Flat rotation curves and the Baryonic Tully-Fisher relation (BTFR) both follow from the Radial Acceleration Relation (RAR). In Mistele et al. (2024b) we emphasize the exciting aspects of the former; these follow from the RAR in the Mistele et al. (2024a). It is worth understanding the connection.
First, the basic result:
Figure 2 from Mistele et al. (2024a). The RAR from weak lensing data (yellow diamonds) is shown together with the binned kinematic RAR from Lelli et al. (2017, gray circles). The solid line is Newtonian gravity without dark matter (gobs = gbar). The shaded region at gbar < 10−13 m/s2 indicates where the isolation criterion may be less reliable according to the estimate by Brouwer et al. (2021). Our results suggest that late type galaxies (LTGs) may be sufficiently isolated down to gbar ≈ 10−14 m/s2. We shade this region where LTGs may still be reliable in a lighter color.
The RAR of weak lensing extends the RAR from kinematics to much lower accelerations. How low we can trust we’ll come back to, but certainly to gbar ≈ 10−13 m/s2 and probably to gbar ≈ 10−14 m/s2. For the mass of the typical galaxy in the KiDS sample, this corresponds to a radius of 300 kpc and 1.1 Mpc, respectively. Hence our claim that the effective gravitational potentials of isolated galaxies are consistent with rotation curves that remain flat indefinitely far out: a million light years at least, and perhaps a million parsecs.
Note that the kinematic and lensing data overlap at log(gbar) = -11.5. These independent methods give the same result. Moreover, this region corresponds to the regions in galaxies where atomic gas rather than stars dominates the baryonic mass budget, which minimizes the systematic uncertainty due to stellar population mass estimates. The lensing results still depend on these, but they agree with the gas-dominated portion of the RAR, and merge smoothly into the star-dominated portion of the kinematic data when the same stellar pop models are used for both. To wit: the agreement is really good.
A flat rotation curve projects into the log(gobs)-log(gbar) plane as a line with slope 1/2. The data adhere closely to this slope, so I knew as soon as I saw the lensing RAR that the implied rotation curves remained flat indefinitely. How far, in radius, depends on galaxy mass, since for a point mass (a good approximation at radii beyond 100 kpc), gbar = GMbar/R2. We can split the lensing data into different mass bins, for which the RAR looks like
Figure 5 from Mistele et al. (2024a). The RAR implied by weak lensing for four baryonic mass bins. The dashed line has the slope a flat rotation curve has when projected into the acceleration plane. That different masses follow the same RAR implies the Baryonic Tully-Fisher relation.
Most dark matter models that I’ve seen or constructed myself predict a mass-dependent shift in the RAR, if they predict a RAR at all (many do not). We see no such shift. But the math is such that the flat rotation speed implied by the slope 1/2 RAR varies with mass in such a way that they only fall on the same RAR, as observed, if there is a Baryonic Tully-Fisher relation with slope 4. So I knew from examination of the above figure that the BTFR was sure to follow, but that’s because I’ve been working on these things for a long time. It isn’t necessarily obvious to everyone else, so it was worth explicitly showing.
Our result differs from the original of Brouwer et al. in two subtle but important ways. The first is that we use stellar population models that are the same as we use for the kinematic data. This self-consistency is important to the continuity of the data. We (especially Jim Schombert) took a deep dive into this, and the models used by Brouwer et al. are consistent with ours for late type (spiral) galaxies (LTGs). However, ours are somewhat heavier^ for early type galaxies (ETGs). That’s part of the reason that they find an offset in the RAR between morphological types and we do not.
Another important difference is the strictness of the isolation criterion. We are trying to ascertain the average gravitational potential of isolated galaxies, those with no big neighbors to compound the lensing signal. Brouwer et al. required that there be no galaxies more than a tenth of the luminosity of the primary within 3 Mpc. That seems reasonable, but we explored lots of variations on both aspects of that limit. It seems to be fine for LTGs, but insufficient for ETGs. That in itself is not surprising, as ETGs are known to be more strongly clustered than LTGs, so it is harder to find isolated examples.
