The following is a guest post by Indranil Banik, Moritz Haslbauer, and Pavel Kroupa (bios at end) based on their new paper

Modifying gravity to save cosmology

Cosmology is currently in a major crisis because of many severe tensions, the most serious and well-known being that local observations of how quickly the Universe is expanding (the so-called ‘Hubble constant’) exceed the prediction of the standard cosmological model, ΛCDM. This prediction is based on the cosmic microwave background (CMB), the most ancient light we can observe – which is generally thought to have been emitted about 400,000 years after the Big Bang. For ΛCDM to fit the pattern of fluctuations observed in the CMB by the Planck satellite and other experiments, the Hubble constant must have a particular value of 67.4 ± 0.5 km/s/Mpc. Local measurements are nearly all above this ‘Planck value’, but are consistent with each other. In our paper, we use a local value of 73.8 ± 1.1 km/s/Mpc using a combination of supernovae and gravitationally lensed quasars, two particularly precise yet independent techniques.

This unexpectedly rapid local expansion of the Universe could be due to us residing in a huge underdense region, or void. However, a void wide and deep enough to explain the Hubble tension is not possible in ΛCDM, which is built on Einstein’s theory of gravity, General Relativity. Still, there is quite strong evidence that we are indeed living within a large void with a radius of about 300 Mpc, or one billion light years. This evidence comes from many surveys covering the whole electromagnetic spectrum, from radio to X-rays. The most compelling evidence comes from analysis of galaxy number counts in the near-infrared, giving the void its name of the Keenan-Barger-Cowie (KBC) void. Gravity from matter outside the void would pull more than matter inside it, making the Universe appear to expand faster than it actually is for an observer inside the void. This ‘Hubble bubble’ scenario (depicted in Figure 1) could solve the Hubble tension, a possibility considered – and rejected – in several previous works (e.g. Kenworthy+ 2019). We will return to their objections against this idea.

Figure 1: Illustration of the Universe’s large scale structure. The darker regions are voids, and the bright dots represent galaxies. The arrows show how gravity from surrounding denser regions pulls outwards on galaxies in a void. If we were living in such a void (as indicated by the yellow star), the Universe would expand faster locally than it does on average. This could explain the Hubble tension. Credit: Technology Review

One of the main objections seemed to be that since such a large and deep void is incompatible with ΛCDM, it can’t exist. This is a common way of thinking, but the problem with it was clear to us from a very early stage. The first part of this logic is sound – assuming General Relativity, a hot Big Bang, and that the state of the Universe at early times is apparent in the CMB (i.e. it was flat and almost homogeneous then), we are led to the standard flat ΛCDM model. By studying the largest suitable simulation of this model (called MXXL), we found that it should be completely impossible to find ourselves inside a void with the observed size and depth (or fractional underdensity) of the KBC void – this possibility can be rejected with more confidence than the discovery of the Higgs boson when first announced. We therefore applied one of the leading alternative gravity theories called Milgromian Dynamics (MOND), a controversial idea developed in the early 1980s by Israeli physicist Mordehai Milgrom. We used MOND (explained in a simple way here) to evolve a small density fluctuation forwards from early times, studying if 13 billion years later it fits the density and velocity field of the local Universe. Before describing our results, we briefly introduce MOND and explain how to use it in a potentially viable cosmological framework. Astronomers often assume MOND cannot be extended to cosmological scales (typically >10 Mpc), which is probably true without some auxiliary assumptions. This is also the case for General Relativity, though in that case the scale where auxiliary assumptions become crucial is only a few kpc, namely in galaxies.

MOND was originally designed to explain why galaxies rotate faster in their outskirts than they should if one applies General Relativity to their luminous matter distribution. This discrepancy gave rise to the idea of dark matter halos around individual galaxies. For dark matter to cluster on such scales, it would have to be ‘cold’, or equivalently consist of rather heavy particles (above a few thousand eV/c2, or a millionth of a proton mass). Any lighter and the gravity from galaxies could not hold on to the dark matter. MOND assumes these speculative and unexplained cold dark matter haloes do not exist – the need for them is after all dependent on the validity of General Relativity. In MOND once the gravity from any object gets down to a certain very low threshold called a0, it declines more gradually with increasing distance, following an inverse distance law instead of the usual inverse square law. MOND has successfully predicted many galaxy rotation curves, highlighting some remarkable correlations with their visible mass. This is unexpected if they mostly consist of invisible dark matter with quite different properties to visible mass. The Local Group satellite galaxy planes also strongly favour MOND over ΛCDM, as explained using the logic of Figure 2 and in this YouTube video.

Figure 2: the satellite galaxies of the Milky Way and Andromeda mostly lie within thin planes. These are difficult to form unless the galaxies in them are tidal dwarfs born from the interaction of two major galaxies. Since tidal dwarfs should be free of dark matter due to the way they form, the satellites in the satellite planes should have rather weak self-gravity in ΛCDM. This is not the case as measured from their high internal velocity dispersions. So the extra gravity needed to hold galaxies together should not come from dark matter that can in principle be separated from the visible.

