This Thanksgiving, I’d highlight something positive. Recently, Bob Sanders wrote a paper pointing out that gas rich galaxies are strong tests of MOND. The usual fit parameter, the stellar mass-to-light ratio, is effectively negligible when gas dominates. The MOND prediction follows straight from the gas distribution, for which there is no equivalent freedom. We understand the 21 cm spin-flip transition well enough to relate observed flux directly to gas mass.

In any human endeavor, there are inevitably unsung heroes who carry enormous amounts of water but seem to get no credit for it. Sanders is one of those heroes when it comes to the missing mass problem. He was there at the beginning, and has a valuable perspective on how we got to where we are. I highly recommend his books,

In bright spiral galaxies, stars are usually 80% or so of the mass, gas only 20% or less. But in many dwarf galaxies,  the mass ratio is reversed. These are often low surface brightness and challenging to observe. But it is a worthwhile endeavor, as their rotation curve is predicted by MOND with extraordinarily little freedom.

Though gas rich galaxies do indeed provide an excellent test of MOND, nothing in astronomy is perfectly clean. The stellar mass-to-light ratio is an irreducible need-to-know parameter. We also need to know the distance to each galaxy, as we do not measure the gas mass directly, but rather the flux of the 21 cm line. The gas mass scales with flux and the square of the distance (see equation 7E7), so to get the gas mass right, we must first get the distance right. We also need to know the inclination of a galaxy as projected on the sky in order to get the rotation to which we’re fitting right, as the observed line of sight Doppler velocity is only sin(i) of the full, in-plane rotation speed. The 1/sin(i) correction becomes increasingly sensitive to errors as i approaches zero (face-on galaxies).

The mass-to-light ratio is a physical fit parameter that tells us something meaningful about the amount of stellar mass that produces the observed light. In contrast, for our purposes here, distance and inclination are “nuisance” parameters. These nuisance parameters can be, and generally are, measured independently from mass modeling. However, these measurements have their own uncertainties, so one has to be careful about taking these measured values as-is. One of the powerful aspects of Bayesian analysis is the ability to account for these uncertainties to allow for the distance to be a bit off the measured value, so long as it is not too far off, as quantified by the measurement uncertainties. This is what current graduate student Pengfei Li did in Li et al. (2018). The constraints on MOND are so strong in gas rich galaxies that often the nuisance parameters cannot be ignored, even when they’re well measured.

To illustrate what I’m talking about, let’s look at one famous example, DDO 154. This galaxy is over 90% gas. The stars (pictured above) just don’t matter much. If the distance and inclination are known, the MOND prediction for the rotation curve follows directly. Here is an example of a MOND fit from a recent paper:

This is terrible! The MOND fit – essentially a parameter-free prediction – misses all of the data. MOND is falsified. If one is inclined to hate MOND, as many seem to be, then one stops here. No need to think further.

If one is familiar with the ups and downs in the history of astronomy, one might not be so quick to dismiss it. Indeed, one might notice that the shape of the MOND prediction closely tracks the shape of the data. There’s just a little difference in scale. That’s kind of amazing for a theory that is wrong, especially when it is amplifying the green line to predict the red one: it needn’t have come anywhere close.

Here is the fit to the same galaxy using the same data [already] published in Li et al.:

Now we have a good fit, using the same data! How can this be so?

I have not checked what Ren et al. did to obtain their MOND fits, but having done this exercise myself many times, I recognize the slight offset they find as a typical consequence of holding the nuisance parameters fixed. What if the measured distance is a little off?

Distance estimates to DDO 154 in the literature range from 3.02 Mpc to 6.17 Mpc. The formally most accurate distance measurement is 4.04 ± 0.08 Mpc. In the fit shown here, we obtained 3.87 ± 0.16 Mpc. The error bars on these distances overlap, so they are the same number, to measurement accuracy. These data do not falsify MOND. They demonstrate that it is sensitive enough to tell the difference between 3.8 and 4.1 Mpc.

One will never notice this from a dark matter fit. Ren et al. also make fits with self-interacting dark matter (SIDM). The nifty thing about SIDM is that it makes quasi-constant density cores in dark matter halos. Halos of this form are not predicted by “ordinary” cold dark matter (CDM), but often give better fits than either MOND of the NFW halos of dark matter-only CDM simulations. For this galaxy, Ren et al. obtain the following SIDM fit.

This is a great fit. Goes right through the data. That makes it better, right?

Not necessarily. In addition to the mass-to-light ratio (and the nuisance parameters of distance and inclination), dark matter halo fits have [at least] two additional free parameters to describe the dark matter halo, such as its mass and core radius. These parameters are highly degenerate – one can obtain equally good fits for a range of mass-to-light ratios and core radii: one makes up for what the other misses. Parameter degeneracy of this sort is usually a sign that there is too much freedom in the model. In this case, the data are adequately described by one parameter (the MOND fit M*/L, not counting the nuisances in common), so using three (M*/L, Mhalo, Rcore) is just an exercise in fitting a French curve. There is ample freedom to fit the data. As a consequence, you’ll never notice that one of the nuisance parameters might be a tiny bit off.

In other words, you can fool a dark matter fit, but not MOND. Erwin de Blok and I demonstrated this 20 years ago. A common myth at that time was that “MOND is guaranteed to fit rotation curves.” This seemed patently absurd to me, given how it works: once you stipulate the distribution of baryons, the rotation curve follows from a simple formula. If the two don’t match, they don’t match. There is no guarantee that it’ll work. Instead, it can’t be forced.

As an illustration, Erwin and I tried to trick it. We took two galaxies that are identical in the Tully-Fisher plane (NGC 2403 and UGC 128) and swapped their mass distribution and rotation curve. These galaxies have the same total mass and the same flat velocity in the outer part of the rotation curve, but the detailed distribution of their baryons differs. If MOND can be fooled, this closely matched pair ought to do the trick. It does not.

Our failure to trick MOND should not surprise anyone who bothers to look at the math involved. There is a one-to-one relation between the distribution of the baryons and the resulting rotation curve. If there is a mismatch between them, a fit cannot be obtained.

