in the sense of collisions (as with the gas whirling around SMBHs) and gravitational interaction between widely separated objects, namely stars?

          1. Yes. Hydrodynamics separates the behavior of gas from stars, chiefly as it interacts with itself, e.g., in the bullet cluster. In spiral galaxies, hydrodynamics matters to issues of disk stability and the behavior of spiral arms. There is no additional dynamical friction, though. Dynamical friction happens when an object moves through a medium of other gravitating objects. In galaxies, these can be treated as fluids, be they gas are lots of point-mass stars. The collective effect is the same.

  7. Regarding your reply to a comment on the related post: ‘Orbits do not become more expansive further out; spiral arms do not result from stars emanating from the nucleus. Near-circular orbits are measured by Gaia using proper motions as well as Doppler radial velocities.’ These assertions were exactly what I was trying to probe and get beyond. Reference to Gaia is beside the point when I was expressly asking about galaxies further than the Milky Way. And I am not ‘pushing’ anything.

    Nor was I saying anything about a relation between the stellar part of the RC and the light distribution (Renzo’s rule) but rather wondering about a possible relation between the stellar part of the RC and stellar orbits, suggesting, on the evidence of the spiral arms in which most stars reside, that the orbits might be curving outwards.

    It is difficult to know what part of Binney & Tremaine’s 800-page tome is considered relevant to my brief comment. Those authors explain spiral structure in relation to density waves. As I mentioned, the concept is not well-supported. Likewise I don’t know is meant by ‘arguments obviously unworkable forty years ago’. In some respects even a textbook published 16 years ago can be out of date. I was referring to images from JWST released this year, which in my opinion current ideas of spiral galaxy formation don’t begin to account for. A couple of months ago a press release referred to streams approaching ‘the black hole little by little, and in a spiral, similar to the way the water swirls down a drain’. Is that your idea? I’ve never seen a drain or whirlpool consisting of two arms.

    Your latest work detects gas rotating round galaxies at a distance of 1 Mpc. That is greater than the width of entire galaxy clusters. The gas includes C and O, so presumably originated from the galaxy; it’s certainly not primordial. Do you have an explanation for that? How, if you accept that it originates from the galaxy, does it end up in a circular rotational orbit?

    Only a few months ago you were remarking on how the outer edges of HI discs, still tracking spiral structure, appear to be hard. How does that fit in? Astronomy is ultimately an observational science, and new observations continue to challenge.

    1. This comment got hung up in WordPress’s spam-checker while I was traveling and unable to approve it remotely.

      You raise a bunch of issues, each of which is worth a post in itself, though few of them are interesting enough to warrant it. As I said earlier, I am not going to attempt to teach the equivalent of a large fraction of a graduate course on the subject in the comments of this blog.

      I will address the one point you had previously raised, about the circularity of stellar orbits. The Milky Way is a bog-standard spiral galaxy, so Gaia constraints on orbits are relevant insofar as we believe the Copernican Principle: it would be pretty weird for the Milky Way to differ from every other spiral in the universe, and vice-versa. Moreover, you had objected to the use of the Doppler effect; Gaia proper motions show that this objection is not warranted. If we use the Doppler effect, then there is evidence (e.g., from the DiskMass survey) that stars in other spirals orbit in nearly circular orbits. If we don’t believe that Doppler traces velocity (there is no good reason not to) then physics is broken at such a fundamental level that this discussion is rendered moot.

  8. Hi @Tritonstation excellent post as usual.

    sorry my comment is a bit off topic but there has been two groundbreaking discoveries that have major implications for MOND (or quantized inertia), that I believe, have passed under the radar.

    Their findings are >4sigmas and probe things that have rarely been looked at before:

    1. The degree of ellipticity in the orbit of local groups. It is found to be much higher than expected and in lcdm leads to a much younger universe. I wonder if MOND or some modification of it could explain such ellipticity and what would the resulting age of the universe be.

