There are a number of galaxies that have been reported to lack dark matter. This is weird in a universe made of dark matter. It is also weird in MOND, which (if true) is what causes the inference of dark matter. So how can this happen?

In most cases, it doesn’t. These claims not only don’t make sense in either context, they are simply wrong. I don’t want to sound too harsh, as I’ve come close to making the same mistake myself. The root cause of this mistake is often a form of static thinking in dynamic situations that the here and now is always a representative test. The basic assumption we have to make to interpret observed velocities in terms of mass is that systems are in (or close to) gravitational equilibrium so that the kinetic energy is a measure of the gravitational potential. In most places, this is a good assumption, so we tend to forget we even made it.

However, no assumption is ever perfect. For example, Gaia has revealed a wealth of subtle non-equilibrium effects in the Milky Way. These are not so large as to invalidate the basic inference of the mass discrepancy, but neither can they be entirely ignored. Even maintaining the assumption of a symmetric but non-smooth mass profile in equilibrium complicates the analysis.

Since the apparent absence of dark matter is unexpected in either theory, one needs to question the assumptions whenever this inference is made. There is one situation in which it is expected, so let’s consider that special case:

Tidal dwarf galaxies

Most dwarf galaxies are primordial – they are the way they are because they formed that way. However, it is conceivable that some dwarfs may form in the tidal debris of collisions between large galaxies. These are tidal dwarf galaxies (TDGs). Here are some examples of interacting systems containing candidate TDGs:

Fig. 1 from Lelli et al. (2015): images of interacting systems with TDG candidates noted in yellow.

I say candidate TDGs because it is hard to be sure a particular object is indeed tidal in origin. A good argument can be made that TDGs require such special conditions to form that perhaps they should not be able to form at all. As debris in tidal arms is being flung about in the (~ 200 km/s) potential well of a larger system, it is rather challenging for material to condense into a knot with a much smaller potential well (< 50 km/s). It can perhaps happen if the material in the tidal stream is both lumpy (to provide a seed to condense on) and sufficiently comoving (i.e., the tidal shear of the larger system isn’t too great), so maybe it happens on rare occasions. One way to distinguish TDGs from primordial dwarfs is metallicity: typical primordial dwarfs have low metallicity while TDGs have the higher metallicity of the giant system that is the source of the parent material.

A clean test of hypotheses

TDGs provide an interesting test of dark matter and MOND. In the vast majority of dark matter models, dark matter halos are dynamically hot, quasi-spherical systems with the particles that compose the dark matter (whatever it is) on eccentric, randomly oriented orbits that sum to a big, messy blob. Arguably it has to be this way in order to stabilize the disks of spiral galaxies. In contrast, the material that composes the tidal tails in which TDGs form originates in the baryonic material of the dynamically cold spiral disks where orbits are nearly circular in the same direction in the same thin plane. The phase space – the combination of position x,y,z and momentum vx,vy,vz – of disk and halo couldn’t be more different. This means that when two big galaxies collide or have a close interaction, everything gets whacked and the two components go their separate ways. Starting in orderly disks, the stars and gas make long, coherent tidal tails. The dark matter does not. The expectation from these basic phase space considerations is consistent with detailed numerical simulations.

We now have a situation in which the dark matter has been neatly segregated from the luminous matter. Consequently, if TDGs are able to form, they must do it only* with baryonic mass. The ironic prediction of a universe dominated by dark matter is that TDGs should be devoid of dark matter.

In contrast, one cannot “turn off” the force law in MOND. MOND can boost the formation of TDGs in the first place, but if said TDGs wind up in the low acceleration regime, they must evince a mass discrepancy. So the ironic prediction here is that, in ignorance of MOND, MOND means that we would infer that TDGs do have dark matter.

Got that? Dark matter predicts TDGs with no dark matter. MOND predicts TDGs that look like they do have dark matter. That’s not confusing at all.

Clean in principle, messy in practice

Tests of these predictions have a colorful history. Bournaud et al. (2007) did a lovely job of combining simulations with observations of the Seashell system (NGC 5291 above) and came to a striking conclusion: the rotation curves of TDGs exceeded that expected for the baryons alone:

Fig. 2 from Bournaud et al. (2007) showing the rotation curves for the three TDGs identified in the image above.