We also attempted to play this same trick on dark matter. The standard dark matter halo fitting function at the time was the pseudo-isothermal halo, which has a constant density core. It is very similar to the halos of SIDM and to the cored dark matter halos produced by baryonic feedback in some simulations. Indeed, that is the point of those efforts: they  are trying to capture the success of cored dark matter halos in fitting rotation curve data.

Dark matter halos with a quasi-constant density core do indeed provide good fits to rotation curves. Too good. They are easily fooled, because they have too many degrees of freedom. They will fit pretty much any plausible data that you throw at them. This is why the SIDM fit to DDO 154 failed to flag distance as a potential nuisance. It can’t. You could double (or halve) the distance and still find a good fit.

This is why parameter degeneracy is bad. You get lost in parameter space. Once lost there, it becomes impossible to distinguish between successful, physically meaningful fits and fitting epicycles.

Astronomical data are always subject to improvement. For example, the THINGS project obtained excellent data for a sample of nearby galaxies. I made MOND fits to all the THINGS (and other) data for the MOND review Famaey & McGaugh (2012). Here’s the residual diagram, which has been on my web page for many years:

These are, by and large, good fits. The residuals have a well defined peak centered on zero.  DDO 154 was one of the THINGS galaxies; lets see what happens if we use those data.

The first thing one is likely to notice is that the THINGS data are much better resolved than the previous generation used above. The first thing I noticed was that THINGS had assumed a distance of 4.3 Mpc. This was prior to the measurement of 4.04, so lets just start over from there. That gives the MOND prediction shown above.

And it is a prediction. I haven’t adjusted any parameters yet. The mass-to-light ratio is set to the mean I expect for a star forming stellar population, 0.5 in solar units in the Sptizer 3.6 micron band. D=4.04 Mpc and i=66 as tabulated by THINGS. The result is pretty good considering that no parameters have been harmed in the making of this plot. Nevertheless, MOND overshoots a bit at large radii.

Constraining the inclinations for gas rich dwarf galaxies like DDO 154 is a bit of a nightmare. Literature values range from 20 to 70 degrees. Seriously. THINGS itself allows the inclination to vary with radius; 66 is just a typical value. Looking at the fit Pengfei obtained, i=61. Let’s try that.

The fit is now satisfactory. One tweak to the inclination, and we’re done. This tweak isn’t even a fit to these data; it was adopted from Pengfei’s fit to the above data. This tweak to the inclination is comfortably within any plausible assessment of the uncertainty in this quantity. The change in sin(i) corresponds to a mere 4% in velocity. I could probably do a tiny bit better with further adjustment – I have left both the distance and the mass-to-light ratio fixed – but that would be a meaningless exercise in statistical masturbation. The result just falls out: no muss, no fuss.

Hence the point Bob Sanders makes. Given the distribution of gas, the rotation curve follows. And it works, over and over and over, within the bounds of the uncertainties on the nuisance parameters.

One cannot do the same exercise with dark matter. It has ample ability to fit rotation curve data, once those are provided, but zero power to predict it. If all had been well with ΛCDM, the rotation curves of these galaxies would look like NFW halos. Or any number of other permutations that have been discussed over the years. In contrast, MOND makes one unique prediction (that was not at all anticipated in dark matter), and that’s what the data do. Out of the huge parameter space of plausible outcomes from the messy hierarchical formation of galaxies in ΛCDM, Nature picks the one that looks exactly like MOND.

It is a bad sign for a theory when it can only survive by mimicking its alternative. This is the case here: ΛCDM must imitate MOND. There are now many papers asserting that it can do just this, but none of those were written before the data were provided. Indeed, I consider it to be problematic that clever people can come with ways to imitate MOND with dark matter. What couldn’t it imitate? If the data had all looked like technicolor space donkeys, we could probably find a way to make that so as well.

Cosmologists will rush to say “microwave background!” I have some sympathy for that, because I do not know how to explain the microwave background in a MOND-like theory. At least I don’t pretend to, even if I had more predictive success there than their entire community. But that would be a much longer post.

For now, note that the situation is even worse for dark matter than I have so far made it sound. In many dwarf galaxies, the rotation velocity exceeds that attributable to the baryons (with Newton alone) at practically all radii. By a lot. DDO 154 is a very dark matter dominated galaxy. The baryons should have squat to say about the dynamics. And yet, all you need to know to predict the dynamics is the baryon distribution. The baryonic tail wags the dark matter dog.

But wait, it gets better! If you look closely at the data, you will note a kink at about 1 kpc, another at 2, and yet another around 5 kpc. These kinks are apparent in both the rotation curve and the gas distribution. This is an example of Sancisi’s Law: “For any feature in the luminosity profile there is a corresponding feature in the rotation curve and vice versa.” This is a general rule, as Sancisi observed, but it makes no sense when the dark matter dominates. The features in the baryon distribution should not be reflected in the rotation curve.

The observed baryons orbit in a disk with nearly circular orbits confined to the same plane. The dark matter moves on eccentric orbits oriented every which way to provide pressure support to a quasi-spherical halo. The baryonic and dark matter occupy very different regions of phase space, the six dimensional volume of position and momentum. The two are not strongly coupled, communicating only by the weak force of gravity in the standard CDM paradigm.

One of the first lessons of galaxy dynamics is that galaxy disks are subject to a variety of instabilities that grow bars and spiral arms. These are driven by disk self-gravity. The same features do not appear in elliptical galaxies because they are pressure supported, 3D blobs. They don’t have disks so they don’t have disk self-gravity, much less the features that lead to the bumps and wiggles observed in rotation curves.

Elliptical galaxies are a good visual analog for what dark matter halos are believed to be like. The orbits of dark matter particles are unable to sustain features like those seen in  baryonic disks. They are featureless for the same reasons as elliptical galaxies. They don’t have disks. A rotation curve dominated by a spherical dark matter halo should bear no trace of the features that are seen in the disk. And yet they’re there, often enough for Sancisi to have remarked on it as a general rule.