To extend MOND to cosmology, we used what we call the νHDM framework (with ν pronounced “nu”), originally proposed by Angus (2009). In this model, the cold dark matter of ΛCDM is replaced by the same total mass in sterile neutrinos with a mass of only 11 eV/c2, almost a billion times lighter than a proton. Their low mass means they would not clump together in galaxies, consistent with the original idea of MOND to explain galaxies with only their visible mass. This makes the extra collisionless matter ‘hot’, hence the name of the model. But this collisionless matter would exist inside galaxy clusters, helping to explain unusual configurations like the Bullet Cluster and the unexpectedly strong gravity (even in MOND) in quieter clusters. Considering the universe as a whole, νHDM has the same overall matter content as ΛCDM. This makes the overall expansion history of the universe very similar in both models, so both can explain the amounts of deuterium and helium produced in the first few minutes after the Big Bang. They should also yield similar fluctuations in the CMB because both models contain the same amount of dark matter. These fluctuations would get somewhat blurred by sterile neutrinos of such a low mass due to their rather fast motion in the early Universe. However, it has been demonstrated that Planck data are consistent with dark matter particles more massive than 10 eV/c2. Crucially, we showed that the density fluctuations evident in the CMB typically yield a gravitational field strength of 21 a0 (correcting an earlier erroneous estimate of 570 a0 in the above paper), making the gravitational physics nearly identical to General Relativity. Clearly, the main lines of early Universe evidence used to argue in favour of ΛCDM are not sufficiently unique to distinguish it from νHDM (Angus 2009).

The models nonetheless behave very differently later on. We estimated that for redshifts below about 50 (when the Universe is older than about 50 million years), the gravity would typically fall below a0 thanks to the expansion of the Universe (the CMB comes from a redshift of 1100). After this ‘MOND moment’, both the ordinary matter and the sterile neutrinos would clump on large scales just like in ΛCDM, but there would also be the extra gravity from MOND. This would cause structures to grow much faster (Figure 3), allowing much wider and deeper voids.

Figure 3: Evolution of the density contrast within a 300 co-moving Mpc sphere in different Newtonian (red) and MOND (blue) models, shown as a function of the Universe’s size relative to its present size (this changes almost linearly with time). Notice the much faster structure growth in MOND. The solid blue line uses a time-independent external field on the void, while the dot-dashed blue line shows the effect of a stronger external field in the past. This requires a deeper initial void to match present-day observations.

We used this basic framework to set up a dynamical model of the void. By making various approximations and trying different initial density profiles, we were able to simultaneously fit the apparent local Hubble constant, the observed density profile of the KBC void, and many other observables like the acceleration parameter, which we come to below. We also confirmed previous results that the same observables rule out standard cosmology at 7.09σ significance. This is much more than the typical threshold of 5σ used to claim a discovery in cases like the Higgs boson, where the results agree with prior expectations.

One objection to our model was that a large local void would cause the apparent expansion of the Universe to accelerate at late times. Equivalently, observations that go beyond the void should see a standard Planck cosmology, leading to a step-like behaviour near the void edge. At stake is the so-called acceleration parameter q0 (which we defined oppositely to convention to correct a historical error). In ΛCDM, we expect q0 = 0.55, while in general much higher values are expected in a Hubble bubble scenario. The objection of Kenworthy+ (2019) was that since the observed q0 is close to 0.55, there is no room for a void. However, their data analysis fixed q0 to the ΛCDM expectation, thereby removing any hope of discovering a deviation that might be caused by a local void. Other analyses (e.g. Camarena & Marra 2020b) which do not make such a theory-motivated assumption find q0 = 1.08, which is quite consistent with our best-fitting model (Figure 4). We also discussed other objections to a large local void, for instance the Wu & Huterer (2017) paper which did not consider a sufficiently large void, forcing the authors to consider a much deeper void to try and solve the Hubble tension. This led to some serious observational inconsistencies, but a larger and shallower void like the observed KBC void seems to explain the data nicely. In fact, combining all the constraints we applied to our model, the overall tension is only 2.53σ, meaning the data have a 1.14% chance of arising if ours were the correct model. The actual observations are thus not the most likely consequence of our model, but could plausibly arise if it were correct. Given also the high likelihood that some if not all of the observational errors we took from publications are underestimates, this is actually a very good level of consistency.

Figure 4: The predicted local Hubble constant (x-axis) and acceleration parameter (y-axis) as measured with local supernovae (black dot, with red error ellipses). Our best-fitting models with different initial void density profiles (blue symbols) can easily explain the observations. However, there is significant tension with the prediction of ΛCDM based on parameters needed to fit Planck observations of the CMB (green dot). In particular, local observations favour a higher acceleration parameter, suggestive of a local void.

Unlike other attempts to solve the Hubble tension, ours is unique in using an already existing theory (MOND) developed for a different reason (galaxy rotation curves). The use of unseen collisionless matter made of hypothetical sterile neutrinos is still required to explain the properties of galaxy clusters, which otherwise do not sit well with MOND. In addition, these neutrinos provide an easy way to explain the CMB and background expansion history, though recently Skordis & Zlosnik (2020) showed that this is possible in MOND with only ordinary matter. In any case, MOND is a theory of gravity, while dark matter is a hypothesis that more matter exists than meets the eye. The ideas could both be right, and should be tested separately.

A dark matter-MOND hybrid thus appears to be a very promising way to resolve the current crisis in cosmology. Still, more work is required to construct a fully-fledged relativistic MOND theory capable of addressing cosmology. This could build on the theory proposed by Skordis & Zlosnik (2019) in which gravitational waves travel at the speed of light, which was considered to be a major difficulty for MOND. We argued that such a theory would enhance structure formation to the required extent under a wide range of plausible theoretical assumptions, but this needs to be shown explicitly starting from a relativistic MOND theory. Cosmological structure formation simulations are certainly required in this scenario – these are currently under way in Bonn. Further observations would also help greatly, especially of the matter density in the outskirts of the KBC void at distances of about 500 Mpc. This could hold vital clues to how quickly the void has grown, helping to pin down the behaviour of the sought-after MOND theory.

There is now a very real prospect of obtaining a single theory that works across all astronomical scales, from the tiniest dwarf galaxies up to the largest structures in the Universe & its overall expansion rate, and from a few seconds after the birth of the Universe until today. Rather than argue whether this theory looks more like MOND or standard cosmology, what we should really do is combine the best elements of both, paying careful attention to all observations.


