Missing baryons: LCDM and MOND compared

In the last few posts we’ve discussed the local missing baryon problem in extragalactic objects spanning over ten orders of magnitude in mass from tiny dwarfs to rich clusters of galaxies. This discussion has so far been entirely in the context of LCDM. So – how does LCDM compare with MOND?

As a refresher, these are the data we’re trying to understand:

The flat rotation speed V_f is an indicator of the dynamical mass – that of the dark matter halo and all the baryons it contains in LCDM, and that of all the (presumptively baryonic) mass in MOND. In LCDM, it would be satisfactory for the baryon fraction of each object, m_b = M_b/M₂₀₀, to be equal to the cosmic baryon fraction (f_b = 0.157 according to Planck). For MOND, what you see is supposed to be what you get, so the baryon fraction should be one.

As we saw previously, m_b = f_b for rich clusters of galaxies. There is no local missing baryon problem for galaxy clusters: a satisfactory result. However, as we look at smaller systems, observations depart from this ideal. They do so systematically, with our accounting of baryons falling progressively shorter of our expectation as we examine progressively lower mass objects. This deficit is illustrated by the gray region here:

Everything is fine for clusters at the high mass end (M_b > 10¹⁴ M_☉), and many people reasonably interpret that as corroboration of LCDM. For lower mass groups and bright galaxies, there is a deficit of a factor of two or three: an issue, but nothing too concerning by the standards of extragalactic astronomy, so this is widely ignored outside the community that works on it. The implicit assumption is that it’ll work out. But the magnitude of the problem continues to grow for smaller objects, becoming already an order of magnitude for intermediate mass galaxies. Not tiny dwarfs, just middle of the road spirals. The smallest mass dwarfs are worse off yet, missing over 90% of the baryons, approaching 98% or 99%. That is not satisfactory.

Making a straight-up comparison with MOND is a little tricky because the concept of a baryon fraction is a non-sequitor. There is no dark matter halo to compare against. Instead, we return to the concept of the velocity factor. In LCDM, we relate the observed flat rotation speed to that of the total dynamical mass through V_f = f_vV₂₀₀. Indeed, we can ask what velocity factor we need to explain away the missing baryon problem: maybe there are no missing baryons, just a systematic divergence of the observed V_f from the halo V₂₀₀. This can’t work, but it is useful to think about and provides a direct comparison with MOND.

In MOND, M_b = AV_f⁴ where A is the normalization^& of the BTFR. We can thus define an equivalent to the velocity factor, the residual velocity, taken here to be the ratio of the observed velocity to that expected for the observed mass, Δ_M = V_f,obs/V_f,pred. If the mass is a good predictor of the flat velocity, then Δ_M = 1. This leads to

This is a straight-up comparison between the theories. Both theories suffer a missing baryon problem, but at different scales. The magnitude of each problem is indicated by the area of the shaded regions. (There is a dearth of data in our study^* from 10¹³ < M_b < 10¹⁴ M_☉, so we’ll just ignore that here.)

LCDM is spot on for clusters over the range 10¹⁴ < M_b < 10¹⁵ M_☉: f_v = 1 suffices to explain the data. Outside of that range, f_v must increase systematically to make up for what we previously attributed to missing baryons. In effect, we’re making the dark matter halos smaller so that the baryon fraction works out. As noted before, this can’t work, as rotation curve fits restrict the viable range of the velocity factor to 1 < f_v < 1.4, but we need it to grow to f_v = 5. That’s silly: at that point, the dark matter halo is contributing so little to the observed dynamics that we wouldn’t infer its existence at all.

MOND is spot on over the range 5 x 10⁵ < M_b < 5 x 10¹² M_☉: the data are consistent with Δ_M = 1. It falls short for rich clusters, where the observed mass of baryons in the intracluster medium (ICM) and the stars in galaxies predicts only ~80% of the observed velocity. This is the residual mass discrepancy in MOND.