It gets worse still. One of the original motivations for invoking dark matter was to stabilize galactic disks: a purely Newtonian disk of stars is not a stable configuration, yet the universe is chock full of long-lived spiral galaxies. The cure was to place them in dark matter halos.

The problem for dwarfs is that they have too much dark matter. The halo stabilizes disks by  suppressing the formation of structures that stem from disk self-gravity. But you need some disk self-gravity to have the observed features. That can be tuned to work in bright spirals, but it fails in dwarfs because the halo is too massive. As a practical matter, there is no disk self-gravity in dwarfs – it is all halo, all the time. And yet, we do see such features. Not as strong as in big, bright spirals, but definitely present. Whenever someone tries to analyze this aspect of the problem, they inevitably come up with a requirement for more disk self-gravity in the form of unphysically high stellar mass-to-light ratios (something I predicted would happen). In contrast, this is entirely natural in MOND (see, e.g., Brada & Milgrom 1999 and Tiret & Combes 2008), where it is all disk self-gravity since there is no dark matter halo.

The net upshot of all this is that it doesn’t suffice to mimic the radial acceleration relation as many simulations now claim to do. That was not a natural part of CDM to begin with, but perhaps it can be done with smooth model galaxies. In most cases, such models lack the resolution to see the features seen in DDO 154 (and in NGC 1560 and in IC 2574, etc.) If they attain such resolution, they better not show such features, as that would violate some basic considerations. But then they wouldn’t be able to describe this aspect of the data.

Simulators by and large seem to remain sanguine that this will all work out. Perhaps I have become too cynical, but I recall hearing that 20 years ago. And 15. And ten… basically, they’ve always assured me that it will work out even though it never has. Maybe tomorrow will be different. Or would that be the definition of insanity?

52 thoughts on “Hypothesis testing with gas rich galaxies”

1. Ron Smith says:

“Or would that be the definition of insanity?”
Are you referring to the definition that says doing the same thing over and over but expecting a different result is insanity?

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2. This reminds me of the recently discovered Ant 2 galaxy (https://arxiv.org/abs/1811.04082). According to the authors, it is “by far the lowest surface brightness system known”. The authors further state that explaining this galaxy “may even require alternatives to cold dark matter”. They do not consider MOND, however. This seems like it would be an ideal test for MOND. Are we going to see a velocity curve prediction any time soon?

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3. Thanks for this blogpost, I gained some new insight, in particular structure formation with self- gravity in dwarf galaxies.
My question is concerned with the resolution of the velocity field. Having the weak principle of equivalence in mind we expect the same velocity of a star /individual H- atom ( irrespective of their masses) at a given radius. In optical discs ( HSB galaxies) velocities of individual stars can be discerned, basically the velocity of a point mass. In contrast the velocity field of H gas ( HSB or LSB galaxies) is averaged over a greater area ( at a given radius). How does the quality of the velocity data ( star versus gas) compare? Is there a significantly greater spread of the gas- velocities in comparison to the star- velocities, at a given radius ?
I am interested in this point because I consider the possibility that objects of different mass have different velocity spreads, in deviance of WEP ( at least in the DML).
Thanks Frank

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4. Enrico says:

Dr. McGaugh,

You wrote:
“I do not know how to explain the microwave background in a MOND-like theory. At least I don’t pretend to, even if I had more predictive success there than their entire community.”

In that paper, you combined MOND and dark matter and predicted neutrino with mass around 1 eV/c^2. It’s too heavy. The three flavors of neutrino have combined mass around 0.3 eV/c^2. If you are predicting sterile neutrino, it’s beyond the Standard Model. Neutrino dark matter does not affect galaxy dynamics because neutrinos move at light speed. Too fast to form dark matter halo in galaxies.

I suppose you applied MOND’s acceleration scale on cosmic scale. In CDM cosmology, the ratio of dark matter to baryonic matter = 26.8/4.9 = 5.47. What is this ratio in MOND cosmology?

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5. Just to clarify my above, a bit convoluted question : does a gravitational bound object ( star) have a better defined, measureable velocity than an extended, diffuse gas in the DML, at a given radius? In case of the star we measure the centre of mass velocity of star? This corresponds to what quantity in gas of the gas? Thanks

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6. Martin R H S says:

The funny (or sad) point is, that the Astrophysics proponents of (C)DM dismiss MOND not at last because they assume it to be a fit – the term ‘Modified’ suggests this more than any short consideration of MOND can overcome. For a very long time I dismissed MOND for just this reason.
Now you show, that the actual situation with real data is just the other way round…
Another illfated term in MOND is the ‘Newtonian’, since this appears immediately obsolete – at that point I’m afraid many physicists don’t think further.
To improve the situation I think it is important to stress (as you do in your article) that MOND is not a fit but a constrained algorithm delivering structured results.
The other point might be to avoid the (misguiding) false duopole: either DM or MOND is the solution. On this dichotomy MOND looses, because it alone is no closed theory anyway and DM works well with cosmology.
I guess the point is, that MOND is a valid description on galactic scales, that cannot easily be dismissed or explained as result from some straightforward DM theory (like CDM). Complementarily a theory working for cosmology and covering MOND behavior has to be found – may be some not simple (interacting?) form of DM or respective Fields.
First of all, the ignorance of the meanwhile convincingly established ‘MOND phenomenology’ has to be overcome and I see this article as a good means for that.