This was a strange, intermediary result. TDGs had more dark matter than the practically zero expected in LCDM, but less than comparable primordial dwarfs as expected in MOND. That didn’t make sense in either theory. They concluded that there must be a component of some other kind of dark matter that was not the traditional dark halo, but rather part of the spiral disk to begin with, perhaps unseen baryons in the form of very cold molecular gas.

Gentile et al. (2007) reexamined the situation, and concluded that the inclinations could be better constrained. When this was done, the result was more consistent with the prediction of MOND and the baryonic Tully-Fisher relation (BTFR. See their Fig. 2).

Fig. 1 from Gentile et al. (2007): Rotation curve data (full circles) of the 3 tidal dwarf galaxies (Bournaud et al. 2007). The lower (red) curves are the Newtonian contribution Vbar of the baryons (and its uncertainty, indicated as dotted lines). The upper (black) curves are the MOND prediction and its uncertainty (dotted lines). The top panels have as an implicit assumption (following Bournaud et al.) an inclination angle of 45 degrees. In the middle panels the inclination is a free parameter, and the bottom panels show the fits made with the first estimate for the external field effect (EFE).

Clearly there was room for improvement, both in data quality and quantity. We decided to have a go at it ourselves, ultimately leading to Lelli et al. (2015), which is the source of the pretty image above. We reanalyzed the Seashell system, along with some new TDG candidates.

Making sense of these data is not easy. TDG candidates are embedded in tidal features. It is hard to know where the dwarf ends and the tidal stream begins, or even to be sure there is a clear distinction. Here is an example of the northern knot in the Seashell system:

Fig. 5 from Lelli et al. (2015): Top panels: optical image (left), total H I  map (middle), and H I  velocity field (right). The dashed ellipse corresponds to the disc model described in Sect. 5.1. The cross and dashed line illustrate the kinematical centre and major axis, respectively. In the bottom-left corner, we show the linear scale (optical image) and the H I  beam (total H I  map and velocity field) as given in Table 6. In the total H I  map, contours are at ~4.5, 9, 13.5, 18, and 22.5 M⊙ pc-2. Bottom panels: position-velocity diagrams obtained from the observed cube (left), model cube (middle), and residual cube (right) along the major and minor axes. Solid contours range from 2σ to 8σ in steps of 1σ. Dashed contours range from −2σ to −4σ in steps of −1σ. The horizontal and vertical lines correspond to the systemic velocity and dynamical centre, respectively.

Both the distribution of gas and the velocities along the tidal tail often blend smoothly across TDG candidates, making it hard to be sure they have formed a separate system. In the case above, I can see what we think is the velocity field of the TDG alone (contained by the ellipse in the upper right panel), but is that really an independent system that has completely decoupled from the tidal material from which it formed? Definite maybe!

Federico Lelli did amazing work to sort through these difficult-to-interpret data. At the end of the day, he found that there was no need for dark matter in any of these TDG candidates. The amplitude of the apparent circular speed was consistent with the enclosed mass of baryons.

Figs. 11 and 13 from Lelli et al. (2015): the enclosed dynamical-to-baryonic mass ratio (left) and baryonic Tully-Fisher relation (right). TDGs (red points) are consistent with a mass ratio of unity: the observed baryons suffice; no dark matter is inferred. Contrary to Gentile et al., this manifests as a clear offset from the BTFR followed by normal galaxies.

Taken at face value, this absence of dark matter is a win for a universe made of dark matter and a falsification of MOND.

So we were prepared to say that, and did, but as Federico checked the numbers, it occurred to him to check the timescales. Mergers like this happen over the course of a few hundred million years, maybe a billion. The interactions we observe are ongoing; just how far into the process are they? Have the TDGs had time to settle down into dynamical equilibrium? That is the necessary assumption built into the mass ratio plotted above: the dynamical mass assumes the measured speed is that of a test particle in an equilibrium orbit. But these systems are manifestly not in equilibrium, at least on large scales. Maybe the TDGs have had time to settle down?

