The Fat One – a test of structure formation with the most massive cluster of galaxies

A common objection to MOND is that it does not entirely reconcile the mass discrepancy in clusters of galaxies. This can be seen as an offset in the acceleration scale between individual galaxies and clusters. This is widely seen as definitive proof of dark matter, but this is just defaulting to our confirmation bias without checking if it is really any better: just because MOND does something wrong doesn’t automatically mean that LCDM does it right.

The characteristic acceleration (in units of Milgrom’s constant a₀) of extragalactic objects as a function of their baryonic mass, ranging from tiny dwarf galaxies to giant clusters of galaxies. Clusters are offset from individual galaxies, implying a residual missing mass problem for MOND. From Famaey & McGaugh (2012).

I do see clusters as a problem for MOND, and there are some aspects of clusters that make good sense in LCDM. Unlike galaxies, cluster mass profiles are generally consistent with the predicted NFW halos (modulo their own core problem). That’s not a contradiction to MOND, which should do the same thing as Newton in the Newtonian regime. But rich clusters also have baryon fractions close to that expected from cosmology. From that perspective, it looks pretty reasonable. This success does not extend to lower mass clusters; in the plot above, the low mass green triangles should be higher than the higher mass gray triangles in order for all clusters to have the cosmic baryon fraction. They should not parallel the prediction of MOND. Within individual clusters, baryons are not as well mixed with dark matter as expected: they tend to have too much unseen mass at small radius, which is basically the same problem encountered by MOND.

There are other tests, one of which is the growth of clusters. Structure is predicted to form hierarchically in LCDM: small objects form first, and pile on to make bigger ones, with the largest clusters being the last to form. So there is a test in how massive a cluster can get as a function of redshift. This is something for which LCDM makes a clear prediction. In MOND, my expectation is that structure forms faster so that massive objects are in place at higher redshift than expected in LCDM. This post is mostly about clusters in LCDM, so henceforth all masses will be conventional masses, including the putative dark matter.

Like so many things, there is a long history to this. For example, in the late ’90s, Megan Donahue reported a high temperature of ~ 12 keV for the intracluster gas in the cluster MS1054-0321. This meant that it was massive for its redshift: 7.4 x 10¹⁴ h^-1 M_☉ (dark matter and all) at z = 0.829, when the universe was only about half its current age. (Little h is the Hubble constant in units of 100 km/s/Mpc. Since we’re now pretty sure h < 1, the true mass is higher, more like 10¹⁵ M_☉.) That’s a lot of solar masses to assemble in the available time. In 1997, this was another nail in the coffin of SCDM, which was already a zombie theory by then. But the loss of Ω_m = 1 was still raw for some people, I guess, because she got a lot of grief for it. Can’t be true! Clusters don’t get that big that early! At least they shouldn’t. In SCDM.

Structure formation in SCDM was elegant in that in continues perpetually: as the universe expands, bigger and bigger structures continue to form; statistically, later epochs look like scaled-up versions of earlier epochs. In LCDM, this symmetry is broken by the decline in density as the universe expands. Consequently, structure forms earlier in LCDM: the action has to happen when there is still some density to work with, and the accelerated expansion provides some extra time (what’s a few billion years among cosmologists?) for mass to get together. Consequently, MS1054-0321 is not problematic in LCDM.

The attitude persisted, however. In the mid-’00s, Jim Schombert and I started using the wide field near-IR camera NEWFIRM to study high redshift clusters. Jim had a clever way of identifying them, which turned out not to be particularly hard, e.g., MS 1426.9+1052 at z = 1.83. This is about 10 Gyr ago, and made the theorists squirm. That didn’t leave enough time for a cluster to form. On multiple occasions I had the following conversation with different theorists:

me: Hey, look at this clusters at z = 1.8.
theorist: That isn’t a cluster.
me: Sure it is. There’s the central galaxy, which contains a bright radio source (QSO). You can see lots of other galaxies around it. That’s what a cluster looks like.
theorist: Must be a chance projection.
me: There are spectra for many of the surrounding galaxies; they’re all at the same redshift.
theorist: …
me: So… a cluster at z = 1.8. Pretty cool, huh?
theorist: That isn’t a cluster.