Indranil Banik is a Humboldt postdoctoral fellow in the Helmholtz Institute for Radiation and Nuclear Physics (HISKP) at the University of Bonn, Germany. He did his undergraduate and masters at Trinity College, Cambridge, and his PhD at Saint Andrews under Hongsheng Zhao. His research focuses on testing whether gravity continues to follow the Newtonian inverse square law at the low accelerations typical of galactic outskirts, with MOND being the best-developed alternative.

Moritz Haslbauer is a PhD student at the Max Planck Institute for Radio Astronomy (MPIfR) in Bonn. He obtained his undergraduate degree from the University of Vienna and his masters from the University of Bonn. He works on the formation and evolution of galaxies and their distribution in the local Universe in order to test different cosmological models and gravitational theories. Prof. Pavel Kroupa is his PhD supervisor.

Pavel Kroupa is a professor at the University of Bonn and professorem hospitem at Charles University in Prague. He went to school in Germany and South Africa, studied physics in Perth, Australia, and obtained his PhD at Trinity College, Cambridge, UK. He researches stellar populations and their dynamics as well as the dark matter problem, therewith testing gravitational theories and cosmological models.

Link to the published science paper.

YouTube video on the paper


Indranil Banik’s YouTube channel.

49 thoughts on “Big Trouble in a Deep Void

  1. I would be a little concerned if a model required us to be living close to the centre of a large void as it would violate the Copernican Principle. With enough supernovae redshifts we should be able to measure the Hubble constant in a range of different directions which would place some severe restrictions on how far from the centre of a void that we are.

    On the other hand, if being in a high density area meant many, many more galactic interactions, it might be that life could not get going and us being in the middle of a void might just be a reflection of the Anthropic Cosmological Principle.

    A very thought-provoking paper.


    1. I can’t see a conflict with the Copernican Principle. Copernicus is the move from the centre of the known world to any place in the universe. And this place can be situated even right in the centre of a large void, with possibly numerous large voids in the universe.


  2. Laurence,

    Thanks for your comment. The article will be published soon, but in the meantime you can find the preprint here:

    The issue of us being close to the void centre was one of my major concerns going into this project. It has been handled in the article, and is included in the 2.53 sigma figure mentioned in the blog. In particular, we considered the likelihood that the Local Group moves at ‘only’ 630 km/s relative to the CMB. Since our model is supposed to solve the Hubble tension, it raises the local H_0 by about 7 km/s/Mpc. This means the peculiar velocity is below the observed value within 90 Mpc of the void centre. As the KBC void radius is about 300 Mpc, this means our position close to the void centre is special at a significance level of (90/300)^3, which is not really very special.

    The paper looks into this in much more detail. It also mentions that there is evidence for directionality in the observed H_0, though to get enough accuracy, usually measurements from e.g. all supernovae have to be combined, losing information on any possible dipole. In fact, I was at a talk just now where Subir Sarkar claimed to have found strong evidence for such a dipole. This could be evidence for our off-centre position in the void, and/or for non-sphericity of the void.

    Anyway, I don’t think our model is unlikely because we need to be at a special place in the void to make it work. Our way of quantifying this particular aspect indicates that this concern is somewhat problematic, but only causes a 2.34 sigma tension.


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  3. Thanks Indranil. I will read your preprint with interest. I remember back in 2012 visiting Roger Angel’s mirror casting facility at the Steward Observatory in Tucson, where they cast the primary/tertiary mirror for what was then called the LSST, now named after Vera Rubin, and being impressed by the effect it will have on the rate of supernovae discoveries. It may just provide enough additional data when it comes into full operation next year.


  4. Very nice way of diminishing not only the Hubble tension, but also the “gravity model” tension. Nothing is better than explaining all the observables with a model


    1. Mauricio,

      I assume by the gravity law tension you mean the tight radial acceleration relation in galaxies and the high internal velocity dispersions of satellite galaxies in the Local Group satellite planes. Indeed, solving these very serious problems simultaneously with the KBC void and Hubble tension is the main advantage of this work. I also know its compromise of having both MOND and dark matter is very likely the right one merely because it annoys Pavel and the LCDM community at the same time, the former because we used dark matter and the latter because we used MOND. I don’t know for sure what the right physics is, but if it was suddenly revealed, it would surely surprise & annoy everyone, which this at least seems to have achieved.


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  5. Thank you for sharing this post/paper. I’m an interested layman when it comes to cosmology. If I’m reading the ‘KBC void’ paper correctly (linked to early in your post) then the SN data used to infer dark energy (lambda) could be because we are observing from this more rapidly expanding bubble… this is fascinating! And to this layman living in an expanding low density void seems like a simpler explanation than ‘dark energy’ everywhere. I’m still working through the rest of the post.



    1. George,

      Thank you for your interest in our work. The link hasn’t been activated yet by MNRAS, so in the meantime, please use the Arxiv preprint:

      We did indeed consider whether the entirety of the apparently accelerated expansion could be due to the void. The slowdown in structure formation at late times could be because to probe late times, we have to look nearby – and if we are in a void, then the lower matter density would cause less structure than a more typical region of the universe.

      However, the KBC void would only be a significant underdensity out to redshifts of about 0.15, while dark energy starts to dominate below redshifts of about 0.6. In addition, the expansion history affects the angular diameter distance to the CMB. Essentially, we know the universe expanded 1100x since then, and if it did so slower, then the CMB would be further away as the light from it must have been travelling for longer. The ages of the oldest stars are too large for this scenario. Importantly, the hot and cold spots in the CMB would have a different angular size than observed – while dark energy does not affect the physics of the CMB, it does affect how far away it is from us. For these reasons, we did not challenge the notion of dark energy.