We can ask how long it takes to make an orbit at the observed speed, which is low by the standards of such systems (hence their offset from Tully-Fisher). To quote from the conclusions of the paper,

These [TDG] discs, however, have orbital times ranging from ~1 to ~3 Gyr, which are significantly longer than the TDG formation timescales (≲1 Gyr). This raises the question as to whether TDGs have had enough time to reach dynamical equilibrium.

Lelli et al. (2015)

So no, not really. We can’t be sure the velocities are measuring the local potential well as we want them to do. A particle should have had time to go around and around a few times to settle down in a new equilibrium configuration; here they’ve made 1/3, maybe 1/2 half of one orbit. Things have not had time to settle down, so there’s not really a good reason to expect that the dynamical mass calculation is reliable.

It would help to study older TDGs, as these would presumably have had time to settle down. We know of a few candidates, but as systems age, it becomes harder to gauge how likely they are to be legitimate TDGs. When you see a knot in a tidal arm, the odds seem good. If there has been time for the tidal stream to dissipate, it becomes less clear. So if such a thing turns out to need dark matter, is that because it is a TDG doing as MOND predicted, or just a primordial dwarf we mistakenly guessed was a TDG?

We gave one of these previously unexplored TDG candidates to a grad student. After much hard work combining observations from both radio and optical telescopes, she has demonstrated that it isn’t a TDG at all, in either paradigm. The metallicity is low, just as it should be for a primordial dwarf. Apparently it just happens to be projected along a tidal tail where it looks like a decent candidate TDG.

This further illustrates the trials and tribulations we encounter in trying to understand our vast universe.

*One expects cold dark matter halos to have subhalos, so it seems wise to suspect that perhaps TDGs condense onto these. Phase space says otherwise. It is not sufficient for tidal debris to intersect the location of a subhalo, the material must also “dock” in velocity space. Since tidal arms are being flung out at the speed that is characteristic of the giant system, the potential wells of the subhalos are barely speed bumps. They might perturb streams, but the probability of them being the seeds onto which TDGs condense is small: the phase space just doesn’t match up for the same reasons the baryonic and dark components get segregated in the first place. TDGs are one galaxy formation scenario the baryons have to pull off unassisted.

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.

29 thoughts on “Non-equilibrium dynamics in galaxies that appear to lack dark matter: tidal dwarf galaxies”

  1. Maybe we should check other tests where MOND predictions are clear cut, and good data can confirm them.
    Then return to these systems and assume MOND is correct to distinguish between those that are in equilibrium and those that are not.
    Then do other science on them now that they are segregated. I mean, in both case there is an assumption, but at our level of confidence in MOND perhaps it would be more fruitful to change the way we look at these TDGs?

  2. Very complicated it seems. Certainly dwarf galaxies, and in particular TDGs, are not isolated systems. The relationships observed in isolated systems between metallicity, age, baryonic and MOND/Dark Matter could be different for dwarfs due to evolution of different entropy conditions.

  3. That’s a good paper Stacy! Thank you ! I am glad to see that basics of system dynamical equilibrium are circulating in the community. Astrophysics is not an easy job !

  4. Over 50 years ago Alar and Juri Toomre did the first computer modelling of colliding galaxies, producing tidal tails. I am fairly sure that their models didn’t include dark matter which was suggested around the same time (in fact I think they just had a central mass and a number of massless test points). With the far greater computer power available these days, would it be possible to repeat this with a sufficient resolution to resolve TDGs in the tidal tails. At least you could do a comparison between MOND and pure Newtonian dynamics and see whether they predicted different TDG statistics. That wouldn’t depend on them being in dynamical equilibrium, just on how well the density distribution correlated with what we see.

    1. Yes, many simulations like this have since been done. Mostly in the context of dark matter, but some in MOND. The links to works by Bournaud et al. and Tiret & Combes are specific to TDGs in DM & MOND respectively. T&C may have done both, I forget, but the upshot was that TDGs were more likely to form in MOND for reasons that I hope are obvious. This seems anecdotally more consistent with what we see (it is hard to get TDGs to form at all in many DM simulation), but I’m not aware of a statistical analysis. I’m not entirely sure how one would even go about that, since every encounter is unique. Bilek et al (https://ui.adsabs.harvard.edu/abs/2018A%26A…614A..59B/abstract) have argued that the dwarf satellites of the Local Group and their orbital characteristics may be the result of an ancient interaction between the Milky Way and Andromeda.

