The effective force law in galaxies is MOND. At high accelerations, this is the same as Newton’s inverse square law: g=GM/r2g = GM/r^2. At low accelerations, the deep MOND limit is g=a0GM/rg = \sqrt{a_0GM}/r. The transition between the regimes occurs at g=a0=1.2×10−10ms−2g = a_0 = 1.2 \times 10^{-10}\;\mathrm{m}\,\mathrm{s}^{-2}.

The effective force law in galaxies (data) looks like MOND (line).

This is well established in galaxies. The question naturally arises whether this holds on larger scales. A recent paper by Gallardo et al. says no:

On scales from 30 – 230 megaparsecs, we constrain the gravitational acceleration between pairs of halos$ at separation r to be g ∝ 1/rn with n=2.1±0.3

so Newton’s n = 2, not the n = 1 of the deep MOND limit. Here is their result for the kinematic SZ effect*.

Fig. 1 from Gallardo et al.: Pairwise kSZ measurements [μK] as a function of the physical separation of galaxy clusters [Mpc].

Here the blue line represents Newton’s n = 2 with the boost of dark matter as appropriate for LCDM. This matches the data better than the yellow line with n = 1 representing MOND. But is that the right representation?

Although the bulk of our analysis is model independent, we have also shown the theoretical curves appropriate for MOND. This test is the largest-scale direct test of MOND to date. Our formalism is an excellent approximation to MOND in the low-acceleration regime

This much is good. It’s a nice, general approach to represent 1/r force laws like the deep MOND limit. But is that the right thing to do here?

that said, we have not included the external field effect (EFE) in this analysis. This can modify the acceleration law in the case where the baryonic gravity of an object is less than the gravity of its larger environment, and has been used to explain# the velocity dispersion of satellite galaxies. However, this effect likely does not affect our analysis.

because the EFE absolutely affects this analysis.

They go on to say words about how the EFE is a thing that applies on small scales (hence the allusion to dwarf galaxies) but they’re looking at large scales so it shouldn’t matter. If only that were true.

The length scale does not matter in MOND. The acceleration scale matters. Is the chunk of the universe over which they’re integrating in the low acceleration regime? Yes. Is the EFE relevant on these scales? Also yes.

Some EFE from large scale structure is unavoidable. Everything feels the tug of everything else. In the deepest depths of the intergalactic medium, the EFE is tiny (maybe 1% of a0) but also ubiquitous. In the absence of a prominent mass, it dominates. That’s exactly the regime this experiment probes.

The force in the MOND EFE regime looks like a boosted version of Newton: g≈(a0/aEFE)GM/r2g \approx (a_0/a_{\mathrm{EFE}})\,GM/r^2. The boost factor a0/aEFEa_0/a_{\mathrm{EFE}} is what we interpret as dark matter: the total mass that we need in excess of what we observe.

So, what Gallardo et al. falsify is a straw man version of MOND in which the force law has n = 1 forever and always. That’s only true in the absence of the EFE, for which the prediction is n = 2 – as observed.

There may be a test in the amplitude of the boost factor. We already know that is in the ballpark that’s required for the cosmic dark matter, Ωm/Ωb≈a0/aEFE\Omega_m/\Omega_b \approx a_0/a_{\mathrm{EFE}}, as this is one of the first things I checked when I was surprised to encounter MOND in the previous century. So I don’t see much hope in distinguishing between the two this way.

Reality is more complicated. The amplitude of the EFE throughout intergalactic space depends on the cosmic distribution of mass. This was calculated by Chae et al. (2021), who found it to vary with both distance (from us) and position on the sky:

Fig. 5 from Chae et al. (2021): Variation of eN,env (the Newtonian% amplitude of the EFE in units of a0) with distance for the galaxies in the NSA and Karachentsev catalogs. Individual galaxies are color-coded by right ascension (R.A.). The black lines show the mean trend (solid) and standard deviation (dashed) in bins of distance.

It’s even worse than this, because most of the baryons are in the IGM. It makes a difference to the amplitude of the EFE how clumpy these are. They’re probably somewhat clustered into filaments and walls, but we don’t really have a great empirical map of that. So: the EFE is definitely there at a level that matters, but precisely what that level might be is rather hard to say. My best guess today is ~2% of a0, but it could be more, and probably is in places. Could be less in the midst of the deepest voids.