This work became part of Jay Frank’s thesis. He found evidence for more structure at even higher redshift. A lot of this apparent clustering probably is not real… the statistics get worse as you push farther out: fewer galaxies, worse data. But there were still a surprising number of objects in apparent association up to and beyond z > 5. That’s pretty much all of time, leaving a mere Gyr to go from the completely homogeneous universe that we see in the CMB at z = 1090 to the first stars around z ~ 20 to the first galaxies to big galaxies to protoclusters – or whatever we want to call these associations of many galaxies in the same place on the sky at the same redshift.

Jay did a lot of work to estimate the rate of false positives. Long story short, we expect about 1/3 of the protoclusters he identified to be real structures. That’s both bad and good – lots of chaff, but some wheat too. One thing Jay did was to analyze the Millennium simulation in the same way as the data. This allows us to quantify what we would see if the universe looked like an LCDM simulation.

The plot below shows the characteristic brightness of galaxies at various redshifts. For the pros, this is the knee in the Schechter function fit to the luminosity distribution of galaxies in redshift bins. We saw the same thing in protoclusters and in the field: galaxies were brighter than anticipated in the simulation. Between redshifts 3 < z < 4, the characteristic magnitude is expected to be 23. That’s pretty faint. In the data, it’s more like 21. That’s also faint, but about a factor of 6 brighter than they should be. That’s a lot of stars that have formed before they’re supposed to, in galaxies that are bigger than they should yet be, with some of them already clustering together ahead of their time.

The characteristic magnitude of galaxies in the Spitzer 4.5 micron band as a function of redshift in the Millennium simulation (black squares) and in reality (circles). This is a classic backwards astronomical plot in which larger magnitudes are fainter sources. At high redshift, simulations predict that galaxies should not yet have grown to become as bright as they are observed to be. From Franck (2017).

This has been the observer’s experience. Donahue wasn’t the first, and Franck won’t be the last. Every time we look, we see more structure in place sooner than had been expected before it was seen. I don’t hear people complaining about our clusters at z = 1.8 anymore; those have been seen enough to become normalized. Perhaps they have even been explained satisfactorily. But they sure weren’t expected, much less predicted.

So, just how big can a cluster get? Mortonson et al. (2011) set out to answer this question. The graph below shows the upper limit they predict for the most massive cluster in the universe as a function of redshift. This declines as redshift increases because we’re looking back in time; high redshift clusters haven’t had time time to assemble more mass than the upper most line. They project this into what would be discovered in an all-sky survey, and more realistic surveys of finite size. Basically nothing should exist above these lines.

The predicted maximum mass of galaxy clusters as a function of redshift from Mortonson et al. (2011). Each line is the predicted upper limit for the corresponding amount of sky surveyed. The green line illustrates the area of the sky in which El Gordo was discovered. The points show independent mass estimates for El Gordo from Menanteau et al. (2012) and Jee et al. (2014). These are significantly above the predicted upper limit.

Their prediction was almost immediately put to the test by the discovery of El Gordo, a big fat cluster at z = 0.87 reported by Menanteau et al. (2012), who published the X-ray image above. It is currently the record holder for the most massive known object that is thought to be gravitationally bound, weighing in at 2 or 3 x 10¹⁵ M_☉, depending on who you ask. That’s about a thousand Milky Ways, plus a few hundred Andromedas. Give or take.

El Gordo straddles the uppermost line in the graph above. A naive reading of the first mass estimate suggests that it’s roughly a 50/50 proposition whether the entire observable universe should contain exactly one El Gordo. However, El Gordo was discovered in something less than a full sky survey. The appropriate comparison is to the green line, which El Gordo clearly exceeds – by about 3 sigma. This is the case for both of the illustrated mass estimates as the high mass point has a larger error bar. They both exceed the green line by a hair less than 3 sigma. Formally, this means that the chance of finding El Gordo in our universe is only a few percent.

A few percent is not good. Neither is it terrible – I’ve often commented here on how the uncertainties are larger than they seem. This is especially true of the tails of the distribution. So maybe a few percent is pessimistic; sometimes that’s how the dice roll. On the other hand, the odds aren’t better than 10%: El Gordo is not likely to exist however we slice the uncertainties. Whether we should be worried about it is just a matter of how surprising it is. A similar situation arises with the collision velocity of the Bullet cluster, which is either absurdly unlikely (about 1 chance in 10 billion) or merely unusual (maybe 1 in 10). So I made the above plot by adding El Gordo to the predictions of Mortonson et al., and filed it away under

Recently, Elena Asencio, Indranil Banik, and Pavel Kroupa have made a more thorough study. They have their own blog post, so I won’t repeat the technical description. Basically, they sift through a really big LCDM simulation to find objects that could be (or become) like El Gordo.