To illustrate this, we show the deviation of the data from the kinematic RAR fit as a function of the isolation criterion:
Figure 4 from Mistele et al. (2024a). Top: the difference between the radial accelerations inferred from weak lensing and the RAR fitting function, measured in sigmas, as a function of how isolated the lenses are, quantified by Risol. We separately show the result for ETGs (red) and LTGs (blue) as well as for small (triangles with dashed lines) and large accelerations (diamonds with solid lines). LTGs are mostly unaffected by making the isolation criterion stricter. In contrast, ETGs do depend on Risol, but tend towards with increasing Risol. Middle and bottom: the accelerations behind these sigma values for Risol = 3 Mpc/h70 and Risol = 4 Mpc/h70.
The top panel shows that LTGs do not deviate from the RAR as we vary the radius of isolation. In contrast, ETGs deviate a lot for small Risol. This is what Brouwer et al. found, and it would be a problem for MOND if LTGs and ETGs genuinely formed different sequences: it would be as if they were both obeying their own version of a similar but distinct MOND-like force law rather than a single universal force law.
That said, the ETGs converge towards the same RAR as the LTGs as we make the isolation criterion more strict. The distinction between ETGs and LTGs that appears to be clear for the Risol = 3 Mpc/h70 used by Brouwer et al. (middle panel) goes away when Risol = 4 Mpc/h70 (bottom panel). The random errors grow because fewer galaxies+ meet the stricter criterion, but this seems a price well worth paying to be rid of the systematic variation seen in the top panel. This also dictates how far out we can trust the data, which show no clear deviation from the RAR until below the limit gbar = 10−14 m/s2.
Regardless of the underlying theory, the data paint a consistent picture. This can be summarized by three empirical laws of galactic rotation:
- Rotation curves become approximately* flat at large radii and remain so indefinitely.
- The amplitude of the flat rotation speed scales with the baryonic mass as Mbar ~ Vf4 (the BTFR).
- The observed centripetal acceleration follows from that predicted by the baryons (the RAR).
These are the galactic analogs of Kepler’s Laws for planetary motion. There is no theory in these statements; their just a description of what the data do. That’s useful, as they provide an empirical touchstone that has to be satisfactorily explained by any theory for it to be considered viable. No dark matter-based theory currently does that.
^The difference is well within the expected variance for stellar population models. We can reproduce their numbers if we treat ETGs as if they were just red LTGs. I don’t know if that’s what they did, but it ain’t right.
+For the record, the isolated fraction of the entire sample is 16%: most galaxies have neighbors. As a function of mass, the isolation criterion leaves a fraction of 8%, 18%, 30%, and 42% of LTG lenses and 9%, 14%, and 22% of ETG lenses, respectively, in each mass bin. The fraction of isolated LTGs is generally higher than ETGs, as expected. There is also a trend for the isolation fraction to increase as mass decreases. In part this is real; more luminous galaxies are more clustered. It may also be that it is easier for objects that exceed 10% of the primary mass (really luminosity) to evade detection as the primaries get fainter so 10% of that is harder to reach.
*Some people take “flat” way too seriously in this context. While it is often true that rotation curves look pretty darn flat over an extended radial range, I say approximately flat because we never measure, and can never measure, exactly a slope of dV/dR = 0.000. As a practical matter, we have adopted a variation of < 5% from point to point as a working definition. The scatter in Tully-Fisher naturally goes up if one adopts a weaker criterion; what one gets for the scatter is all about data quality.
@tritonstation,
Just wanted to thank you for letting me (mr nobody) be a part of your
historic work.
I am somebody when I get your emails with all your hard work, thoughts
and ideas laid out for me
to ingest. I follow you and Pavel Kroupa and the rest of the trail
blazers with great excitement.
THNX so much,
Best Regards,
Thanks. I appreciate your kind words.
Great work showing one more time that empirical evidence reigns supreme.
But still Peebles, P. J. E. claim that:
“The standard LambdaCDM cosmology passes demanding tests that establish it as a good approximation to reality. It is incomplete, with open questions and anomalies”
https://ui.adsabs.harvard.edu/abs/2024arXiv240518307P/abstract
What else do these people need to acknowledge that their model has been falsified?
If I understood correctly, several theories try to reproduce these empirical laws by explaining that at low accelerations, the effective gravitational mass of a body increases. The higher mass yields a higher gravitational attraction.
How could one imagine such an effect? More precisely, how could one imagine the cosmological constant to play a role? Imagining the cosmological constant as a sort of noise in space, even if frictionless, how could it increase gravitational mass?
Do people imagine this like “heavy electrons” in conductors, i.e. like conduction electrons with a large effective mass? Or are there different approaches?