      In a MOND context, there is also the cosmic coincidence of MOND, namely that the acceleration below which a gravitational field has less energy density than the dark energy also happens to be within an order of magnitude of a_0. This could be an indication of very small quantum gravity effects which are nonetheless important as the issue here is exactly how strong gravity is at the tenth decimal point. I explain this in the first few minutes of this lecture I gave, which focuses on the satellite planes and galaxy scale evidence:



      1. “For these reasons, we did not challenge the notion of dark energy.”

        Yes, that is an important point. While the void idea might explain the Hubble-constant tension, there are so many lines of evidence (age of the Universe, CMB, BAO, supernova cosmology, large-scale structure formation) for the cosmological constant that trying to explain it away with a local bubble just don’t work. Kudos for not throwing the baby out with the bathwater, which in my experience is one reason MOND is not as popular as it is. Even if your scientific opponents are fools, it does not help to state that fact, but actually they are not fools, even if they might not have everything completely correct.

        Many on both sides of the debate are sceptical about hybrid solutions, but I’ve never seen a reason to rule out the possibility of both something like MOND and something like dark matter.


  6. So what this paper does is slap MOND and the evocatively named, sterile neutrinos, together into yet another tasty theoretical model that has absolutely no calories, or general informational value, and bears no resemblance to physical reality. It’s all just math and metaphysical posturing. It is yet another floundering variant of the “expanding universe” model.

    The “expanding universe” model (inclusive of all its variants, the most popular being LCDM) has nothing to do with the cosmos we observe; its depiction of physical reality is physically absurd. This “expanding universe” model is more properly labeled a creation myth. However, this myth is the essence of the standard model of cosmology.

    Modern cosmology won’t even begin making sense, physically and logically, until it jettisons the faulty assumptions at the heart of the standard model. It is an assumption (and only an assumption) of the orthodox mythology that 1) the Cosmos is a unitary entity capable of behaving in a coordinated, simultaneous fashion (i.e. expanding or contracting), and 2) the cosmological redshift is caused by some form of recessional velocity. Neither of those assumptions are supported by any direct empirical evidence.

    LCDM describes a universe that looks nothing like the Cosmos we observe. The Cosmos we observe does not contain a Big Bang creation event, an inflation event and inflaton field, substantival spacetime, dark matter, or dark energy. Those are, however, the defining elements of the standard model of cosmology, and without them LCDM does not even exist as a theory.

    Modern cosmology is nothing but a primitive belief system that has been awarded a scientific pedigree by its true believers, despite the model having absolutely no scientific merit. Modern cosmology is a pre-modern, unscientific mess. This pre-modern cosmology cannot be fixed; it needs to be scrapped. The original assumptions need to be set aside for cause; they lead only to nonsensical models that inaccurately describe and mischaracterize, the observed Cosmos. Can the model be mathematically massaged to fit actual observations? Sure, look no further than this paper, but so what? That kind of self-deluding mathematicism is as old as Ptolemy.

    Science is supposed to be a self-correcting enterprise. To the extent that modern cosmologists seem incapable (likely, for socio-economic reasons) of reconsidering the foundational assumptions of their failed model(s), modern cosmology cannot be considered a science; in its current form, it is only a primitive belief system.


    1. The very scientific “method” is flawed in this math driven science. A mathematical model is discovered and refined, that does in fact work or at least approximate the observed Cosmos. The big flaw is that then, out of an infinite number of possibilities, an arbitrary physical model is chosen and stapled onto it, that apparently the mathematical model “proves” by some sort of magical induction.

      An example is General Relativity. The concept of “curved space” is absurd. Space is an outer dimension and does not have qualities. “Curved space” is as ridiculous a concept as “blue space”. All GR really tells us is that gravity causes behavior AS IF space was curved, that if we pretend that space is curved, we end up with a consistent math and predictions concerning the movement of bodies in space. It’s a flying leap to say that GR proves that space IS curved. It does no such thing. Ditto for array of esoteric physical behaviors noted in SR, quantum mechanics, and cosmology.

      Another critical flaw is complete lack of humility. We have yet to discover any causative mechanism for gravity. There is a lot of speculation of course, but no graviton nor other physical mechanism has ever been identified. To this day we do not know what gravity is, we only know how it behaves and interacts between bodies. Yet CDM arises by making the flawed assumption that we know everything there is to know about gravity and that therefore, the only explanation for galactic phenomena that goes against this “supreme knowledge” is that 95% of the universe must be comprised of undiscovered invisible dark matter and energy. If there was ever a tail wagging a dog, it’s right there with the claim that 95% of the universe MUST BE esoteric invisible stuff.

      A final criticism that you hint about, is the eerie similarity modern cosmology bears to pre-modern creation myths! That should be eyebrow raising to say the least and science should have to go many extra miles to prove itself not influenced by such a thing when the resemblance is so clear.

      Science is and can be corrupted in many ways, by politics and money. It is telling when you see defenders of cosmology say that “if the competing models were any good, they’d get grant money”.


      1. @LeeC
        When I first started to read your post I thought “Oh dear!”, but as I read through it I found myself completely agreeing with your points, though I might have used a bit gentler phrasing.
        Yes, many, if not most, scientists have let Ego become their central value.
        However, if you look at the details of the history of science you will see this is nothing new. It is seldom mentioned in science texts because those that adopt such a narrow minded approach are doomed to eventually shrink into obscurity, even if their early careers were quite brilliant. The real breakthroughs in any branch of science are made by people who abandon the existing assumptions and boldly seek different paths.
        Unfortunately that is also the way to becoming a true crackpot, and maybe less then 1 in 10 that take that path actually do make a real breakthrough.
        So it becomes a tough choice for someone wishing to be a career scientist: do I take the dull path that will allow me to make some minor contributions and achieve some degree of status and security, or do I take the high risk path that might lead me to a participate in a new and exciting breakthrough, but is more likely to end in ostracism and/or disgrace?