The short answer is that it doesn’t happen, similar to big voids. They estimate that the odds of El Gordo existing are a bit less than one in a billion. I’m sure one can quibble with details, but we’re not going to save LCDM with factors of two in a probability that starts this low. El Gordo just shouldn’t exist.

The probability is lower than in the graph above because it isn’t just a matter of mass. It is also the mass ratio of the merging clumps (both huge clusters in their own right), their collision speed, impact parameter, and morphology. As they are aware, one must be careful not to demand a perfect match, since there is only one reality. But neither is it just a matter of assembling mass; that understates the severity of the problem. This is where simulations are genuinely helpful: one can ask how often does this happen? If the answer is never, one can refine the query to be more lenient. The bottom line here is that you can’t be lenient enough to get something like El Gordo.

Here is their money plot. To be like El Gordo, an object would have to be up on the red line. That’s well above 5 sigma, which is the threshold where we traditionally stop quibbling about percentiles and just say Nope. Not an accident.

Logarithmic mass as a function of expansion factor [how big the universe is. This is inversely related to redshift: a = 1/(1+z)]. The color scale gives the number density of objects of a given mass as a function of how far the universe has expanded. The solid lines show the corresponding odds (in sigma) of finding such a thing in a large LCDM simulation. Figure from Asencio et al (2020).

In principle, this one object falsifies the LCDM structure formation paradigm. We are reluctant to put too much emphasis on a single object (unless it is the bullet cluster and we have clickbait to sell) as its a big universe, so there can always be one unicorn or magnetic monopole somewhere. Ascencio et al note that a similar constraint follows for the Bullet cluster itself, which also should not exist, albeit at a lower significance. That’s two unicorns: we can’t pretend that this is a one-off occurrence. The joint probability of living in a universe with both El Gordo and the Bullet cluster is even lower than either alone.

Looking at Ascencio’s figure, it strikes me as odd not only that we find huge things at high redshift, but also that we don’t see still bigger objects at low redshift. There were already these huge clusters ramming into each other when the universe had only expanded to half its present size. This process should continue to build still bigger clusters, as indicated by the lines in the plot. The sweet spot for finding really massive clusters should be about z = 0.5, by which time they could have reached a mass of nearly 10¹⁶ M_☉ as readily (or not!) as El Gordo could reach its mass by its observed redshift. (The lines turn down for the largest expansion factors/lowest redshifts because surveys cover a fixed area on the sky, which is a conical volume in 3D. We reside at the point of the cone, and need to see a ways out before a volume large enough to contain a giant cluster has been covered.)

I have never heard a report of a cluster anywhere near to 10¹⁶ M_☉. A big cluster is 10¹⁵ M_☉. While multiple examples of clusters this big are known, to the best of my knowledge, El Gordo is the record holding Fat One at twice or thrice that. The nearest challenger I can readily find is RX J1347.5-1145 at z=0.451 (close to the survey sweet spot) weighing in at 2 x 10¹⁵ M_☉. Clusters just don’t seem to get bigger than that. This mass is OK at low redshift, but at higher z we shouldn’t see things as big as El Gordo. Given that we do see them at z = 0.87 (a = 0.535), why don’t we see still bigger ones at lower redshift? Perhaps structure formation saturates, but that’s not what LCDM predicts. If we can somehow explain El Gordo at high z, we are implicitly predicting still bigger clusters at lower redshift – objects we have yet to discover, if they exist, which they shouldn’t.

Which is the point.

The image featured at top is an X-ray image of the hot gas in the intracluster medium of El Gordo from NASA/CXC/Rutgers/J. Hughes et al.

Big Trouble in a Deep Void

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

Modifying gravity to save cosmology

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Authors

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

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

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

Link to the published science paper.

YouTube video on the paper

Contact: ibanik@astro.uni-bonn.de.

Indranil Banik’s YouTube channel.