Gallardo et al. do not cite Chae et al. (2021), nor evince awareness& that there are relevant constraints on the EFE on the scales they probe. Note that the distance range of the figure from Chae et al. (2021) goes out to 150 Mpc, which is where the relevant data of Gallardo et al. are (their last two points are largely irrelevant). So we are talking about very much the same length scales, which does matter to the integration they do. That integration per force averages over any real variations in the EFE.

There is nowhere you can go to completely escape the EFE. A rather profound fact that appears not to be widely appreciated is that there is a minimum cosmic acceleration of order ∼10−12ms−2\sim 10^{-12}\;\mathrm{m}\,\mathrm{s}^{-2}.

$Saying “halos” here is quite the linguistic bias since there are no dark matter halos in the theory they’re testing.

*There are less obscure ways to do this, but this is what they chose to do and I’m not going to attempt to unravel it here. Ask them.

#Not just explain, but predict. Repeatedly. Until the same ability to predict kinematics in advance of observation is demonstrated by LCDM, I’m going to remain underwhelmed by post-facto tests that claim to favor LCDM over MOND.

%If I recall the notation correctly, the actual EFE is a0eN,env\sqrt{a_0\,e_{\mathrm{N,env}}} and in these units a0=1a_0 = 1 so log⁡(eN,env)=−3\log(e_{\mathrm{N,env}}) = -3 corresponds to an EFE that is 3% of a0a_0. This plot was made assuming maximal clumping of the IGM baryons which gives an upper limit, so the reality is probably less; see Fig. 6 of Chae et al.

&I wonder what the refereeing process looked like. I can imagine there being no mention of the EFE initially, with a referee (or perhaps one of the many coauthors) asking if they ought to maybe worry about it and the majority choosing to wave their hands through it. That’s what it looks like.

This is not the first time I’ve encountered the misconception that the EFE can be ignored on cosmic scales. It happened in the submitted version of Aguirre et al. (2001). In that case, I was the referee, and pointed out that the EFE had a profound effect on what they were saying about Lyman-alpha absorbers. To their credit, they listened and corrected it. I think they still kinda low-balled the amplitude of the EFE, but it went from a factor of tens problem to a small problem or maybe not a problem at all.

While on the topic of refereeing, I note that papers that find MOND wanting generally get less scrutiny than those which find that it works. Hardly surprising, once framed that way. I only mention this because there are certain toxic science communicators who rush to social media to denounce the incompetence of the referee any time a MOND-positive paper appears. Funny how they can be sure of the competence of the referee in a process they are not in any way privy to. If you take the time to think about it, you can more likely presume that the MOND paper has been held to a higher, not lower, standard, and weigh its credence appropriately.

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.

62 thoughts on “The External Field Effect and Tests of MOND on Cosmological Scales”

  1. Am curious how you think the current apparent standoff between the Chae et al camp and the Banik et al camp on the EFE interpretation will resolve? I have been following it a bit but confess that the technical debate on uncertainties and systematics are beyond my understanding. My hazy takeaway was “looks like they are waiting for more and better data” Thanks

    1. I have been careful not to take a position on this as I don’t want to be seen to be putting my thumb on the balance. Chae has been proactive about obtaining better data; but I’m not aware of all efforts.

  2. > that said, we have not included the external field effect (EFE) in this analysis.

    I confess my jaw dropped when I read this. I was sure you were pulling my leg.

    3. It is like supernova researchers, who do not include corrections for dust or for selection-effects in calculating values for the Hubble constant, knowing full-well that doing so increases Hubble tension.

  3. @tritonstation

    > we have not included the external field effect (EFE) in this analysis

    This is a wild statement to write in your paper.

    Even wilder that it, along with the justification, got through referees.

  4. I’ve been thinking about your post about falsifying dark matter. If dark matter since the earliest moments after the Big Bang is neither created nor destroyed, if the apparent dark matter content of the universe was found to change over time, would that falsify dark matter? Thanks

    1. That would indeed be weird if the amount of dark matter changed over time. It is not impossible, however. One can imagine (and people have) that dark matter is some particle that is not stable, and decays over time. I don’t give this much credence even as an idea, because one must fine-tune the decay time to be of order a Hubble time: if it is much shorter, it is already gone; if it is much longer, it doesn’t matter and can be treated as stable for all practical purposes.
      It also matters what it decays into. If it is other particles then the mass is still around. It could be some other kind of dark particles, or it could decay into Standard Model particles, in which case the baryon density would go up as the DM density goes down. It could also decay into energetic but massless particles like photons, in which case its contribution to the expansion differs. There are probably theories out there blaming oddities in the expansion history of the universe on this effect.