        1. I have chosen the second option. If I can work on the correct theory for even just five years, this would be much more valuable than working on the wrong theory for fifty years. And this at least is always possible – with a four year PhD and one postdoc afterwards, usually one can get to slightly over five years. If one is keen, then one also has to add having a full time job outside astronomy for the next forty-five years with a few percent of it (spare time) spent on the right theory. This will surely get the total spent on the right theory over five years full time equivalent, which would have a fair bit of societal value. But fifty years working on the wrong theory is just a waste of time, even if it pays well.

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            1. Well, General Relativity is one of the foundational assumptions of the standard cosmological model. So I accept your challenge. Indeed you are almost certainly correct that there is a fundamental problem, so tinkering around the edges (e.g. replacing cold dark matter with axions) will not help.


              1. In and of itself, General Relativity does not present a cosmological problem. Rather it is the naïve assumption that GR, a local theory, which assumes that a universal frame does not exit, could nonetheless, be applied to a “universal” metric (FLRW). Couple that assumption with the misinterpretation of the cosmological redshift as being caused by some form of distance-dependent recessional velocity, and a Big Bang type model is as inevitable as it is ludicrous.


        2. @LeeC,

          Substantival spacetime that causally interacts with matter and energy is not a part of Einstein’s General Relativity but an addendum attributable to the mid-century GR revisionists, Dicke, Wheeler, et al. GR is not, generally speaking, the problem with modern cosmology. Rather, the current mess is a consequence of the misapplication of GR by theorists who are only capable of interpreting its mathematical structure (derived in the context of the Solar System) as a literal description of physical reality. Einstein did not have that problem; he was a scientist, not a mathematicist.


          1. I’ve heard it claimed that Albert once said something like:
            “I used to understand General Relativity, until the mathematicians got ahold of it”.


  7. Super interesting article. Are you worried that by combining MOND and Dark Matter – you loss some of the strength and predictive power of the “core” of MOND? I just finished reading David Merritt’s book “A Philispophical Approach to MOND”, and I was struck by his comment in the summary to this effect. Do you think a workable relativistic model might not solve MOND’s issues without needing to add DM?


    1. Regarding the strength and predictive power of MOND, this is still retained for galaxies. Where we lose it is for galaxy clusters and the CMB/expansion rate history. This loss is however no worse than the loss of predictive power faced by LCDM in the same situations. So a theory with no predictive power in clusters but a lot in galaxies is still better than one with no predictive power in either. However, interestingly enough, the amount of sterile neutrinos needed in MOND fits to galaxy clusters is not completely arbitrary – it seems to reach the so-called Tremaine-Gunn limit at the centre if the neutrino has the assumed mass:

      This suggests that one could begin with the assumption that sterile neutrinos would typically be accreted by a cluster until they built up to a high density, when ultimately quantum degeneracy pressure (similar to the Pauli Exclusion Principle) would prevent further build up. Of course, this might not work if e.g. you look early in the history of the Universe when this process has not yet had time to operate. But today you could start with sterile neutrinos at the Tremaine-Gunn limit at the centre, work outwards, and see if the resulting density profile accounts for the dynamics under MOND gravity. The above paper indicates that essentially the procedure would work. So we are not at a complete loss to understand the dynamics of galaxy clusters in MOND. If anybody else comes up with another way to predict galaxy cluster dynamics, that would be good – but ideas like the EMOND of Famaey & Zhao really do not work well (this is where the a_0 parameter is changed as a function of the potential depth). Even their original proponents agree on this point, which is generally a bad sign for the theory. One reason is that you need a substantial modification with quite a shallow cluster potential, and some galaxies can probe such potentials. The only consistent theory I have seen for how classical purely baryonic MOND works so well in galaxies yet fails in clusters is that there is additional collisionless matter in clusters.

      Which brings me on to your last question. I would like a relativistic MOND theory to explain all the observations without dark matter. The Skordis (2020) model does seem to mostly do that – it can get the CMB and expansion rate history correct with only the baryonic matter. However, I have to go with the available information – this model is not published, and the authors indicated to me that large scale structure would be much the same as in LCDM, thus not solving the KBC void and Hubble tension. My most major concern is that the dynamics of galaxy clusters are in my understanding not likely to come out right, since there is no good reason for the theory to behave like classical MOND in a galaxy (as is needed to get the rotation curves right) yet differently in a cluster (where MOND needs some sort of extra mass). I would be delighted if I was proved wrong on this one, and everything could be understood without dark matter. My personal opinion is that there is a 10% chance of this, and a 90% chance that additional mass will still be required – as in the model I advocated based on the work of Angus (2009). But if we discover the sterile neutrinos, then this would just be ordinary matter, or at least matter within the standard model of particle physics. There are also some works in progress which I can’t discuss right now but are strongly in favour of the Angus (2009) model. Certainly it is not my intention to advocate for dark matter if I can avoid it, but I have found this to be the path of least resistance against the data. While for the gravity law, the Milgromian one is contrary to less data. While any discrepancies it faces with observations can generally be reconciled by adding a certain form of collisionless matter, the same is not true of General Relativity, or otherwise the LCDM paradigm would be able to fit the observations.


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      1. . If anybody else comes up with another way to predict galaxy cluster dynamics, that would be good – but ideas like the EMOND of Famaey & Zhao really do not work well (this is where the a_0 parameter is changed as a function of the potential depth). Even their original proponents agree on this point, which is generally a bad sign for the theory.

        maybe find a better way modify MOND or gravity specific to galaxy cluster dynamics?