Cosmology, then and now

I have been busy teaching cosmology this semester. When I started on the faculty of the University of Maryland in 1998, there was no advanced course on the subject. This seemed like an obvious hole to fill, so I developed one. I remember with fond bemusement the senior faculty, many of them planetary scientists, sending Mike A’Hearn as a stately ambassador to politely inquire if cosmology had evolved beyond a dodgy subject and was now rigorous enough to be worthy of a 3 credit graduate course.

Back then, we used transparencies or wrote on the board. It was novel to have a course web page. I still have those notes, and marvel at the breadth and depth of work performed by my younger self. Now that I’m teaching it for the first time in a decade, I find it challenging to keep up. Everything has to be adapted to an electronic format, and be delivered remotely during this damnable pandemic. It is a less satisfactory experience, and it has precluded posting much here.

Another thing I notice is that attitudes have evolved along with the subject. The baseline cosmology, LCDM, has not changed much. We’ve tilted the power spectrum and spiked it with extra baryons, but the basic picture is that which emerged from the application of classical observational cosmology – measurements of the Hubble constant, the mass density, the ages of the oldest stars, the abundances of the light elements, number counts of faint galaxies, and a wealth of other observational constraints built up over decades of effort. Here is an example of combining such constraints, and exercise I have students do every time I teach the course:

Observational constraints in the mass density-Hubble constant plane assembled by students in my cosmology course in 2002. The gray area is excluded. The open window is the only space allowed; this is LCDM. The box represents the first WMAP estimate in 2003. CMB estimates have subsequently migrated out of the allowed region to lower H0 and higher mass density, but the other constraints have not changed much, most famously H0, which remains entrenched in the low to mid-70s.

These things were known by the mid-90s. Nowadays, people seem to think Type Ia SN discovered Lambda, when really they were just icing on a cake that was already baked. The location of the first peak in the acoustic power spectrum of the microwave background was corroborative of the flat geometry required by the picture that had developed, but trailed the development of LCDM rather than informing its construction. But students entering the field now seem to have been given the impression that these were the only observations that mattered.

Worse, they seem to think these things are Known, as if there’s never been a time that we cosmologists have been sure about something only to find later that we had it quite wrong. This attitude is deleterious to the progress of science, as it precludes us from seeing important clues when they fail to conform to our preconceptions. To give one recent example, everyone seems to have decided that the EDGES observation of 21 cm absorption during the dark ages is wrong. The reason? Because it is impossible in LCDM. There are technical reasons why it might be wrong, but these are subsidiary to Attitude: we can’t believe it’s true, so we don’t. But that’s what makes a result important: something that makes us reexamine how we perceive the universe. If we’re unwilling to do that, we’re no longer doing science.

A Significant Theoretical Advance

The missing mass problem has been with us many decades now. Going on a century if you start counting from the work of Oort and Zwicky in the 1930s. Not quite a half a century if we date it from the 1970s when most of the relevant scientific community started to take it seriously. Either way, that’s a very long time for a major problem to go unsolved in physics. The quantum revolution that overturned our classical view of physics was lightning fast in comparison – see the discussion of Bohr’s theory in the foundation of quantum mechanics in David Merritt’s new book.

To this day, despite tremendous efforts, we have yet to obtain a confirmed laboratory detection of a viable dark matter particle – or even a hint of persuasive evidence for the physics beyond the Standard Model of Particle Physics (e.g., supersymmetry) that would be required to enable the existence of such particles. We cannot credibly claim (as many of my colleagues insist they can) to know that such invisible mass exists. All we really know is that there is a discrepancy between what we see and what we get: the universe and the galaxies within it cannot be explained by General Relativity and the known stable of Standard Model particles.

If we assume that General Relativity is both correct and sufficient to explain the universe, which seems like a very excellent assumption, then we are indeed obliged to invoke non-baryonic dark matter. The amount of astronomical evidence that points in this direction is overwhelming. That is how we got to where we are today: once we make the obvious, imminently well-motivated assumption, then we are forced along a path in which we become convinced of the reality of the dark matter, not merely as a hypothetical convenience to cosmological calculations, but as an essential part of physical reality.

I think that the assumption that General Relativity is correct is indeed an excellent one. It has repeatedly passed many experimental and observational tests too numerous to elaborate here. However, I have come to doubt the assumption that it suffices to explain the universe. The only data that test it on scales where the missing mass problem arises is the data from which we infer the existence of dark matter. Which we do by assuming that General Relativity holds. The opportunity for circular reasoning is apparent – and frequently indulged.