    2. They measure the orbit of satellite galaxies and find it match a younger universe in LCDM.

    https://arxiv.org/abs/2406.19612

    https://arxiv.org/abs/2401.10342


    1. I don’t have much to say about these. The result appears to be that groups of galaxies look messier in reality than expected from simulations, and similarly for their velocities. Those have both been known at some level for a while: putting on a theory hat, one wants to assume that groups are a spherical cow that is in dynamical equilibrium, but from the observers’ view there is no such thing as a smooth cluster in equilibrium: they almost always have significant substructure in both position and velocity space.

      While it doesn’t surprise me that groups don’t look like simulations say they should, I don’t see how that leads to a younger universe. MOND makes things move faster than LCDM on these scales so maybe it plays a role in this, but then that alters the interpretation of the age.

  9. Wonderful to have discovered this blog just a week before I am to introduce an audience of elderly Mainiac non-scientists to both Dark Matter and MOND. I’m neither a particle physicist nor an astronomer, just a planetary scientist with an ideological ax to grind (pro MOND, anti LCDM). I’ve been on Milgrom’s side for 15 years. It was Chae’s 2023 paper that made me believe MOND will “win” in the end, and that it was time to share my perspective with more than a few friends. My talk will certainly mention the recent Mistele findings as well.

  10. Stacy

    2 papers on mond could you comment

    arXiv:2407.03413 (astro-ph)[Submitted on 3 Jul 2024]Simulations of cluster ultra-diffuse galaxies in MONDSrikanth T. Nagesh, Jonathan Freundlich, Benoit Famaey, Michal Bílek, Graeme Candlish, Rodrigo Ibata, Oliver Müller

    Ultra-diffuse galaxies (UDGs) in the Coma cluster have velocity dispersion profiles that are in full agreement with the predictions of Modified Newtonian Dynamics (MOND) in isolation. However, the external field effect (EFE) from the cluster seriously deteriorates this agreement.

    Astrophysics > Astrophysics of Galaxies
    arXiv:2406.08872 (astro-ph)
    [Submitted on 13 Jun 2024]
    Testing MOND using the dynamics of nearby stellar streams
    Orlin Koop, Amina Helmi

    The observational constraints provided by the streams, which MOND fails to reproduce in its current formulation, could potentially also be used to test other alternative gravity models.

    1. I can only offer very quick takes.

      On the first, there are some indications that the EFE is suppressed, and others that it is active. It is theory-specific, with different behaviors possible in modified inertia and modified gravity. This paper necessarily uses a specific implementation of modified gravity, which limits its generality. It is important to start considering non-equilibrium effects during orbits, as dwarfs orbiting more massive systems can have breathing modes where their size and velocity dispersion vary with phase, or even disrupt tidally – see Brada & Milgrom.

      On the second, the primary concern seems to be the short timescale over which the clumps discussed would dissolve. That is a legitimate concern, something I also noted in McGaugh & Wolf. However, I also notice that the dark matter interpretation apparently requires a prolate DM halo, which is itself a problem – such things do not emerge from simulations (disk formation rounds out the halo) and tend not to be stable (which would also pose a timescale problem). I’m old enough to remember when early derivations of satellite orbits not showing evidence of precession were argued to require a spherical halo and exclude MOND; now it sounds the other way around. I’d also note that this is a hard problem: one has to account for the LMC, SMC, Sgr dwarf (as they discuss) and is again theory-specific.

      The PoR code is used in both these studies, which is great, but it is specific to QUMOND. AQUAL as implemented by Brada & Milgrom and later Tiret & Combes seemed to give qualitatively more satisfactory results, but it is harder to code and takes longer to run. And those are both modified gravity theories; modified inertia is even harder to implement. So I’d caution against the generality of these results, good or bad. These are early days for such numerical investigations, and we’re still arguing about how simulations should work in CDM, which we’ve invested many more person-years in studying.

  11. Pingback: The Radial Acceleration Relation to very low accelerations – Triton Station

Comments are closed.