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        1. This is an interesting issue. What seems to work is to shift the value of a0. One sees a MOND-like relation in clusters, it is just shifted from that in galaxies. That shouldn’t happen in pure MOND, but it shouldn’t happen in LCDM either. The simplest solution would be if there really is more mass than meets the eye – some unseen baryons, or something like the sterile neutrinos considered here. I don’t much like these solutions, but it did happen before that there was a lot of normal but unseen mass. When Milgrom introduced MOND in 1983 he speculated that the X-ray gas in clusters might make up the difference, which seemed unlikely at the time. In retrospect, he was right: most of the baryonic mass is in the X-ray emitting gas of the intracluster medium, not in the galaxies (as was then widely believed). Turns out it goes a long way, but not far enough for MOND. It seems unlikely to me that there are large amounts of dark baryons still lurking in clusters, but it happened before so I’m reluctant to say it can’t happen again.

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          1. What seems to work is to shift the value of a0.

            perhaps dark energy causing shift the value of a0 due to size ?

            The simplest solution would be if there really is more mass than meets the eye – some unseen baryons, or something like the sterile neutrinos considered here.

            how about black holes


            1. Yes, it could be black holes. While a logical possibility, I don’t see Why there would be the required (large) number of black holes in clusters of galaxies. It is easy to say there is mass we don’t see; quite another to come up with a plausible idea for what it is.
              This is a microcosm of the whole missing mass problem. We have no idea what’s there, so we imagine something it could be. That’s not good enough; we’d like a way of checking what is there, whether it be black holes, WIMPS, sterile neutrinos, a large congregation of free floating space donkeys, or what have you.

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              1. just in case sterile neutrinos or any dark matter does not exist. if you take EMOND of Famaey & Zhao and combine with RelMOND theory of Skordis & Zlosnik (2020) would that cover both cmb and galaxies cluster

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          2. Am I correct in understanding the method for calculating the velocity dispersion for individual galaxies in a cluster to be a form of the virial theorem that seeks to mimic the Keplerian method?? The method that has been such a spectacular (not to mention embarrassing) failure when used on the galactic scale to calculate expected rotation curves? Really? Is this an accurate account of the current method for calculating the velocity dispersion?!?!?:



            1. The velocity dispersion is measured as the rms variation of the redshifts of cluster member galaxies around the mean redshift of the cluster. The mass inferred from that is obtained with something like the virial theorem. There are more sophisticated estimators than what is described in wikipedia, but it boils down to the same thing within factors of order unity: M ~ R*sigma^2/G. In MOND, M ~ sigma^4/(a0*G), so when I first reviewed this issue (in a 1998 paper), I wasn’t too worried that the MOND masses came out a bit large. A factor of two overestimate in mass follows from a 20% overestimate of the velocity dispersion, so it was easy to imagine that the latter might be inflated somewhat by interlopers – galaxies along the line of sight to the cluster, but not actually in it. That’s really hard to sort out in practice. However, X-ray observations imply a gas temperature that also gives too large a mass, but about the same factor. That’s a completely different method, which is good, but I worry about systematic differences when comparing the two. Still, the problem in clusters seems robust – or at least persistent.


              1. I take that answer to be a yes. Whether you write it like this, M ~ R*sigma^2/G, or as the wiki article has it, GM/R ~ sigma^2, that is essentially the Keplerian method. While perfectly reasonable for a first approximation in the context of the Solar System, where 98% of the mass of the system is concentrated at the center, it is a completely unreasonable technique for estimating rotation curves, dispersion velocities or mass, in galaxies or galactic clusters.

                It makes no physical or logical sense to mathematically treat the vast, dispersed masses of such systems as if they were concentrated at a central point. It is simply wrong to employ the Keplerian method in analyzing galaxy and galaxy cluster dynamics. That can be said unequivocally because, in the context of galaxies and galaxy clusters, the Keplerian approach always yields incorrect results. Always.

                Rather than fix this straightforward analytical issue, the preferred solution – Dark Matter!, has turned a simple problem into an inane farce. Dark matter does not exist; all attempts to locate dark matter have failed. There is no evidence for the existence of dark matter except for the flaccid claim that dark matter makes this poorly thought out mathematical approach work.

                MOND does not address the analytical problem. Rather, it obscures the issue behind an ad hoc mathematical correction to the results that leaves the flawed, underlying analysis intact.

                There is no missing matter in the Cosmos. Matter is only missing in the models, where all the dispersed matter of the physical systems has been mathematically relegated to a central point, which constitutes a complete mathematical misrepresentation of the physical systems supposedly being modeled.

                Science is supposed to be self-correcting. It is long past time for modern cosmology to start that self-correction process. Until it does the standard model will remain a simplistic and absurd creation myth.


  8. My comment ignores LCDM, MOND and DM and it sounds like ignorance, like this:

    Redshift is not caused by velocity.
    Voids are not caused by gravity.
    Gravity is not caused by mass.

    But if you read the paper it solves a lot of today’s cosmological mess.

    (Liebe Gruesse von Berlin nach Bonn und Cleveland)


  9. stacy or anyone

    does magnetic field of the galaxies also have mond like behavior that caused galaxies cluster dynamic ?


  10. I’m just a dumb amateur, at best, or a crackpot at worst, and my opinion is unlikely to be relevant in any way. However to me this paper sounds like the old problem that if you add enough parameters to any theory you can make it work, at least for the moment.
    In other words it is a bit of a Frankenstein’s monster.
    On the other hand I do think that it is a good thing this work was done. It provides a more concise look at what is needed (at least mathematically) for a complete solution than I have seen before.
    So over all I would say it is a step forward, but should be used with caution.