  5. Great write up – I heard about the Gallardo paper from a certain science social media communicator, and he was pretty definite this was a slam dunk MOND is falsified result. Was hoping to see your response!

  6. “The effective force law in galaxies is MOND. … papers that find MOND wanting generally get less scrutiny than those which find that it works.” FUNDAMOND is essential. According to Prof. Edward Witten, “… one really has to stand on one’s head to reconcile MOND with what is well-established about relativistic physics, and the results are pretty obscure and far-fetched looking.”
    “Witten, Milgrom, Brown, and Kroupa on Modified Newtonian Dynamics (MOND)”
    2015, D. Brown
    https://vixra.org/abs/1501.0123
    Google “guendelman ssb”. Assume gravitational energy is conserved.
    I recently concocted the following email:
    header: 6 extra dimensions of space curl up. So what? GUENDELMAN HITS THE JACKPOT!!!!!!!!
    message: If 6 extra dimensions of space curl up, then are there profound physical manifestations of such curling up? YES! YES! YES! The curling up creates 3 dimensions of Milgrom spin & 3 dimensions of Guendelman-Guth spin. That allows 2 new forms of spontaneous symmetry breaking of string vibrations — namely, the dark matter phenomenon & the dark energy phenomenon. Replace Weinbergian gravitons by Guendelmanian gravitons & FUNDAMOND string theory is a huge empirical success.

  7. They only mention the EFE at all (to say they ignored it) because they were testing MOND. But what do people who defend LCDM think about the actual evidence for the EFE, is it seen as likely to exist? It seems to me there are areas where they might concede a little ground, without conceding everything.

    For instance, the idea that individual galaxies are governed by MOND, and not DM, is a very reasonable idea. It can be far more strongly defended than trying to win outright against DM. Good evidence for flat rotation curves extending several times further than the radius of the theoretical halo – how are they going to argue against that? So perhaps we’re asking too much, although of course there’s plenty of evidence for MOND at larger scales. They won’t concede the lot, but going for a little territory – just galaxies – puts pressure where it might actually do something. Then perhaps the hybrid approach that many think will inevitably be needed wafts in a little, and starts to shift the deadlock, and chip away at the prejudice and misinformation.

    This might seem unrealistic. But there are a lot of discussions going on, and perhaps the emphasis could shift towards areas they can’t defend so easily.

    1. The most common response from advocates of LCDM is to concede galaxies entirely while denying that means anything by asserting that MOND does nothing else. That’s not correct, so the first challenge is to agree on what observational reality is. We won’t make progress on the interpretation until we agree on the facts.

      1. So then I guess they would say that the vast majority of dark matter is assumed to exist in intergalactic filaments, and conceding a small fraction as MOND doesn’t significantly change things. That’s when you hit them with the notion that Weyl Transverse Gravity explains away Dark Energy, so its paradigm should be thoroughly investigated in the context of large scale Dark Matter observations.

    2. A third possibility is that DM distributes itself following a mass distribution law instead of a modified force law. In other words – the disk-halo conspiracy is real, and DM and baryon mass distributions are indeed coupled to produce the radial regularities we see today. MOND and CDM could then make the same predictions radially but would make different predictions vertically – and we could directly test both in “predict” instead of “fit” mode.

      1. How? How does DM distribute itself following a mass distribution law?
        We know what CDM should do (an NFW halo) and what we observe is not that. By construction, CDM does not and cannot do what you’re suggesting: there is no mechanism to couple its distribution to that of the baryons in the very specific way that is required by the data.
        We could make up some new form of DM that does; bipolar DM and superfluid DM are two examples. They are in no way standard CDM; they have additional properties that try to make it so.

        1. It would definitely be a different species of DM, possibly a family of particles with specific properties, certainly not “classic” CDM profiles. This does not “solve” the disk-halo conspiracy – it relocates it into requirements for DM carrier properties to produce the mass distribution. The larger point is that force law modification (itself phenomenological) is not the only phenomenological path to observed empirical regularities.

          1. Can’t argue with that as it is one of the logical possibilities I’ve suggested myself. It is not something that is easy to accomplish in practice, nor is it clear that the examples noted above perform as well as MOND. More generally, I have philosophical reservations about making up a theory to capture the accomplishments of a different theory. Even if we succeed, why should we prefer the latter to the theory that made correct predictions in the first place?