      1. Ya know, I thought I cited Tiret & Combes above, but I’m not seeing the link up there. Certainly meant to do so, but perhaps it slipped my mind.
        https://www.youtube.com/watch?v=Uo89jxW9mUI
        Here are some of their papers:
        https://ui.adsabs.harvard.edu/search/fq=%7B!type%3Daqp%20v%3D%24fq_database%7D&fq_database=database%3A%20astronomy&q=author%3A(%22tiret%22%20%22combes%22)&sort=date%20desc%2C%20bibcode%20desc&p_=0
        For example, https://arxiv.org/abs/0712.1459

  5. Probably too rare and too low mass for a weak lensing measurement? I would guess one would need hundreds of thousands of tidal dwarfs ….

    1. For sure. Would be messy to disentangle from the rest of the system as well – would probably need strong lensing for the spatial resolution required, and I don’t think any of these things are above the critical surface density for that even if there were a convenient background quasar.

  6. “In the vast majority of dark matter models, dark matter halos are dynamically hot, quasi-spherical systems with the particles that compose the dark matter (whatever it is) on eccentric, randomly oriented orbits that sum to a big, messy blob.” Is string theory with dark matter particles a mess of unpredictability? String theorists (perhaps justifiably) reject my concept of “inertial forces from alternate universes”. However, they also (unjustifiably) reject MOND. Professor Milgrom of the Weizmann Institute is more than 20 years overdue for a Nobel Prize — the reason that he does not have a Nobel Prize is that his formulation of MOND does not make sense in terms of Green-Schwarz-Witten string theory. However, Guendelman’s new formulation of string theory might provide a way of incorporating MOND inertia into string theory. Has Professor Guendelman of Ben-Gurion University of the Negev found the MONDian completion of string theory?
    “Dynamical String Tension Theories with target space scale invariance SSB and restoration” by Eduardo Guendelman
    https://arxiv.org/abs/2104.08875 March 2025 European Physical Journal C
    Is it possible that Guendelman’s theory that “each string and each brane generates its own tension” will win a Nobel Prize for Guendelman (& Milgrom)?
    Consider the following hypotheses: (1) There are 2 fundamentally different types of string tension: one type generating Newton-Einstein inertia & the other type generating MOND inertia. (2) The dark matter phenomenon is entirely explained by MOND inertia. (3) Gravitational mass-energy, which is equivalent to inertial mass-energy, is predicted by Green-Schwarz-Witten string theory. (4) Quantum entanglement generates MOND inertia but does not generate gravitational mass-energy — this property of quantum entanglement is compatible with Guendelman’s new version of string theory.

  7. The short times since formation of some TDGs is interesting, less than an orbit. If there’s no DM, in PSG an excess of the emitted medium may not have had time to build up, so you just get the basic small-scale graded refractive medium, which leads to Newtonian gravity.

    In the MOND regime, once the system has had time to settle down, you should get the outer pattern, with faster dissipation, which leads to either MOND or DM, depending on ones view. At larger scales there’s a genuine excess, and real DM (though it’s not matter) – collisionless, non self-gravitating etc.

    But the emitted medium needing time to settle down and ‘do its thing’ is a possible aspect of this in PSG, so the time since formation could correlate with something. I’m always hoping something (probably not this particular question), will stick out far enough to fit the data in a clear way. But TDGs seem too messy to pin anything down for certain, and if they’re not in equilibrium all bets may be off.

    Incidentally, the PSG paper will soon be on a preprint site, as its now well and truly finished, for now it’s here https://gwwsdk1.wixsite.com/link/newpreprint-pdf , the new abstract is by far the best summary of it ever.

  8. There is always a refreshing candor in your writing. I find it iffy when MOND predictions are extended to -other unique- systems when the root cause or mechanism for MOND is not on-the-table. The bullet cluster is another example of a mess that no one should expect to dwell in equilibrium.

    1. The external field effect of MOND can definitely be a factor, and was explicitly considered by Gentile et al. It is specific to the field due to the environment in each case, so it is hard to make blanket statements beyond that.

      1. This topic makes me wonder if there is a way to assign a gravitational entropy to a system based on how closely it reproduces MOND.