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7. A relevant recent paper is this one which points out the relationships between the CDM parameters that leap out of the data: Paolo Salucci, “The distribution of dark matter in galaxies” (November 21, 2018) (60 pages, 28 Figures ~220 refs. Invited review for The Astronomy and Astrophysics Review). https://arxiv.org/abs/1811.08843

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8. Lots of good comments & questions here. In order:
– Yes, repeating the same behavior expecting different results is the colloquial definition of insanity I had in mind.
– Antlia 2 I comments on in my Twitter feed. It would be a great test if it were far away. Since it is so diffuse yet relatively close to the Milky Way, in MOND it should be in the throes of tidal disruption – it is more a collection of stars in similar orbits than it is a self-bound galaxy. In this case, the customary and necessary assumption of dynamical equilibrium doesn’t hold, so we can’t really predict the velocity dispersion.
– Stars and gas share the same orbits to the extent that we expect them to do so, which is to say: I’ve seen no indication of a violation of the weak equivalence principle. We don’t expect their orbits to be identical as stars are subject to scattering which results in asymmetric drift (a net lag behind the local circular speed caused by mildly eccentric orbits) while gas can be subject to shocks and other fluid dynamical effects. All that said, the best measurements that I’m aware of are consistent with them behaving as they should in the same gravitational potential. I should add that MOND violates the strong equivalence principle, but not the weak equivalence principle. And to the follow-up question – stars are not better measured, per se, but they do make better ballistic tracers, as their orbits are harder to change once established. But that can cut both ways; gas dissipates, but in doing so tends to settle into an orderly, rotating disk. This is the condition we see in many disk galaxies.
– I mostly agree with what Martin says, and I started from the same perspective. MOND and its phenomenology is an endpoint achieved by galaxies that needs to be explained in any successful theory, regardless of whether that is based in dark matter or some modification of dynamics. Certainly both “modified” and “Newtonian” were unfortunate choices, but it was a long time ago that Milgrom made this choice. But the point of my post is that it is pretty crazy to think this will fall out of a dark matter theory just because dark matter does better on cosmological scales. Sure it does – it is an infinity of free parameters. In terms of the philosophy of science, that’s bad, not good: one can never hope to falsify it – even if it is wrong – because there is always another knob to tweak. So how do we tell if we’re barking up the wrong tree? I do disagree with the assertion that I often see, and remains a widespread misconception among scientists, that MOND does not do a good job of explaining cosmic phenomena. I’ve been through all that, and it is simply incorrect to say dark matter is always better outside of galaxies. Sure, it has shortcomings, but often dark matter only appears to be better because it declines to make a comparably testable prediction. Clusters of galaxies are a great example: MOND is off by a factor of 2 in mass (20% in velocity). Dark matter makes no prediction. Anything over the luminous mass is OK. The rest is dark. It doesn’t matter if that’s a factor of 2 or 5 (the modern value) or 6 (the cosmic value) or ~100 (Zwicky’s value). Being a factor of 2 off only sounds bad because there is a definite prediction! With dark matter we don’t care what that factor is because we have no prediction – at least, not one that can not be fudged.
Still, MOND itself is incomplete. I think we are left with two primary options: (i) MOND is pointing to some deeper theory of dynamics (a change in the law of inertia a possibility; it doesn’t have to be gravity) that has MOND as an approximation in the appropriate limit, or (ii) some kind of dark matter that interacts directly with baryons in a way that gives rise to MOND-like phenomenology. I spent a lot of time over many years trying to invent such stuff, without success. More recent efforts like Blanchet’s dipolar dark matter or Khoury’s superfluid dark matter are viable possibilities that others have invented. SIDM does not fall in this category. It interacts with itself, not the baryons, so my reaction when it was reintroduced in 2000 was that it was dead on arrival: it is constructed to ignore the real problem. I’m less certain now as the math does have a term that involves the baryons, albeit not obviously the right term, so I give it a [very] outside chance that it might work out. What does not have a realistic chance is what the vast majority of my colleagues seem to favor: that the observed phenomenology will somehow emerge from feedback during galaxy formation in the conventional context of cold dark matter. This strikes me as a form of magical thinking. If MOND-like phenomenology were so natural to simulations of galaxy formation, why did it never emerge from such simulations before I told them the answer? I have no doubt that if can be *made* to emerge from such simulations, as simulators are very clever and have an endless number of knobs to twiddle. It then becomes a separate question whether the universe really works like that, with all the knobs turned Just So.
Subtle is the Lord, but malicious He is not.

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9. There was so much to say here, I missed the bit about neutrinos, which is probably worth an entire post in itself. In the paper to which Enrico refers, I did find that it helped to have a somewhat heavy neutrino (~1 eV) to describe the microwave background in a theory devoid of cold dark matter. That doesn’t need to be a literal neutrino: there has to be something new in a more complete theory to address the CMB that “looks like” a neutrino when analyzed conventionally. But there is nothing to preclude the neutrino mass from being heavier than the 0.3 eV Enrico quotes. Not sure where that number comes from. There are two distinct upper limits. The experimental limit on the neutrino mass is around 2 eV. Independently, there is a limit from structure formation in LCDM, which imposes a much stronger upper limit of 0.12 eV on the sum of the 3 neutrino masses according to the latest Planck results. There is also a lower limit: a minimum mass of 0.06 eV is required by neutrino oscillations. Consequently, there is another important test here, as so there is a narrow window in which the neutrino mass must reside (0.06 – 0.12 eV). However, the structure formation limit does not apply in MOND, where it would be convenient if the neutrino mass were somewhat larger, not just for the CMB, but also for clusters, and to damp early structure formation a little. So rather than being excluded, a neutrino as heavy as what I discussed in the context of the CMB remains experimentally possible, and would falsify LCDM if detected.
As for the cosmic ratio mass ratio, yes, I addressed this long ago. The cosmic mass ratio of the apparent total mass density to the baryonic density depends on the acceleration scale where the cosmic value is measured. So one naturally gets a boost like what is observed, from the 0.05 of baryons up to ~0.3 inferred conventionally. Probes that measure Omega_matter in different ways need not exactly agree, as it’ll depend on the mean accelerations to which they’re sensitive. So there need not be a single ratio. This might go some way to explaining the diversity of estimates in the literature (not every measurement is concordant, e.g., https://arxiv.org/abs/1810.06326).

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10. JB says:

@Dr. McGaugh: You wrote, “I think we are left with two primary options: (i) MOND is pointing to some deeper theory of dynamics (a change in the law of inertia a possibility; it doesn’t have to be gravity) that has MOND as an approximation in the appropriate limit, or (ii) some kind of dark matter that interacts directly with baryons in a way that gives rise to MOND-like phenomenology.”
This is very interesting. Do you expect that evidence of one would rule out the other?