            1. True. The only possible reason to prefer it is if it actually leads to a real carrier identification that can recover a_0 and the regularities as a consequence of the carrier microphysics. It is very tough to beat MOND on parsimonious predictivity. I find MOND seductive because it is simple, it is telling us something real and I think it is about the relationship between DM and visible matter. There is huge opportunity lost due to camps disagreeing vs joining forces

  8. Perhaps if it became widely acceptable to think MOND is the correct description for galaxies, even if many think that’s all it does, the wall has been breached. From there it can’t be ignored, so more steps are possible.

  9. I wonder if they had to apply fine tuning of the Dark Matter distribution at the scales they were studying, just as the DM distribution and configuration around galaxies has to be just right to explain their rotation curves? I haven’t read the paper, but will do so to see if perhaps this is the case.

    1. I don’t think so. Their method averages over huge swaths of space, so is sensitive to the index of the force law rather than the details of the distribution.

  10. There is another conséquence of non-linearity: the center of mass theorem does not hold. I suppose this means the shape of the galaxy (or the cluster) plays a role. Is it significant ?

    1. Yes, the geometry matters, as does the theory – e.g., linear momentum isn’t conserved in the original ansatz. One needs a theory derived from a Lagrangian, like AQUAL, to have proper conservation of momentum and energy. One way to tell if you’ve written a MOND code right is to check if the center of mass accelerates away – that’s what happens with one-line changes to ordinary particle pushing codes.

      1. On the theoretical side, you have a formula for the effective acceleration (fonction of the Newtonian one) that should prove enough for that purpose.

  11. “The length scale does not matter in MOND. The acceleration scale matters.” In the context of general relativity, excess acceleration means excess gravitational redshift & excess slowing of time. MOND does not make sense in terms of the prevailing conceptual framework of physics. Because MOND does make many successful predictions, a new framework is needed.
    Google “guendelman ssb”.
    Think about spontaneous symmetry breaking (SSB) in particle physics:
    “An introduction to spontaneous symmetry breaking”, 2019,
    by Beekman, Rademaker, & van Wezel
    https://arxiv.org/pdf/1909.01820
    In the standard form of Einstein’s field equations, the –1/2 represents a symmetry with respect to the geometric tensor. FUNDAMOND string theory with Guendelmanian gravitons might be the most plausible way to mathematically model a FUNDAMOND SSB that is approximated by MOND. Newton-Einstein inertia combined with FUNDAMOND inertia might give a highly plausible FUNDAMOND.

  12. Fact #1: There is now overwhelming empirical evidence supporting MOND.
    Fact #2: At the present time, there is no empirical evidence supporting supersymmetry (SUSY).
    Am I wrong about the 2 preceding alleged facts? It seems to me there are only 2 possibilities: FUNDAMOND string theory or FUNDAMOND non-string theory. What might FUNDAMOND string theory be?
    According to Edward Witten in “Comments on string theory.” arXiv preprint hep-th/0212247 (2002):
    “String theory avoids the ultraviolet infinities that arise in trying to quantize gravity. It is also more predictive than conventional quantum field theory, one aspect of this being the way that it contributed to the emergence of the concept of “supersymmetry” of particle interactions. There are hints from the successes of supersymmetric unified theories of particle interactions that supersymmetry is relevant to elementary particles at energies close to current accelerator energies; if this is so, it will be confirmed experimentally and supersymmetry is then also likely to be important in cosmology, in connection with dark matter, baryogenesis, and/or inflation.”
    https://arxiv.org/abs/hep-th/0212247
    If SUSY occurs in nature, then it seems plausible that the same Higgs field that works for quarks & gluons should also work for squarks & gluinos. Therefore, it is plausible that finding the Higgs field should also quickly lead to finding evidence for SUSY (provided that SUSY really exists). I say that the string theorists should focus on FUNDAMOND instead of SUSY — because we know that MOND makes many successful predictions.
    Another point: Riofrio & Sanejouand seem to dismiss the possibility that inflation & the Big Bang are compatible with a more complicated interpretation of the Riofrio-Sanejouand model. However, changes in hypothetical Guendelman-Guth inertia might account for the phenomenology of the Riofrio-Sanejouand model.

    1. Indeed, there is no evidence for SUSY. One of the things that encouraged the launch of string theory was that gravity “naturally fell out of it.” But that means standard gravity, which is taken for granted. String theory has made no progress because they’re playing solitaire with an incomplete deck: the theory ignores the empirical input of the observed MOND phenomenology.

  13. If you had full control over where the field’s research resources were directed, where would you steer them over the next 12 months?