  10. I have hung my hat on a theory that would predict TDGs should be brighter on the tails furthest from the galactic center. After reading your post, I was going to ask you, but I got even lazier and ask Google AI first:
    “Based on the provided search results, tidal dwarf galaxies (TDGs) are often found as bright, star-forming condensations, particularly towards the tip (the furthest end from the galactic centers) of tidal tails.”
    So then there is that:)
    This would also mean that the non-TDGs masquerading as the same would not have this same uniqueness.

    1. Yes, it is my anecdotal impression that there are frequently star-forming condensations at the tips of tidal tails. Those could just be star-forming regions at the tips of tails, or it could be that TDGs prefer to form there. One can’t tell without detailed kinematic measurements: is the gas there just part of the tail, or has it decoupled to form its own disk?

      Tiret & Combes say “We also show that tidal dwarf galaxies can be naturally formed at the tip of the tidal tails, in MOND. ”
      https://arxiv.org/pdf/0712.1459

  11. How might Tiret & Combes’s 2007 “Interacting Galaxies with MOND”
    https://arxiv.org/pdf/0712.1459 be extended?
    According to the prevailing paradigm for physics, MOND indicates an anomalous gravitational redshift, which is approximately constant in the MOND regime of approximation. Take the big G constant from Einstein’s general relativity theory and consider (1 + MOND-constant) * G.
    Consider MOND-constant in the range (1/5) * 10^–5 to 2 * 10^–5 and break up the range into 100 equal pieces. Then imitate the Tiret-Combes Newtonian analysis for 100 different values of MOND-constant to see which value gives the best fit to empirical data.

  12. Looking at general points about the mass discrepancy, and the odd fact that MOND works very well, there are questions about how well we understand gravity. Is curvature the right kind of description? A lot of physicists think GR is a large-scale approximation, but that curvature is right. Others are open to new theories like Verlinde’s, and getting gravity out of different concepts. Whatever one thinks about MOND, it can apply with or without curvature – either as an extension of GR, or as part of some UT with different concepts.

    So any handle on curvature is interesting. There are a few places where it’s seen as untouchable, and the geodetic effect is one of them. It was measured by Gravity Probe B, the orbiting gyros slowly turned though an angle – it was the right one, near enough. There has never been any explanation other than curvature, so it’s widely seen as conclusive, and the geodetic effect has been dubbed “curvature’s signature effect”. Google AI told me that it must be curvature doing it, and that no flat space theory could possibly explain it. But when I asked if its opinion might reflect any bias in the literature it reads, it said yes, definitely.

    The great thing about mathematics is you can actually determine things. In PSG, ten minutes of checking an equation can tell you a lot. The equation for the geodetic effect was reached by assuming what you always assume in PSG – the simple rule is that matter in a gravity field is slowed by sqrt (1 – 2GM/rc^2). This slows different parts of an orbiting object slightly differently due to different radii, which turns it through an angle over time (as in the introduction of the paper). GR approximates the angle per orbit as 360 GM/rc^2, the PSG equation gives the same result to 14 decimal places. This means it doesn’t have to be curvature, and is the first alternative explanation – so it weakens curvature’s grip on our picture of gravity.

  13. “… the odd fact that MOND works very well …” This year I had a very brief (about 30 seconds) phone conversation with a Nobel laureate who rejects MOND. His rejection of MOND seems to be based on the belief that MOND does not make sense in terms of theoretical physics. It seems to me that anyone who has spent 200 hours or more studying MOND’s empirical successes gets the idea that Milgrom should have a Nobel Prize, a Wolf Prize, & a Breakthrough Prize.

    1. Who? Doesn’t matter; that’s what I thought when I first encountered MOND. The difference between me and your generic Nobel prize winner is that I’ve made the effort to think it through while they have not. The attitude that MOND has to be wrong on theoretical grounds is itself wrong on many levels. Indeed, it is the other way around: that MOND works to the extent it does is the strongest possible indication of new physics beyond the usual blather. Unlike the quantum revolution a century ago, the brilliant theorists of today are gratuitously failing to engage with the difficult experimental prompt.