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11. Depends on the evidence. In general, it will be hard to distinguish between these options. One clear distinction would be the detection of a dark matter particle with the appropriate properties. That is not something current experiments are designed to do, and it isn’t even clear that there is a well-posed prediction for such a particle, much less an experiment to detect it.

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12. Enrico says:

Dr. McGaugh,
The 0.3 eV sum of neutrino masses I quoted came from Planck data of CMB (Battye and Moss, 2014)
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.051303

The sterile neutrino is not in the Standard Model and its mass is unknown. It is a candidate for dark matter so your prediction of 1 eV neutrino mass, if true, does not falsify dark matter since particle mass is a free parameter.

As for your Option 1, Dr. Hossenfelder has an equation that relates MOND acceleration scale and the cosmological constant (see video)
http://backreaction.blogspot.com/2018/11/modified-gravity-demystified-video.html

Here’s my MOND=CC hypothesis where I derive MOND acceleration scale g = 1.2 e-10 m/s^2 from the cosmic acceleration (a) of the expansion of the universe. Age of universe (t) = 13.8 billion years. Without cosmic acceleration, the radius of universe would be 13.8 billion lightyears. But the radius to cosmic horizon R = 46.5 billion ly. Thus the scale factor s = 46.5/13.8 = 3.37 is due to cosmic acceleration

MOND=CC hypothesis postulates that all space must expand by a factor of 3.37. However, space within galaxies is prevented from expansion by galactic gravitational field, which is stronger than the anti-gravitational field (repulsive force) of cosmic expansion. At the outer radii of galaxies, gravitational field is weaker than anti-gravity. Thus this region of space should expand by the scale factor. I prove mathematically that this gravitational field threshold is equivalent to MOND acceleration scale.

From kinematic equations:
R = v t + ½ a t^2 where v = c = speed of light
Solving for a = cosmic acceleration
a = 2 (R – v t) / t^2 = 3.27 e-9 m/s^2
Since a > g = MOND acceleration scale, space should expand. It already did. I calculate the galactic radius before expansion and show cosmic acceleration (a) approximately equals MOND acceleration scale (g)

From Newtonian mechanics:
g = G M / r^2 where: G = gravitational constant, M = mass of galaxy, r = radius of galaxy
We reduce r by the scale factor s = 3.37 to get radius before space expansion
g’ = G M / (r/3.37)^2 = 11.36 G M/r^2 where g’ = gravitational acceleration before expansion
Hence: g’ = 11.36 g
Equating g’ and a:
a = g’ = 11.36 g
a/g = 2.4

Therefore, cosmic acceleration (a) approximately equals MOND acceleration scale (g) by a factor of 2.4. This is pretty close considering estimates of the mass of our galaxy vary by a factor of 5. MOND=CC hypothesis predicts the deviation of galactic orbital speed from Newtonian dynamics will not exceed the scale factor s = 3.37

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13. OK – 0.3 eV sounds right for earlier constraints from Planck on the neutrino mass. The most recent constraint that I’ve seen from them is < 0.12 eV (https://arxiv.org/abs/1807.06209). That leaves the narrow window 0.06-0.12 eV available to LCDM.
Let's be careful about what falsifies what. The Planck constraint Sum[m(nu)] < 0.12 eV *assumes* that LCDM is correct. So the detection of a 1 eV neutrino would falsify this assumption, and with it, LCDM as currently posed. This does not require sterile neutrinos. I am explicitly saying that it remains experimentally permissible for there to be a normal neutrino mass heavier than allowed by LCDM. That would falsify cosmology as we currently understand it. It would not falsify the concept of dark matter, which, as a concept, is not falsifiable. It would nevertheless contradict everything that is currently said to be good about WIMP dark matter, as it would break the currently good fits to the CMB and large scale structure. No harm to the standard model of particle physics is required in the form of sterile neutrinos or other new particles. Such a result would also be a strong indication that structure formation is fueled by a change in the force law rather than dark matter, as already suggested by the early appearance of galaxies as predicted by Bob Sanders.
As for your MOND=CC hypothesis, there are some interesting ideas there. It is already striking that the age of the universe is very nearly 1/H0, as expected for a constant expansion rate (neither decelerating nor accelerating. I consider it a to be a fine-tuning problem that LCDM first decelerates then accelerates to average out to very nearly constant expansion right now when we live even though the entire future is strongly dominated by acceleration: we live at a special time.) I also think there is some merit to considering what happens in the transition between a bound galaxy and the expanding space in which it is embedded. It is also striking that the cosmic acceleration scale is close to the MOND acceleration scale.
All that said, accelerations in galaxies are often observed to fall to 0.1 a0, and sometimes to only a few percent. So whatever the scale factor has to say about this has to accommodate at least this much dynamic range. Does the factor s=3.37 suffice to explain the very low accelerations observed in low surface brightness galaxies?

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14. Enrico says:

In my opinion, WIMP dark matter is unlikely. They have not found sparticles in the LHC. If dark matter particle exists, it’s likely to be neutrinos or some other particles similar to neutrino – very low mass and doesn’t interact with matter.

Correction to MOND=CC hypothesis:
The ratio should be before space expansion:
a/g’ = 2.4

Deviation of orbital velocity (v) from Newtonian dynamics:
Centripetal force = gravitational force
m v^2/R = G M m/R^2
v = (G M/R)^0.5
Calculate v’ = orbital velocity before space expansion, decrease R by scale factor s = 3.37
v’ = (G M/(R/s)^0.5 = s^0.5 (G M/R)^0.5
Ratio of orbital velocities before and after space expansion:
v’/v = s^0.5 = 1.84
Hence, the predicted deviation from Newtonian dynamics will not exceed the square root of the scale factor = 1.84

The acceleration scale (a) is a threshold so it can accommodate lower accelerations (k ao) where 0 < k < 1
Approximation of acceleration scale:
a ≈ g’ = G M / (r/3.37)^2
For lower accelerations, multiply both sides of equation by k:
k a ≈ k G M / (r/3.37)^2
k a ≈ G M / (r/3.37/k^0.5)^2
Hence, lower acceleration is equivalent to increasing the radius r to r/k^0.5
The same scale factor s = 3.37 applies to lower accelerations.