    Broad question, the answer can be as simple as which observable facts you’d prioritize establishing.

    I’m aiming to get your sense of what you see as the most immediate progress we can make, regardless of what theory we camp with.

    1. There will be a whole white paper about that Soon.
      The thing that needs to happen is the development of a community working on MOND rather than investing ever more in chasing dark matter. I hope and expect that progress could accelerate rapidly once a critical mass is obtained.

  14. Off topic. The hype about the Roman telescope states that “Astronomers plan to use Roman’s observations to probe the mysterious nature of dark matter …” Is there anything new that this telescope can do that can distinguish dark matter versus MOND?

    1. Hmm, good question. The data that constrain both ideas is the same; the chief issue is a matter of interpretation – especially as to whether we allow ourselves to think that way. So I can’t offhand think of anything Roman can do that would clearly say one or the other. It will make the sky visible at incredible resolution over a large volume, so perhaps clarity will emerge.

  15. Please excuse me for going off-topic, but I wonder how long it will take to provide an explanation within the LCDM framework?

    C.M. Cheng et al. Hidden mass in early galaxies revealed by bottom-heavy initial mass functions. Nat Astron, published online August 18, 2026; doi: 10.1038/s41550-026-02932-4

    Thanks for your blog!

    1. Interesting. So Cheng et al. argue that the IMF is bottom heavy in elliptical galaxies at high z. That compounds the early galaxy formation problem, which was that there was already too much mass in stars early on. LCDM is barely consistent with the most challenging data if early galaxies somehow turn all their available baryons into stars almost instantaneously. That’s assuming a standard IMF, so it’s worse if the IMF has more low mass stars, so more stellar mass for the observed light. That might well be impossible to cope with for LCDM, though I suppose one could always invoke some DM halos getting more than their fair share of baryons.
      All that said, claims of IMF variations must always be taken with a grain of salt. I expect the consensus will be that this means the IMF doesn’t vary!

  17. “String theory … ignores the empirical input of the observed MOND phenomenology.”
    In Guendelman’s new version of string theory, “… the dynamical string theories can bridge between the low and high energy quantum gravity effects.”
    “Dynamical String Tension Theories with target space scale invariance SSB and restoration”, 2025 by Eduardo Guendelman
    https://arxiv.org/abs/2104.08875
    If we assume that MOND is the low energy approximation to FUNDAMOND string theory, then we need to persuade the younger string theorists to pursue FUNDAMOND (and, possibly, simplify & streamline Guendelman’s string theory).

  18. According to Brian Greene, “From one principle—that everything at its most microscopic level consists of combinations of vibrating strands—string theory provides a single explanatory framework capable of encompassing all forces and all matter.”
    Chapter 1. Tied Up with String, subsection String Theory as the Unified Theory of Everything, “The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory”, 2024, 25th anniversary edition, 1st edition 1999
    https://books.google.com/books?id=4SnjEAAAQBAJ&pg=PT21
    However, according to Nobel Laureate Robert B. Laughlin,
    “… string theory … has failed to account for any measured thing.”
    https://en.wikiquote.org/wiki/Robert_B._Laughlin
    I suggest that Guendelman’s new version of string theory might justify FUNDAMOND string theory as follows:
    Introduce 2 additional worldsheet scalar fields. The 2 new worldsheet scalars might distort the stringy representations of gravitons in some way that allows a spontaneous symmetry breaking of the geometric tensor and a spontaneous symmetry breaking of the energy-momentum tensor. The 2 new spontaneous symmetry breakings might explain the dark matter phenomenon & the dark energy phenomenon. Am I full of sheet?

  19. I was sitting outside listening to the crickets and thinking that the EFE is one of the most telling (hard to produce in other theories) pieces of data. It’s not understood enough as a prediction of Mond. Which is sad mostly.

        1. Thank you for your response to my previous question. On a slightly more relevant note, what do you think of this article, which, according to the authors, benefited from comments by K.-H. Chae and H. Desmond? (I haven’t found an answer online, let alone a published one.)

          https://doi.org/10.1093/mnras/stac2689

          1. I think it confuses the EFE in MOND with extra mass (the two halo term) in LCDM. They say “detection of an EFE-like signal is not, by itself, evidence for physics beyond GR” but I think the opposite is true: the EFE is unique to MOND and does not happen in GR. What can happen in GR+DM is that there is extra force from clumps of dark matter that may be encountered at very large radii/very low accelerations where the EFE in MOND should occur. So they make the argument that LCDM has this as an EFE, albeit with the opposite sign of that in MOND. That’s just a misnomer; it isn’t an EFE, it’s extra mass.