        1. The disproof of dark matter.

          No one is willing to consider MOND until dark matter is falsified. But dark matter can’t be falsified, so we spin our wheels ever deeper into the rut.

          1. The wide binaries can become a strong falsification, and I found your paper with Mistele using weak lensing out to 1 Mpc also a nice falsification.

            Anyhow, the American scientific business gets heavy bashing now, perhaps it will make them reflect better?

            1. It’s so surprising and nonsensical the way they ignore MOND that it’s tempting to say DM is a myth they believe in, like other myths. But that’s not the case. DM does better than MOND with the large scale structure and the CMB, and in clusters there seems to be some collisionless stuff floating around. MOND does better at the galaxy scale, where as you say, DM has really been falsified. The two views are not alternatives – they win, loosely speaking, in two different areas, at different scales. Stacy, you know better than me how accurate that is, I’d be interested to know what you think on it. It seems to me the situation is crying out for a hybrid theory, but people are so entrenched in opposing views, they sometimes consider that less than they might. But there has still been a crop of hybrid theories, as people try to find something that can fit an increasingly detailed set of clues.

              1. Yes, there needs to be something new that incorporates the best of both worlds. That could be a hybrid, or it could be some theoretical extension of MOND that winds up naturally explaining cosmology. Whether these are different possibilities is somewhat a matter of perspective – there are theories where the extra fields that are introduced are rather DM-like (e.g., the ghost condensate of AeST). But we’re not going to get a comprehensive understanding unless we engage with predictive success of MOND. Right now it is an article of faith that this will somehow fall out of simulations; this is a form of magical thinking.

  14. And looking at which is likely to be true, we know from the CMB that space is flat at a large scale. That’s a problem for GR, but not for flat space theories. The only real way out for GR is inflation, but that has had major problems since the Planck satellite data, and Penrose has shown inflation to have a fine tuning problem worse than the one it’s brought in to resolve – in one discussion Penrose said inflation has been falsified.

  15. “… dark matter can’t be falsified …” Is there a FUNDAMOND string theory that is far more empirically convincing than dark matter particles? Can Guendelman’s new version of string theory mathematically justify MOND inertia? Consider some conjectures:
    (1) String vibrations occur in 10-dimensional spacetime combined with 6 dimensions of quantum entanglement.
    (2) String vibrations have two different string tensions: (a) Green-Schwarz-Witten string tension which mathematically justifies the Standard Model of particle physics & general relativity theory and (b) Guendelman string tension which mathematically justifies quantum entanglement and can provide a mathematical justification for a correction to general relativity theory.
    (3) There are two fundamental types of inertia: Newton-Einstein inertia & quantum inertia — quantum inertia results from quantum entanglement & is mathematically approximated by MOND inertia in the MOND regime of approximation.

  16. What a difficult task to distinguish between all the possibilities when trying to determine whether a particular TDG genuinely has a mass discrepancy or not, or even if it’s a TDG. There are so many factors to take into account. The amount of research that has gone into trying to solve these puzzles is truly staggering.

    Meanwhile I’m close to uploading to viXra an amateur theory-model that adheres largely to the Standard Model of particle physics, except that it assigns an exotic form of charge to all fermions that coincides with weak-isospin. This leads to a ripple effect that impacts cosmology and atomic dynamics. Reluctantly, as a MOND purist, it ended up being a “hybrid model”, with dark matter (DM) accumulating principally in galactic clusters. This hypothetical DM is postulated to carry a very specific form of charge that explains its vanishingly small interaction cross-section with ordinary matter.

    It’s off the charts on speculation, and is structurally incomplete. So, admittedly, it’s has some problems. However, the model predicts ‘hidden’ Flavor-Changing-Neutral-Currents (FCNC’s) in 2nd and 3rd generation particle decays. At first I feared the proposed interactions would classify as 3rd or 4th order weak interactions and thus not occur, but was relieved when I reread an old Physics Forums thread “Neutrino to Neutrino Interactions Possible?” The interaction I proposed in that thread was indeed 4th order and thus basically non-existent. But these postulated FCNC events embody only two weak vertexes so should be as probable as the more common weak interactions.

    Please don’t ask any questions here, as I don’t want to interrupt this thread. As soon as I upload to viXra, or some other site, I’ll provide a link.

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