One can have the effective gravitational force increase, or the effective inertial mass decrease so that particles become easier to push around. There are a variety of theories that fall in the former category; the latter category was a possibility for MOND from its inception. Milgrom (1999) speculated about a direct connection between inertia and the vacuum energy represented by the cosmological constant in which particles obtain their inertial mass when accelerated wrt the Unruh radiation of the vacuum. Such an approach has appealing aspects, but it is inherently nonlocal. That makes it hard to come to grips with both intellectually and numerically, as one in principle needs to know where everything was at all times to get the computation right. But one thing I’ve noticed is that the universe does not care about what adheres to our computational convenience.
It seems hard to imagine that inertial mass decreases at low accelerations. It appears more appealing to imagine that gravitational force increases. As if, at low accelerations, the cosmological constant acts as a molasse sticking to mass. But even with entropic gravity or something similar, it seems hard to get an intuition for it. A molasse should reduce force, not increase it.
Yes, that’s what I thought at first, and what others have said: e.g., Jerry Sellwood called modified inertia “a bridge too far.” But we’re in the current state we are because invisible mass seemed more appealing than to imagine that gravitational force increases. So I’ve become pretty open-minded about what the universe may be getting up to.
Maybe it’s time to reconsider Einstein’s original conception that GR unified gravity and inertia in the geodesic equation. See the discussion in the latter part of this paper which also sets the record straight regarding Einstein’s career long rejection of the geometrization of spacetime as an explanation for the gravitational effect.
https://www.sciencedirect.com/science/article/pii/S1355219813000695
If the inertial and gravitational effects coexist but the gravitational effect falls off with the square of the radial distance while the inertial effect remains constant then at large radial distances you would expect the inertial effect to predominate.
Hi Stacy,
Thank you for this post on the “Radial Acceleration Relation”.
Although slightly off topic, perhaps you could write something sometime on modified inertia in the context of MOND. When I look at Milgrom’s papers they all start at a somewhat advanced level. I find myself going back and delving into topics including action, Lagragians, inertial mass, active gravitational mass, passive gravitation mass, and so on. I suspect there is a tutorial out there that takes one from a basic level to the starting point of Milgrom’s papers on MOND & Modified Inertia, but I can’t seem to find one. Thanks.
Ha ha – so do I! If you find such a tutorial, let me know – I’d like to read it too.
is RAR and Banik et al. find purely Newtonian behavior in Wide binary debate and claim to exclude MOND at 19σ results compatible ?
Not according to Banik et al.
Wide binaries are interesting but remain controversial, with different groups making opposite claims. I don’t really care: the RAR we observe in galaxies is clearly in the data, so it is a secondary question why wide binaries might be incompatible with that observational reality.
Jonathan Fay discusses Mach’s writings in their historical context and with respect to various interpretations in a paper at the link,
http://philsci-archive.pitt.edu/22555/1/Mach_s_Principle_and_Mach_s_Hypotheses.pdf
Fay points out that criticism of Newtonian inertia involves both an epistemological question and a causality question. Mach had been aware of this, where his contemporaries had not.
Fay also identifies Mach’s empirical bias:
“However, with his empiricist humility, Mach leaves the task of developing a more precise replacement for the law of inertia to future investigators”
Interesting. This subsequent development still remains wanting. Einstein is reputed to have worked hard to build Mach’s Principle into GR, but ultimately failed. It seems like we are missing something fundamental.
Which makes whole DM vs MOND debate nothing more than inter-tribal bickering. If both camps are missing something fundamental then neither can be correct, right? Or has logic left the building?
What gives MOND high ground is the fact its camp isn’t spending billions trying to prove their point. The fact DM has so many holes it’s infuriating how grown up, educated people can claim it true would then gives you … level ground?
To get fertile ground you’d stop talking to outside world, bunker in, realise this 1980s-string-theory playbook can only get you to the top of The Draft; except there isn’t one, and come up with at least mostly unified idea why MOND works as well as it does.
Only then you can claim equal partnership in this MAWT, where we stack up approximations because that’s clearly the way to enlightenment. But hey, there’s always hope errors will cancel out, right?
@Dr. McGaugh
I suspect that what is missing is a philosophical perspective. Angela Collier produced an edutainment video on differing opinions between Mach and Boltzmann over the existence of atoms,
https://m.youtube.com/watch?v=DM5qBRwU5EU
This history exposes how Mach had straddled empiricism and empirical science. The former has been folded into idealism by 20th century philosophers portraying the unity of science as a choice between idealism and physics as a foundation. In this context, the aspect of Mach not captured by Einstein relates to “witnessability.”