          2. Is a study of this kind partly indicative of concern that the EFE might one day falsify GR? It seems to me there are real reasons to believe in DM, but also sometimes an element of protecting GR from becoming obsolete (just as inflation is used to explain the fine tuning in space being flat at a large scale, even though it has a major fine tuning problem of its own). So if an effect from modified gravity starts to look rather telling, they try to show that ‘…well, DM can do that too’.

  20. Instead of the question
    “Where in the World are SUSY & WIMPS?” by Nima Arkani-Hamed, 2017
    https://www.youtube.com/watch?v=dKVXxcbJ4YY
    I think the question should be: Where in the world are the string theorists who understand the importance of MOND & FUNDAMOND?
    My guess is that the 6-dimensional Calabi-Yau manifold should be replaced by a 12-dimensional Guendelman manifold that puts Milgrom spin and Guendelman-Guth spin on gravitons.

    1. Last I saw Nima, which probably predates this video, he was making the case that the next generation theory would necessarily be nonlocal. I pointed out that this was a feature of modified inertia theories and his response was basically “not THAT nonlocal!”

  21. So if ones acceleration >= a0, then Newtonian
    3% ao <= acc <= ao, then Mond
    acc <= 3% acc, then Newtonian again with modified G, due to background of Universe.
    Mond has an explanation of this, the EFE, but one can understand why some people might find this a little contrived. Or least not what one should expect for a fundamental physical law.

    1. It is odd in that it is one of the harder things to wrap one’s head around, but it is not contrived from the perspective of a nonlinear force law, where such behavior is hard to avoid.

    2. So this is just me, but I find it somewhat satisfying. If it was mond-like forever you get this logarithmic function that goes on forever… and what limits do you stick in your integral. But if at some point you are feeling the mass of the rest of the universe (or whatever it is that causes the lower limit) then the logarithmic divergence is finite. It’s a little like Mach’s principle. Your state depends (at some point) on the entire universe.

        1. So in the graph at the top of this post, there’s a slight downturn in the last (low acceleration) 3-4 data points (in the lensing data). Is that the ‘back to an 1/r^2 force law (energy as 1/r) ? Does that measure the mass of the universe or something?

          1. Maybe. The downturn is intriguing but not highly significant. It may well map the point at which we hit the bedrock EFE, but the exact shape of that depends on the details of theory. Chae & Milgrom had a paper illustrating some of the possibilities – I think it was https://arxiv.org/abs/2201.02109. There is too much slop in both data & theory to make too bold an interpretation.

  22. “… the next generation theory would necessarily be nonlocal … a feature of modified inertia theories …”
    If gravitational energy is conserved, then my main hypothesis is that Weinbergian gravitons need to be replaced by Guendelmanian gravitons. However, Weinbergian gravitons might be the real deal.
    According to the Stephen Wolfram and his co-workers on the Wolfram Physics Project,
    “… we think of space as being made of discrete atoms of space … the raw material of space …”
    “The Structure of Time and Space | Stephen Wolfram”, Closer to Truth, YouTube, July 15, 2026
    https://www.youtube.com/watch?v=r6wYw_A1MKw&t=165s
    String vibrations among Wolframian atoms of space might create a combination of a semi-lattice and a semi-fluid at the Planck scale. The semi-lattice property of the string vibrations might create FUNDAMOND inertia, while the semi-fluid property of the string vibrations might create dark-energy inertia. Pro-MOND astronomers & astrophysicists should try to persuade Wolfram to work on understanding MOND & FUNDAMOND.

  23. The EFE implies that everything else in the universe would change the gravitational dynamics of an isolated system, which is suggestive of Mach’s principle. What if the isolated MOND (no EFE) behavior itself originates from the “interaction” between the universe and a galaxy? This is suggested by the mysterious relation a0 ~ cH0

    1. Yes, this is a very intriguing line of thought. It seems like we’re missing something fundamental analogous to the equivalence principle, and Mach’s principle is an obvious possibility. What is lacking is a mechanism by which it results in the observed phenomenon.
      Einstein is reputed to have worked hard to try to incorporate Mach’s principle into GR, and ultimately gave up. I wonder what he considered along that path. I doubt it included MOND, but if it did, I’m confident he would have rejected it as patently absurd, just as he invented the cosmological constant in rejection of the what at the time must have seemed the absurd GR prediction that the universe could not be static.

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