It should not come as a shock that General Relativity might not be completely sufficient as a theory in all circumstances. This is exactly the motivation for and the working presumption of quantum theories of gravity. That nothing to do with cosmology will be affected along the road to quantum gravity is just another assumption.

I expect that some of my colleagues will struggle to wrap their heads around what I just wrote. I sure did. It was the hardest thing I ever did in science to accept that I might be wrong to be so sure it had to be dark matter – because I was sure it was. As sure of it as any of the folks who remain sure of it now. So imagine my shock when we obtained data that made no sense in terms of dark matter, but had been predicted in advance by a completely different theory, MOND.

When comparing dark matter and MOND, one must weigh all evidence in the balance. Much of the evidence is gratuitously ambiguous, so the conclusion to which one comes depends on how one weighs the more definitive lines of evidence. Some of this points very clearly to MOND, while other evidence prefers non-baryonic dark matter. One of the most important lines of evidence in favor of dark matter is the acoustic power spectrum of the cosmic microwave background (CMB) – the pattern of minute temperature fluctuations in the relic radiation field imprinted on the sky a few hundred thousand years after the Big Bang.

The equations that govern the acoustic power spectrum require General Relativity, but thankfully the small amplitude of the temperature variations permits them to be solved in the limit of linear perturbation theory. So posed, they can be written as a damped and driven oscillator. The power spectrum favors features corresponding to standing waves at the epoch of recombination when the universe transitioned rather abruptly from an opaque plasma to a transparent neutral gas. The edge of a cloud provides an analog: light inside the cloud scatters off the water molecules and doesn’t get very far: the cloud is opaque. Any light that makes it to the edge of the cloud meets no further resistance, and is free to travel to our eyes – which is how we perceive the edge of the cloud. The CMB is the expansion-redshifted edge of the plasma cloud of the early universe.

An easy way to think about a damped and a driven oscillator is a kid being pushed on a swing. The parent pushing the child is a driver of the oscillation. Any resistance – like the child dragging his feet – damps the oscillation. Normal matter (baryons) damps the oscillations – it acts as a net drag force on the photon fluid whose oscillations we observe. If there is nothing going on but General Relativity plus normal baryons, we should see a purely damped pattern of oscillations in which each peak is smaller than the one before it, as seen in the solid line here:

CMB_Pl_CLonly — The CMB acoustic power spectrum predicted by General Relativity with no cold dark matter (line) and as observed by the Planck satellite (data points).

As one can see, the case of no Cold Dark Matter (CDM) does well to explain the amplitudes of the first two peaks. Indeed, it was the only hypothesis to successfully predict this aspect of the data in advance of its observation. The small amplitude of the second peak came as a great surprise from the perspective of LCDM. However, without CDM, there is only baryonic damping. Each peak should have a progressively lower amplitude. This is not observed. Instead, the third peak is almost the same amplitude as the second, and clearly higher than expected in the pure damping scenario of no-CDM.

CDM provides a net driving force in the oscillation equations. It acts like the parent pushing the kid. Even though the kid drags his feet, the parent keeps pushing, and the amplitude of the oscillation is maintained. For the third peak at any rate. The baryons are an intransigent child and keep dragging their feet; eventually they win and the power spectrum damps away on progressively finer angular scales (large 𝓁 in the plot).

As I wrote in this review, the excess amplitude of the third peak over the no-CDM prediction is the best evidence to my mind in favor of the existence of non-baryonic CDM. Indeed, this observation is routinely cited by many cosmologists to absolutely require dark matter. It is argued that the observed power spectrum is impossible without it. The corollary is that any problem the dark matter picture encounters is a mere puzzle. It cannot be an anomaly because the CMB tells us that CDM has to exist.

Impossible is a high standard. I hope the reader can see the flaw in this line of reasoning. It is the same as above. In order to compute the oscillation power spectrum, we have assumed General Relativity. While not replacing it, the persistent predictive successes of a theory like MOND implies the existence of a more general theory. We do not know that such a theory cannot explain the CMB until we develop said theory and work out its predictions.