For perspective, it helps to plot the linear baryon fraction. The astronomical scales of astronomical data oblige us to use logarithmic scales in many circumstances, but this may lead one to under-appreciate the scale of the issue. So here is the baryon fraction again, in both LCDM and MOND, this time with a linear scale:

Individual galaxies and groups of galaxies are missing a lot of baryons in LCDM. This is not a subtle problem. It is not explained by simulations, nor am I aware of a satisfactory^% explanation. Worse, the apparent reason that we infer all these missing baryons is because the BTFR looks like the M_b ~ V_f⁴ of MOND rather than the M₂₀₀ ~ V₂₀₀³ of LCDM. With dark matter, we can accommodate pretty much any power law, or none at all – a lot of scatter would be more natural. So why did it have to be MOND? Even in ignorance of MOND the data pose a fine-tuning problem for LCDM. But it isn’t just a fine-tuning problem; it is a fine-tuning that arises because of MOND. To be successful, a LCDM model must be tuned to look like MOND. If it doesn’t, it’s wrong. If it does, why should we prefer a fine-tuned model to the theory that predicted the correct behavior in the first place?

MOND is not perfect here: it suffers a missing baryon problem in rich clusters. Since M_b ~ V_f⁴, predicting only ~80% of the observed velocity translates to missing ~60% of the mass. That’s a lot! But it could be worse: if, like Zwicky, we had done this experiment before the advent of X-ray observatories, we would be unaware of the mass of gas in the ICM, and infer that MOND was missing practically all (~96%!) the mass. That would seem utterly ridiculous, and we would conclude that MOND is wrong when much of the problem would have been that we were missing an important reservoir of baryons. Perhaps we still are. I do not like this possibility – there is still a lot of ground to make up, and I am not aware of a satisfactory solution. I guess I’m just a skeptic that way.

If we think the residual mass discrepancy problem MOND suffers in rich clusters is serious and perhaps fatal, should we not also conclude the same from the local missing baryon problem in LCDM?

But the bullet cluster double-secret falsifies MOND!

Let’s examine that assertion in the context of what we learned above.

The bullet cluster is composed of two clusters that collided and passed through one another. The collision segregated the gas of the ICM (pink above) from the galaxies. This happens because gas is diffuse and collisional. The gas of the two clusters can’t help smacking into each other, slowing down and forming the shock front visible in the shape of the gas of the smaller cluster on the right. Galaxies, on the other hand, have lots of empty space between them. They are collisionless and pass right by each other. In doing so, they are slowed less than the gas, getting ahead of it, leading to the separation that we observe.

OK, cool. The argument one usually hears against MOND based on this is that the baryonic mass in gas outweighs that in galaxies, so the lensing signal should be centered on the gas: the blue should align with the pink, not with the galaxies. Instead, we see the opposite, so the mass has to be dark matter.

This would be a good argument if the gas were all of the baryonic mass. This is a common assumption that makes sense in LCDM, where the baryon fraction checks out, so most people seem to stop thinking at that point. But each theory needs to be considered in its own context, and it cannot be the case in pure^# MOND that we see all the baryons^## in the picture above. That’s what we learned above. It may be unsatisfactory, but we knew this already before the bullet cluster was discovered (e.g., Sanders & McGaugh 2002). So the only new thing we learn from this aspect of the bullet cluster is that if there is an additional reservoir of baryonic mass, it is collisionless. It didn’t collide like the gas, it passed through like the galaxies. There are lots of candidate baryonic objects that fit that requirement: brown dwarfs, neutron stars, black holes, very small rocks^{^}. There is no requirement that the unseen mass be non-baryonic; we do not need the new physics of a new dark matter particle from beyond the Standard Model of particle physics on top of the new physics of MOND.

Now, as I think I’ve made clear, I am very uncomfortable with the apparent requirement that there is lots of undetected baryonic mass in clusters. If I were the MOND partisan that lots of people seem to assume I am, then I guess I’d portray this as a bold prediction. The dark baryons have to be there, and we should be turning all possible resources to detecting them, rather like we have for WIMPs. But I’m not that person. I am also not a person who sees this missing baryon problem for MOND as automatically worse than the missing baryon problem for LCDM. There is a much bigger deficit to be made up in LCDM, in many more systems^### of very different types over a larger dynamic range in mass. The missing baryon problem in LCDM looks worse to me than that in MOND. Yet the community attitude seems to be largely unaware of it. Those who are seem mostly to presume that it’ll work out. Maybe, but this should not be accepted by assumption, it needs to be demonstrated. It has yet to be.

If you think the missing baryon problem in clusters is a terrible problem for MOND, then you should be similarly worried that LCDM evinces the same kind of problem – one that is objectively larger in amplitude. It seems that, having accepted that there is dark matter, people don’t much care what it is. I do. In effect, LCDM requires two kinds of dark matter: dark baryons mixed in with each and every dark matter halo, and some entirely novel form of particle to be the dark matter halo. These are different things, so the dark matter hypothesis is not as parsimonious as simply saying “dark matter” sounds.