Very simply put, the reliance of applied mathematics on the number line taught to first-graders as an algebraic dimension of time is incompatible with Machian empiricism.
I began looking at citations from Fay’s paper. Barbour appears to be the most significant author. He offers a possible reconstruction of Mach’s principle in the paper,
https://arxiv.org/abs/1007.3368
although a more accessible example of his view is found in a defense at
https://arxiv.org/abs/1108.3057
I cannot judge how a physicist might interpret Barbour’s claim. There is much to his analysis worth considering, but, invoking Poincare is only a partial step in my opinion. All topological notions occurring in linear geometry rely upon mappings into the real number line.
Applied mathematicians have discounted the work of symbolic logicians. Shannon’s information theory uses logarithms “for engineering convenience.” Much later, the results of symbolic logic are “recovered” with billiard ball computing,
https://en.m.wikipedia.org/w/index.php?title=Billiard-ball_computer&diffonly=true
I am very familiar with the stupidity of academic sociology — “believe me, not him.” After enough iterations, one is talking in circles.
The second Barbour paper has an example that speaks to rotations. Symbolic logicians working in the first-order logic have already established the problematic character of rotations. Tarski proved that the theory of closed real fields is decidable; Richardson proved that arbitrary extension of closed real fields with trigonometric functions is undecidable.
Barbour gives a reasonable account of why Newton had to introduce absolute time and space — a meaningful definition for velocity. But, Newton’s Scholia begins with quantities of matter in relation to volumes (physics) differentiated from spatial positions (mathematics), and, Barbour’s analysis says nothing about the significance of Newton’s distinction on this matter — it relates to probability theory and the use of integration.
I apologize for the length of this. My objective, however, is to point out that Milgrom’s modified inertia paper involves entire trajectories as parameters. I am not skilled enough to know if the shape of a trajectory is of essential importance in his equations. However, in so far as a trajectory can be assumed to have two endpoints, inclusion of a trajectory into Milgrom’s equations may be interpretable as a “witnessable change” in Mach’s empricism.
As for the general practice corresponding to Newton’s use of absolute space and absolute time, l’Hospital had written the first calculus book long before 19th century hubris invaded academic studies. His first definition can be found at the link,
https://mathshistory.st-andrews.ac.uk/Biographies/De_LHopital/
and reads:
“Definition 1. Variable quantities are those that increase or decrease continuously while a constant quantity remains the same while other vary.”
To the best that I can tell, Matt Strassler’s explanation of the Higgs field reflects this constancy for quantum field theory, and, Milgrom invokes the Higgs field as an analogy for how he would like to understand modified inertia .
Wikipedia paraphrases: “You are standing in a field looking at the stars. Your arms are resting freely at your side, and you see that the distant stars are not moving. Now start spinning. The stars are whirling around you and your arms are pulled away from your body. Why should your arms be pulled away when the stars are whirling? Why should they be dangling freely when the stars don’t move?”
But isn’t this question easy? By the friction with the atmosphere, absolute rotation stops (except the rotations of the earth itself). And by inertia, the stars can’t move fast when you’re not moving or rotating. For what do we need rotating frames of reference?
I just started wondering: Might a rotating universe be the cause of the Hubble constant? Everything spiraling away from the middle?
In what sense would the universe have a center? What would that even mean?
electrons ‘dont have a radius’ and still have spin.
The ‘spin’ of subatomic particules has nothing to do with rotation.
“has nothing to do with rotation”
is too hard. See Einstein de Haas effect.
On the other hand, YES.We don’t have a good model of electrons.For example, we don’t know what spin is good for an electron.We only know that it has one.
i am going to take it even harder and say quantum spin is a more fundamental description of rotation than classical rotation
I was thinking of something like Godel’s universe (https://en.m.wikipedia.org/wiki/G%C3%B6del_metric) without negative cosmological constant. But it is apparently ruled out or the rotation is so small it has not enough effect on the expansion of the universe.
The most exciting part about this result imo is the downturn of the data at the far low acceleration end. That was in the Brouwer analysis already but now seems to be robust against a different analysis methodology. Since this is the region where the isolation regime is most questionable, shouldn’t this downturn be useful for estimating the external field effect on the sample much like was done in Chae et al. 2020?