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15. OK, good – that’s what I thought – the deviation from Newton is modest. What I am saying is that we know of many cases where the deviation is large – often a factor of 10 in acceleration, sometimes nearly 100.

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16. Enrico says:

The Newtonian equation for gravitational field is valid for orbiting bodies where most of the mass is near the center of mass (or galaxy):
g = G M/R^2
For galaxies with uniform mass distribution, I use mass density formula:
M = D V
Where: M = mass of galaxy, D = mass density, V = volume
For disk galaxies:
V = pi R^2 t
Where: R = radius = distance of orbiting body from galactic center, t = thickness of disk
M = D pi R^2 t
Substituting M from above equation to the gravitational field equation:
g = G (D pi R^2 t)/R^2
g = G D pi t

Hence, gravitational field and gravitational acceleration (g) are independent of R and depend only on mass density and thickness of disk. Using the gravitational field equation, an orbiting body 10x farther would yield a gravitational acceleration 100x weaker. But if the galaxy has uniform mass distribution, the observed acceleration would deviate from Newtonian equation by a factor of 100.

Uniform mass distribution may apply to ISM dominated galaxies. Could this explain the big deviations from Newtonian dynamics?

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17. Enrico says:

MOND=CC hypothesis states that space in the outer edges of galaxies where gravitational field is weaker than cosmic acceleration has expanded by the scale factor = 3.37. As a result, stars in these regions moved away 3.37x farther than their original radii (r). Their orbital velocities (v) remain the same because spacetime expansion includes both space and time.
v = r/t
After spacetime expansion: v = 3.37 r/(3.37 t) = r/t
Hence, kinetic energy remains the same but potential energy increased since radius has increased.

Conservation of potential energy (PE) and kinetic energy (KE):
-PE = KE
-G M m/r = ½ m v^2
The negative sign means that increase in KE must be balanced by decrease in PE and vice versa.
After spacetime expansion:
-G M m/(3.37 r) = ½ m v^2
The equation is not balanced. This violates the conservation of energy. I will show that energy is conserved when I include quantum mechanical effect.

From De Broglie equation:
p = h/y
where: p = momentum, h = Planck constant, y = wavelength of matter-wave
KE in momentum form:
KE = ½ p v
Substitute De Broglie equation into p:
KE = ½ (h/y) v
After spacetime expansion, all lengths increased by the scale factor including wavelength:
KE = ½ h/(3.37 y) v
Substitute KE to energy equation:
-G M m/(3.37 r) = ½ h/(3.37 y) v
-G M m/r = ½ (h/y) v = ½ p v = ½ m v^2
The equation is balanced. Energy is conserved after including quantum mechanical effect.

However, there is a paradox here. Momentum decrease by 1/3.37 but mass and velocity remain the same. Next I will resolve the paradox by invoking dark energy.

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18. OK, so, going back to your equations above, like g = G D pi t – these are dimensionally correct, but their application requires integration that you appear to be missing. I realize this is not a good forum for detailed math, but the acceleration isn’t just the enclosed surface density; it is the derivative of the gravitational potential that follows from the surface density by integration of the Poisson equation. See the text “Galactic Dynamics” by Binney & Tremaine for details. Lots and LOTS of essential details.

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19. You make some intriguing points:

1. “…galaxy disks are subject to a variety of instabilities that grow bars and spiral arms. These are driven by disk self-gravity.”

2. “…a purely Newtonian disk of stars is not a stable configuration, yet the universe is chock full of long-lived spiral galaxies.”

3. “In many dwarf galaxies, the rotation velocity exceeds that attributable to the baryons (with Newton alone) at practically all radii.”

4. “…this is entirely natural in MOND…where it is all disk self-gravity…”

Given 1-3, why is the Newtonian calculation for rotation curves employed at all. If Newton alone can’t calculate galactic disk structure, why should it be expected to accurately predict the rotation curves of those disks? Shouldn’t it be necessary to incorporate disk self-gravity into the calculations?

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20. Regarding point #4 in my previous comment:

Are you saying that MOND does incorporate disk self-gravity? I have never previously heard it said that MOND was derived based on any such a qualitative consideration.

BTW, another great post. If the DM zombie is ever finally laid to rest, your work will have played a significant part.

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21. Enrico says:

Ok now I use Gauss’s law and Gaussian surface for cylindrical mass distribution (disk is a flat cylinder) Gravitational field in a disk with mass M at radius r:
g = G M/r
Gravitational force on orbiting body with mass m:
F = G M m/r
Centripetal force on body with orbital velocity v:
Fc = m v^2/r
To stay in orbit:
Fc = F
m v^2/r =G M/r
v = (G M)^0.5

M = D V = D pi r^2 t
Where: D = mass density, V = volume inside radius, t = thickness of disk
v = (G D pi t)^0.5 r

Orbital velocity should increase with r. But if D and t decrease with r, they can compensate to yield a flat rotation curve for disk galaxies.

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22. Enrico says:

Resolution to the momentum paradox in MOND=CC hypothesis. The energy equation with quantum mechanical effect shows momentum of orbiting star has decreased by the scale factor but its mass and velocity remain the same. This constant mass pertains to rest mass but its relativistic mass must decrease. To invoke dark energy, I shift the reference frame from galactic center to cosmic horizon. The velocity of star with respect to cosmic horizon is very fast. This is the source of its relativistic momentum.