    1. Ronald,

      I agree that there are a lot of assumptions, mainly MOND and hot dark matter. But basically this is just saying we haven’t got either the right gravity law, or a complete list of particles with cosmologically significant mass/energy density. This seems conservative to me – why should we expect to have got either of these things right, when the Solar System is not representative of the Universe in so many ways? The paper is an attempt to use observations to try and pin down in what way we may have got these aspects wrong, or at least how we might need to extend our Solar System-based understanding in order to fit data from the rest of the Universe.

      I agree about being able to match the data with enough free parameters. Readers will have to judge for themselves whether we explain many more observations than the free parameters we put in. Our model seems to explain or be consistent with:

      Sub-atomic: neutrino oscillations, which suggest an extra sterile species, and no evidence for supersymmetric or other WIMPs/cold dark matter candidates (the very large part of parameter space already ruled out surely lowers the likelihood of this being correct)

      Terrestrial and Solar System: MOND effects small at high accelerations

      It is possible to have a relativistic MOND theory where gravitational waves travel at the speed of light (Skordis & Zlosnik 2019). The ones detected so far are produced in very dense high-acceleration environments, so there should be little difference with classical General Relativity.

      Wide binaries (parsec scales): this test has not yet been done, and is discussed further here: mnras/sty2007

      kpc scales are mainly probed by galaxies, where MOND works very well:

      The high prevalence of thin disk galaxies (Kormendy+ 2010) also argues against galaxies merging very often, which is required if they have large cold dark matter halos. These halos also cause severe problems with properties of the bars in spiral galaxies (work in progress).

      Mpc: MOND better explains the Local Group satellite planes and the high velocity of NGC 3109, which are very problematic in LCDM (MNRAS, 473, 4033). Moti has also written about MOND getting the correct velocity dispersions for galaxy groups.

      Galaxy clusters work OK in MOND with sterile neutrinos of the proposed mass:

      The formation of massive high-redshift galaxy clusters like El Gordo is also problematic in LCDM, but less so in MOND (work in progress).

      100 Mpc scales were addressed in our work, but additional evidence for supervoids being more prevalent than in LCDM is given here:

      1 Gpc is the approximate size of the region used to infer the local Hubble constant from linear regression/model-fitting to the supernovae Hubble diagram/other probes. We all know what happened to LCDM here, while our publication addresses the corresponding results in the nuHDM MOND-based cosmological framework.

      A large local void could also explain these results if we are off-centred, since some of the supposedly z > 1 sources are at much lower redshifts due to measurement errors:

      Our model recovers a standard expansion rate history.

      In the early universe, the CMB and Big Bang nucleosynthesis should work in the standard way, with small undetectable differences. Though interestingly there is some evidence that the CMB is actually 1% hotter than commonly thought, as expected in our model because CMB photons have to climb a potential hill to reach our location within the void, in addition to the usual cosmologcial redshift from an expanding universe:

      The main assumptions are:
      Inflationary hot Big Bang

      Dark energy

      Sterile neutrinos with mass between (11 – 300) eV/c^2 as hot dark matter, most likely at the bottom of this range

      Milgromian gravity (MOND).

      I believe that the assumptions explain enough data that our paper is in line with Occam’s Razor, which is about making as few assumptions as possible to fit the data. It is currently not possible to explain the observations if one or other of these assumptions is removed. I would also argue that adding dark matter and modifying gravity involve relaxing assumptions – they are not really ‘assumptions’ in the sense of picking one particular outcome among many possibilities, with some justification provided. Currently, the data can only be understood by relaxing the assumptions that we detected all the mass and that GR is the right gravity law. Relaxing either on its own is not sufficient, though there are attempts to relax only the GR assumption and match all the observations. Certainly most things can be matched this way, but a problem seems to remain in clusters and possibly for large scale structure. It will be very interesting to see how all this develops.

      Liked by 1 person

  11. @budrap – our limit on nested replies has run out.
    Yes but no. As I said, there are more elaborate formulations that take into account the non-centrality of the mass distribution. You are correct that this is important. But it does not change the basic problem – the corrections are of order unity (a factor of 1.4 in velocity at most) for an order of magnitude problem.
    If we could have self-corrected our way out of this, we would have. This was suggested and discussed at length circa 1980. It does not suffice.


    1. Are these more elaborate formulations accessible online? I’d certainly like to take a look at them. Incorporating the distributed masses into a model would be a necessary first step, but only a first step. In addition, it would also be necessary to account properly for any varying gravitational effects unique to distributed mass systems.

      For instance, in typical disk galaxies the bulge component would fit the Keplerian method with a simple declining, centripetal force effect, but the disk component would contribute non-centripetal forces, the effects of which should become more prominent, at significant radial distances, far from the center of mass, as the Keplerian bulge effect drops off.

      Whether you call it gravitational viscosity, after Zwicky, or disk self-gravity, it does not appear that we have a robust mathematical tool that properly captures that effect. MOND only addresses the second order acceleration and cannot be considered scientifically satisfying, because it fails to capture the underlying causal mechanism.

      It would seem far more reasonable to initiate a concerted effort to devise the required mathematics than continue the destined-to-be-fruitless snipe hunt that constitutes the search for dark matter. Such an effort need not be large. A few competent astrophysicists/astronomers like yourself working with a few capable mathematicians should suffice. Any success would be greatly resisted by the dark matter search industry, of course, but such is the nature of the modern scientific establishment.

      In the case of large galactic clusters, with a preponderance of the system mass in the form of high energy plasma, gravitational analysis can hardly be sufficient. For the overall cluster dynamics, the thermodynamic and electrodynamic forces of the plasma need to be properly accounted for. If you know of any such work being done, I’d be very interested in seeing it.

      Were the discussions of 40 years ago documented?


      1. > Are these more elaborate formulations accessible online?