There is the communal failure of objectivity about this. The thought process is both transparent and simple: MOND doesn’t explain clusters; it requires dark matter. Therefore dark matter^#### exists and it is silly to think about MOND. That would make sense if it weren’t a logical fallacy. Instead, it provides a permission structure to remain ignorant of what MOND gets right. I get that; there’s a lot to know. But I would also suggest that ignorance does not strong provide a strong basis for drawing scientific conclusions, especially for a subject so rife with confirmation bias and cognitive dissonance.

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^&The normalization is related to Newton’s constant and Milgrom’s constant through A = ζ/(a₀G) where ζ is a factor of order unity that depends on the geometry of the system. It is one for spheres, and always approaches the limit ζ → 1 at sufficiently large radii, but observations are usually obtained at radii where the flattened geometry of disk galaxies is relevant, so in practice ζ ≈ 0.8. This can be derived from the geometry (all purely conventional; nothing to do with MOND) or one can obtain it empirically by comparing A = 50 M_☉ km^-4 s⁴ from fitting the BTFR to data for galaxies with known a₀; for a₀ = 1.2 x 10^-10 m s^-2, (a₀G)^-1 = 63 M_☉ km^-4 s⁴, so ζ =A(a₀G) = 50/63 = 0.8.

^*There remains room for improvement for poor clusters (here I call 10¹³ < M_b < 10¹⁴ M_☉ objects “poor clusters” because astronomical terminology can always be made worse). A particular issue is the quantity of intracluster gas, which dominates rich clusters (and is readily detected in X-rays), but seems to be absent in the smallest groups. There has to be a transition in between, but is it smooth so that all poor clusters have the same amount, or is there a huge variation in ICM mass among poor clusters? I have seen anecdotal indications that poor clusters that are detected in X-rays extend the trend of rich clusters while those that aren’t don’t, as if the residual mass discrepancy MOND evinces in clusters is somehow related to the presence of X-ray gas.

^%There are lots of unsatisfactory explanations. Some sound more plausible than others, but all fail to engage with the underlying prompt: why do the data look like MOND if we live in a universe made of dark matter?

^#It is possible that the problem MOND faces in clusters might not be one of missing mass, but rather it could be an indication of a deeper theory that is not exactly like pure MOND.

^##If there is additional mass in clusters, it doesn’t necessarily have to be baryonic. It could, in part, be neutrinos or sterile neutrinos or other more exotic beasts of the unknown meagerie of our enormous universe. However, there is no requirement that the unseen mass be anything other than mundane, ordinary matter.

^{^}Though an amusing thought, very small rocks do not make a viable candidate dark matter object any more than witches float because they weigh the same as a duck.

^###I have heard otherwise brilliant scientists dismiss the successes of MOND as a fluke. MOND has made too many successful predictions for that to be a reasonable assertion; it is a good example of what Putnam meant by “no miracles.” Yet the same scientists will cite the consistency of the baryon fraction in clusters to the cosmic baryon fraction as something that cannot be a fluke, ergo LCDM must be right. So which fluke is worse? I do not have patience to list all of MOND’s successful predictions here, though there are many reviews that do so and there will be a long paper soon that does more. What I will note here, having just done the exercise, is that the cluster baryon fraction is more likely to be a fluke. In order to estimate a baryonic mass for each cluster, we extrapolate the so-called beta profile that describes the distribution of X-ray gas. That’s a reasonable thing to do, and when we do it, we get an answer that is satisfactory in LCDM. However, it is not a small extrapolation. We are inferring a lot of baryonic mass at large radii from the fit of the beta profile at smaller radii. That’s the obvious thing to do, and I think it is probably correct, but it is also something that could go badly wrong. We experimented with other plausible gas mass profiles, and the answer can vary a lot, often leading to considerably fewer baryons than the cosmic fraction. That would be bad for LCDM, and also make the problem MOND suffers (too few baryons) worse, so it doesn’t help anything. But if there is a fluke here, it is more likely to be the coincidence of the cluster baryon fraction with the cosmic baryon fraction than is the consistency of the observed BTFR with the prediction of MOND for most of the rest of the universe.

^####This is where sloppy terminology leads to a logical fallacy: people equate “dark matter” with non-baryonic cold dark matter. The latter is a subset of the former; the unseen mass in MOND need not be the same as the non-baryonic stuff that we commonly assume the dark matter is.