Assume distance to cosmic horizon = 46.5 Gly = 14,257 Mpc
which gives a scale factor = s = 46.5/13.8 = 3.37. Let Hubble constant = 71 km/s/Mpc
Velocity of star = v’ = 14,257 (71) = 10^6 km/s
Convert Newtonian velocity (v’) to relativistic velocity (v):
v’ = v L where L is Lorentz factor, L = 1/(1 – (v/c)^2)^0.5
Divide both sides of equation by c = speed of light:
v’/c = (v/c) L since v’/c = s = 3.37
Substituting values:
s = (v/c) 1/(1 – (v/c)^2)^0.5 = 3.37
Solving above equation gives:
v/c = 0.9587, L = 3.516
Hence, the relativistic momentum (p) of star:
p = m L v = m L (v/c) c where m = rest mass

I call this cosmological momentum because all celestial bodies in an expanding universe possess this relativistic momentum. There is a corresponding relativistic kinetic energy that I call cosmological kinetic energy. Next I will show the cosmological kinetic energy is equivalent to dark energy.

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1. JB says:

Is there a particularly novel idea that is required to understand this, or must we just accept that ultimately a body tends to go round and round in circles regardless of the perceived potential?

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23. I’m just back from giving a talk at the physics department at the University of Minnesota, so trying to catch up a little.

– Enrico: If you have a hypothesis, I suggest you write it up for publication as a cohesive paper in a scientific journal. The comments section of a blog is not an appropriate venue for this.

– bud rap: MOND is sourced from the visible baryonic matter, so by construction it is all disk self-gravity for baryons arranged in a disk like spiral galaxies. It has even been shown (independently by Brada & Milgrom and by Tiret & Combes) to provide about the right balance between stability and fragility, the latter allowing some spiral structure and bars even in low surface brightness disks without tearing them apart.
The deeper question you ask, “why is the Newtonian calculation for rotation curves employed at all?” has a simple answer: we assume that Newton holds in galaxies. It seems like a good assumption, but it is an assumption and nothing more. For “why should [Newton] be expected to accurately predict the rotation curves of those disks?” It doesn’t. We invoke dark matter because Newton plus the stuff we can see does not work. Once we invoke dark matter, predictive power virtually disappears. Whatever isn’t normal matter must be dark matter; it has proven very difficult to predict what we can’t see. But if that is indeed the way the universe works, it is exceedingly strange that MOND successfully predicts anything at all, let alone the many things it does correctly. It is a great frustration to me that so many professional scientists fail to grasp this simple fact and its obvious consequences.

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24. Dimitri says:

a.) “I spent a lot of time over many years trying to invent such stuff, without success.”

Are any of those unsuccessful tries published or uploaded somewhere?

I think it may be interesting for others to know what an established scientist has tried and why he thinks it doesn’t work.
(e.g. perhaps someone has a different idea and can take that work further…)

b.) “It is a great frustration to me that so many professional scientists fail to grasp this simple fact and its obvious consequences.”

Professionalism is like inertia when it comes to changing our mind 🙂

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25. I’d love to hear your comments on a new preprint that builds upon your 2016 paper. James G. O’Brien, et al., “Radial Acceleration and Tully-Fisher Relations in Conformal Gravity” (December 7, 2018). https://arxiv.org/abs/1812.03152

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26. I have not had time to review this paper. At a quick glance, it looks like they’ve included many galaxies that have duplicate observations to what is already in SPARC. So I’m not sure what value there is to add there.

As for my attempts to sort out what didn’t work, many remain unpublished. This is unfortunate, because there should be a record of what doesn’t work. Makes a lousy paper though, and it takes a looooong time to write such a thing. I did try to provide a broad overview of what all went wrong in https://xxx.lanl.gov/abs/astro-ph/9801123 and a shorter version at https://xxx.lanl.gov/abs/astro-ph/9812327. In many respects, the history of galaxy formation theory for the last 20 years has been pursuing in bitter detail ideas I rejected as obviously unworkable in these papers.

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27. Dimitri says:

Great, thanks a lot for the links!

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28. Ron Smith says:

Have you perhaps written, or are you perhaps aware of, any papers that do a review of various modified gravity theories to see if/how well, they account for MoND? For example TEVES, MOG, Vanlinden’s Emergent Gravity?

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29. There is an extensive review of relativistic theories in this context at https://arxiv.org/abs/1112.3960. Emergent gravity is too recently to have covered there; we made a first test at https://arxiv.org/abs/1702.04355 but I’d say the jury is still out on that. Indeed, it needs more theoretical development, even at the level of obtaining the appropriate solutions in the cylindrical geometry appropriate to disk galaxies.

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30. David Schroeder says:

There’s a recent paper by Jamie Farnes, proposing that negative-energy matter could be behind the Dark Sector. It’s being discussed over at Backreaction, and also on PhysicsForums. I have to admit I find it fascinating, and posted a few comments on Backreaction regarding it. https://arxiv.org/abs/1712.07962

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31. This paper by Farnes is earnest, well written, touches on many salient issues, and is obviously wrong. The explanation for rotation curves isn’t an explanation at all. It doesn’t even demonstrate an awareness of what needs to be explained. In that, it is far from unique.

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32. Dr McGaugh,

“MOND is sourced from the visible baryonic matter, so by construction it is all disk self-gravity for baryons arranged in a disk like spiral galaxies.”

Sure, MOND captures the results of disk self-gravity – by construction. The problem is that our analytical model which we use to calculate the expected rotation curves does not incorporate disk self-gravity.

“…we assume that Newton holds in galaxies.”

The math holds but the analytical structure, which only accounts for the centripetal force, is not adequate – by construction. Galactic structures are not similar to the solar system structure that underlies the Keplerian/Newtonian analytics and therefore that analytical structure is inadequate to calculating the expected rotation curves of galaxies. The fault is not in the galaxies (DM) or the math (MOND), it is in the analytics.

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33. budrap;

Here’s your chance to become famous: show us how to do this “correctly”.

sean s.