        Yes, almost all of science done after ~1970 is available online. You can use google scholar, arxiv, nasa ads or scopus to search for scientific articles. Books you can find using library catalogs, gooble books and goodreads. Getting a copy of whatever you found can be done via any decent university library, both on paper and digitally. If material is open access or a preprint is available you can usually find the pdf link right on the page of the search result. Most material is behind paywalls but usually you can just email the authors for a pdf. If you are going to read a lot I strongly suggest not bothering authors with constant requests and just get a library subscription. If for some reason that is not an option for you (if you are in the third world for example) you can try Sci-hub for papers and Library Genesis for books.

        For an introduction to rotation curves the two links are a good place to start. The first is a broad review by Sofue and Rubin from 2001. The second is one of Milgrom’s early 1983 papers. It goes through the maths of galaxy rotation curves in MOND (the purely Newtonian methodology is also discussed of course). In practice the analytic solutions will have to be found numerically but if you understand the theory and know how to do numerical calculations it should be trivial to whip up something accurate enough.

        Click to access 0010594.pdf

        Click to access nph-iarticle_query

        The links probably don’t work directly on wordpress but if you just copy and paste the entire url it should take you where you want to go.

        > For the overall cluster dynamics, the thermodynamic and electrodynamic forces of the plasma need to be properly accounted for. If you know of any such work being done, I’d be very interested in seeing it.

        You’re not one of those “Electric Universe” nuts are you?


        1. If you just want a few more lines of algebra to get beyond the point mass approximation to the case of a thin exponential disk, then I wrote a short document explaining this several years back:

          It should be easy enough to follow with only basic knowledge of maths and astronomy. But it shows why rotation curves have the shapes they do, at least from a MOND perspective, purely based on differences in the central surface density of an exponential disk galaxy.


          1. Indranil, I’m afraid your paper is a good illustration of the problem I’m complaining about. The following paragraph represents exactly the type of faulty physical reasoning that requires MOND (or alternatively dark matter) as a corrective:

            The disk is assumed to have spherical symmetry, allowing us to use the shell theorem
            to ignore mass at larger radii and consider all mass at smaller radii as if it were at the
            centre of the galaxy. It is known that this approximation works fairly well. However,
            real rotation curves will have higher speeds than in this calculation because the mass
            is closer than assumed to the point where we are calculating the force.
            There is also a
            large amount of cancellation of forces in spherical symmetry, because only the force
            towards the centre matters. Forces in all other directions cancel. In a disk, the force
            from any other part of the disk is more closely aligned with the inward radial direction
            than for a sphere.

            I added the boldface to highlight the only sentence in that paragraph that makes physical sense. Essentially, that sentence says that the shell theorem analysis being employed here does not work properly in the context of the physical system that is being modeled (a disk galaxy).

            The initial assumption, wherein a disk is said to have spherical symmetry is simply illogical. A disk is not a sphere; it cannot, therefore, have spherical symmetry. All following statements that depend on that assumption are consequently incorrect.

            Newton derived the shell theorem in the context of a perfectly-spherical-solid. In as much as, a galactic disk is not a sphere, nor is it a solid, I am at a loss to understand why you choose to assume that the shell theorem is applicable, while in your own (boldfaced) words you acknowledge that it does not, in fact, work at all for a galactic disk. The whole thing is profoundly illogical, physically nonsensical, and scientifically disturbing.


        2. @ Mark Huisjes, may I suggest that, in the future, if you don’t know the answer to a question I’ve asked, you refrain from saying anything at all, lest you look a fool.


          1. Keep it civil, people. Mark and Indranil were trying to help. If you didn’t get the answer you were looking for, perhaps you need to rephrase the question. Don’t spit on their efforts.

            The maths you seek are in textbooks like Galactic Dynamics by Binney & Tremaine. It is not a trivial text, so no one is going to reproduce it here.

            The issue of disk geometry was specifically suggested by Kalnajs as a solution to the flat rotation curve issue at IAU Symposium 100 in the early ’80s. It was an instructive moment, and if I find time maybe I’ll scan the relevant page. Right now, I am focussed on prepping to explain the cosmic microwave background to my students, which is my actual job. For now, I’ll just say that yes, you make a valid point, which is exactly the same point that Kalnajs made. Doing these things right, as he motivated us to do, was an important step forward, and that step was taken – long ago. While a real concern, 40 years ago, it turns out not to make that big a difference: it is an important detail, but it is not something that gets us out of the missing mass problem.

            Liked by 1 person

  12. I have a question about Figure 4.
    In the key is listed a symbol for “Exponential Profile”, but I do not see that symbol on the figure. Is this an oversite, or would the symbol actually lie off of the chart?


      1. Yes, that is correct. If you look very carefully, you can see the inverted triangle peeking out from behind the square. This is explained in the paper.


  13. I am very grateful to Stacy for adding a link below the blog to a video me and the lead author Moritz made about our paper:

    It is for the CosmologyTalks channel of Shaun Hotchkiss, who was the host and prepared the video. It covers similar material to the blog, though a video might be more fun, especially due to us having to answer questions about our work. There’s also some particularly exciting stuff at the end about where things might be headed.

    Liked by 1 person

  14. If anybody is interested, me and Moritz will be giving a talk on 7th December about the work described in this blog:

    Presumably this will be similar to the YouTube video linked at the bottom of the blog, but I guess you can ask questions if it’s ‘live’. The structure will also be a little different, as I’ll be introducing MOND a bit more carefully and going through the classical arguments for MOND in more detail, similarly to the first parts of the talks on my YouTube channel. The main classical arguments are the rotation curve regularities and the Local Group satellite planes – cosmological evidence like the KBC void and hubble tension are very new lines of evidence indeed. Moritz will be covering that in his portion of the talk.


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