Yes, the downturn at very low acceleration is an intriguing hint of the EFE. It is also where we have to worry the most about systematic effects, and it is hard to estimate what we should expect here since the galaxies were selected to be isolated. So it will take a lot more work to tease out how much of this aspect of the signal to believe.
But yes, taken at face value, this looks like the EFE.
Stacy McGaugh
thoughts on arXiv:2407.11139 (astro-ph)[Submitted on 15 Jul 2024]Globular cluster orbital decay in dwarf galaxies with MOND and CDM: Impact of supernova feedbackM. Bílek, F. Combes, S. T. Nagesh, M. Hilker
Dynamical friction works very differently for Newtonian gravity with dark matter and in modified Newtonian dynamics (MOND). While the absence of dark matter considerably reduces the friction in major galaxy mergers, analytic calculations indicate the opposite for very small perturbations, such as globular clusters (GCs) sinking in dwarf galaxies
However, we find that supernovae cannot prevent massive GCs (M≥4×105M⊙) from sinking in MOND. The resulting object looks similar to a galaxy with an offset core, which embeds the sunk GC. The problem is much milder in the Newtonian simulations. This result thus favors Newtonian over QUMOND gravity,
Sounds fair, insofar as it is specific to QUMOND and the simulation they do. I wouldn’t read more into it than that, especially as the program for CDM for the past 30 years has been to add MORE FEEDBACK every time it fails.
thanks you are the only person i can ask about MOND
That’s easy to believe. One thing we are lacking is a community of critical mass to discuss these things.
the paper says “However, we find that supernovae cannot prevent massive GCs (M≥4×105M⊙) from sinking in MOND.”
then “This result thus favors Newtonian over QUMOND gravity,”
so MOND or QUMOND is not consistent with reality ?
I guess what I meant before is that feedback doesn’t really do all the things it is advertised to do for Newton + dark matter, so that it also does not work for QUMOND is profoundly uninteresting to me.
Full sentence:
“This result thus favors Newtonian over QUMOND gravity, but we note that it relies on the correctness of the difficult modeling of baryonic feedback.”
arXiv:2407.12482 (cross-list from astro-ph.GA) [pdf, html, other]Exploring Milky Way rotation curves with Gaia DR3: a comparison between ΛCDM, MOND, and General Relativistic approachesWilliam Beordo, Mariateresa Crosta, Mario Gilberto LattanziComments: 18 pages, 8 figures, 2 tables. Submitted to JCAPSubjects: Astrophysics of Galaxies (astro-ph.GA); Cosmology and Nongalactic Astrophysics (astro-ph.CO); General Relativity and Quantum Cosmology (gr-qc)
We find that all models, including MOND and ΛCDM, are statistically equivalent in representing the observed rotational velocities. However, ΛCDM, characterized by an Einasto density profile and cosmological constraints on its parameters, assigns more dark matter than the model featuring a Navarro-Frenk-White profile, with the virial mass estimated at 1.5-2.5×1012M⊙ – a value significantly higher than recent literature estimates. Beyond 10-15 kpc, non-Newtonian/non-baryonic contributions to the rotation curve are found to become dominant for all models consistently.
win for MOND?
This is interesting – and too involved to comment on with confidence after a quick skim.
Just quickly, looking at their Fig. 4, all the models overshoot the Gaia data at R > 20 kpc where others claim a Keplerian downturn. This is what I’ve said before: said downturn is incompatible with both MOND and plausible dark matter models. Their own analysis of the Gaia data doesn’t extend that far – the numbers of stars does peter out, so maybe they are wise to stop where they do. It also appears that their Gaia stellar rotation curve is higher than that of others around R = 15 kpc and also inside the solar circle (R < 8 kpc). The latter would be more consistent with the terminal velocities observed in the interstellar gas.
Also in Fig. 4, it looks like the best fit LCDM and MOND models are indistinguishable. That means those are the best-fit lines to their data, nothing more. One has to look at the assumptions that go into those models to judge further.
They appear to have done a good job, albeit in the usual parametric way. i.e., at the level of my 2008 MW paper, which they cite. They miss the more recent non-parametric papers that predict a rotation curve similar to what they derive.
It’ll take a lot more time to say anything more about this, but I find it encouraging that the literature seems to be iterating in the right direction.
i’m looking forward to that thanks