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34. This paper reports a very big outlier to MOND with an atypical seemingly very cuspy DM halo. I think that there are several ways it could be off and clues that there might be an issue. One is that the SMBH is independently estimate is 25 times larger relative to the total size of the galaxy than the one in the Milky Way. Another is that this is almost edge on and inclination may mess up the estimate. The distance from us estimate could also be wrong and the rotational velocity naively seems high. But, I’d welcome your thoughts about what is going on. https://arxiv.org/abs/1812.05918

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35. You are correct that the rotation curve looks odd. This is often a sign of some problem in the analysis.
NGC 4321 is very face on. This is problematic, as the uncertainty in the mass in MOND goes as the inverse fourth power of the sin of the inclination [1/sin(i)]^4. For this reason, people who do mass modeling generally avoid galaxies that are this face-on. From experience, even small inclination errors make a huge difference when the inclination is as small as it is here (see, e.g., the case of UGC 1230 in https://arxiv.org/abs/astro-ph/9805120).
Beyond that, they don’t build a mass model, so they don’t really test MOND. This makes it hard to assess what might be going on. They show the light profile, so they have the information necessary to do it, but they don’t do it. Instead, they restrict themselves to vague statements about the enclosed mass at some radii. That’s fine, but doesn’t test MOND. Their argument seems merely to be that the velocity is big – too big for their crude MOND estimate – so they need dark matter in the middle.
I have lost count of the number of times arguments like this fail on mild scrutiny. Extrapolating that experience, I expect this one will too. I have better things to do that fact check every half-baked arxiv posting.
Let us suppose, for argument, that they are correct, and that this one galaxy is an outlier from MOND. That means it is also an outlier from both the baryonic Tully-Fisher relation and the radial acceleration relation. That means that this object is also a contradiction to any dark matter-based explanation of those relations. So we can either explain those things with DM, or we can believe that this one galaxy falsifies MOND, but we can’t, in honesty, have it both ways.

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1. JB says:

Let’s assume matter-energy curves space-time, in a way not fully understood on large scales. Now we have a perceived discrepency in our understanding of their interaction based on long range observations of spatially and temporally localized EM frequency signals.

After much searching, there is no discovery of the hopeful dark matter in our local quantum field experiments.

Perhaps either the local experiments or the hopeful dark matter is not the right approach.

We still have energy, space and time to investigate – but I guess we don’t know how to test them?

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2. JB says:

Any references that clearly rule out the possiblity that a0 is related to a fundamental limit on the amount of information we can get about a discrete frequency signal, given the need to localize the signal in time and space relative to the other baryonic signal components?

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36. Right now there are discrepancies from standard GR+what we can see that crop up at a particular acceleration scale. (Large scales are not the right way to think about the problem, even though our brains seem to think first in terms of size. The problem is at small accelerations.) We don’t understand the quantum nature of space-time nor gravity nor the origin of inertial mass. So the first problem is conceptual. It is hard to conceive an experiment in the absence of a hypothesis to test. Which is to say, we need to make some theoretical advances more than we need to improve on current experiments. That’s always a good thing to do, and doing that has revealed these unexpected problems – problems to which all hypotheses to date fail pretty miserably, albeit in different ways.

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37. Perhaps a0 is related to some holographic principle. Bekenstein put some thought into that. I myself have no idea.

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38. JB says:

I was thinking more along the lines of Gabor’s uncertainty principle, where for any signal the product of the time uncertainty and the frequency uncertainty must be at least 1/(4*pi), and also perhaps in consideration of a decreasing signal population at lower accelerations, or some other parameter related to the baryonic distibution.

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39. Shantanu says:

Stacy, how uncertain is the baryonic mass profile (in spiral galaxies such as our own)? Briefly (in a sentence or two) how is it determined from observations? is it possible some baryons are hidden (do not emit/absorb light at any wavelengths)? Of course this would be inconsistent with BBN and CMB. But I am still curious whether a purely “baryonic dark matter” is viable on galactic scales.

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40. The baryonic mass profiles of spiral galaxies are determined from the azimuthally averaged run of starlight and 21cm emission (atomic gas). The largest uncertainty is in the stellar mass-to-light ratio. The luminosity profiles are well measured; converting light to mass is the trick. There is plenty of room in the BBN budget to allow for lots of baryonic DM in galaxies, as galaxies do not add up to much (see https://tritonstation.wordpress.com/2016/07/30/missing-baryons/ and https://tritonstation.wordpress.com/2016/09/09/what-is-the-baryon-density-anyway/ ). While there are certainly some baryons in extended coronas around galaxies, there is very little room for dark baryons in spiral disks themselves, as this would mess up the baryonic Tully-Fisher relation – see eq. 4 of https://xxx.lanl.gov/abs/astro-ph/0003001 .

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41. Enrico says:

RE: Holographic principle
You think invisible dimensions are more believable than invisible particles? (a.k.a. dark matter)

Dr. McGaugh,
What do you think of this new paper that claims observational evidence of dark matter heating in dwarf galaxies? Did they actually observe dark matter or just their fantasy?
https://www.surrey.ac.uk/news/dark-matter-move

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42. I think I commented on this previously. But in brief, they certainly didn’t observe dark matter. What they do have is a quasi-plausible way in which dark matter might be affected by star formation, and an apparent correlation (or at least dichotomy) between dwarfs that have suffered from this and dwarfs that have not: those with younger stellar populations tend to show better evidence for cores in their dark matter profiles, rather than the expected cusps, which remain a viable possibility in dwarfs with old stars.
From the perspective of standard dark matter, this is a decent step forward. From my perspective, it is a tiny step that doesn’t really address what I see to be the fundamental, underlying problem: that the distribution of baryons dictates that of the dark matter.

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43. Thanks, yes, I heard of it. I have not had the opportunity to check if this new case is consistent with MOND. Presumably the EFE applies, as you note.

At some point, one tires of fact-checking. Whether it works out or not in this particular case says nothing about all the cases where it already did. I’m more concerned with the forest than with the occasional outlying tree.

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44. One of the notable aspects of DF-4 is that even though it seems to be further in the field of view from NGC1052 than DF-2, it has lower velocity dispersion. Of course, it is hard to know what to make of that because we have a lot more precision orthogonal to the line of vision than we do in terms of depth, so we don’t know what is really closest and it isn’t easy to precisely model the EFE’s in the system for the same reason.

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