That said, it is a tall order. One needs a theory that provides a significant driving term without a large amount of excess invisible mass. Something has to push the swing in a universe full of stuff that only drags its feet. That does seem nigh on impossible. Or so I thought until I heard a talk by Pedro Ferreira where he showed how the scalar field in TeVeS – the relativistic MONDian theory proposed by Bekenstein – might play the same role as CDM. However, he and his collaborators soon showed that the desired effect was indeed impossible, at least in TeVeS: one could not simultaneously fit the third peak and the data preceding the first. This was nevertheless an important theoretical development, as it showed how it was possible, at least in principle, to affect the peak ratios without massive amounts of non-baryonic CDM.

At this juncture, there are two options. One is to seek a theory that might work, and develop it to the point where it can be tested. This is a lot of hard work that is bound to lead one down many blind alleys without promise of ultimate success. The much easier option is to assume that it cannot be done. This is the option adopted by most cosmologists, who have spent the last 15 years arguing that the CMB power spectrum requires the existence of CDM. Some even seem to consider it to be a detection thereof, in which case we might wonder why we bother with all those expensive underground experiments to detect the stuff.

Rather fewer people have invested in the approach that requires hard work. There are a few brave souls who have tried it; these include Constantinos Skordis and Tom Złosnik. Very recently, the have shown a version of a relativistic MOND theory (which they call RelMOND) that does fit the CMB power spectrum. Here is the plot from their paper:

Note that black line in their plot is the fit of the LCDM model to the Planck power spectrum data. Their theory does the same thing, so it necessarily fits the data as well. Indeed, a good fit appears to follow for a range of parameters. This is important, because it implies that little or no fine-tuning is needed: this is just what happens. That is arguably better than the case for LCDM, in which the fit is very fine-tuned. Indeed, that was a large point of making the measurement, as it requires a very specific set of parameters in order to work. It also leads to tensions with independent measurements of the Hubble constant, the baryon density, and the amplitude of the matter power spectrum at low redshift.

As with any good science result, this one raises a host of questions. It will take time to explore these. But this in itself is a momentous result. Irrespective if RelMOND is the right theory or, like TeVeS, just a step on a longer path, it shows that the impossible is in fact possible. The argument that I have heard repeated by cosmologists ad nauseam like a rosary prayer, that dark matter is the only conceivable way to explain the CMB power spectrum, is simply WRONG.

The Hubble Constant from the Baryonic Tully-Fisher Relation

The distance scale is fundamental to cosmology. How big is the universe? is pretty much the first question we ask when we look at the Big Picture.

The primary yardstick we use to describe the scale of the universe is Hubble’s constant: the H₀ in

v = H₀ D

that relates the recession velocity (redshift) of a galaxy to its distance. More generally, this is the current expansion rate of the universe. Pick up any book on cosmology and you will find a lengthy disquisition on the importance of this fundamental parameter that encapsulates the size, age, critical density, and potential fate of the cosmos. It is the first of the Big Two numbers in cosmology that expresses the still-amazing fact that the entire universe is expanding.

Quantifying the distance scale is hard. Throughout my career, I have avoided working on it. There are quite enough, er, personalities on the case already.

No need for me to add to the madness.

Not that I couldn’t. The Tully-Fisher relation has long been used as a distance indicator. It played an important role in breaking the stranglehold that H₀ = 50 km/s/Mpc had on the minds of cosmologists, including myself. Tully & Fisher (1977) found that it was approximately 80 km/s/Mpc. Their method continues to provide strong constraints to this day: Kourkchi et al. find H₀ = 76.0 ± 1.1(stat) ± 2.3(sys) km s^-1 Mpc^-1. So I’ve been happy to stay out of it.

Until now.

I am motivated in part by the calibration opportunity provided by gas rich galaxies, in part by the fact that tension in independent approaches to constrain the Hubble constant only seems to be getting worse, and in part by a recent conference experience. (Remember when we traveled?) Less than a year ago, I was at a cosmology conference in which I heard an all-too-typical talk that asserted that the Planck H₀ = 67.4 ± 0.5 km/s/Mpc had to be correct and everybody who got something different was a stupid-head. I’ve seen this movie before. It is the same community (often the very same people) who once insisted that H₀ had to be 50, dammit. They’re every bit as overconfident as before, suffering just as much from confirmation bias (LCDM! LCDM! LCDM!), and seem every bit as likely to be correct this time around.

So, is it true? We have the data, we’ve just refrained from using it in this particular way because other people were on the case. Let’s check.

The big hassle here is not measuring H₀ so much as quantifying the uncertainties. That’s the part that’s really hard. So all credit goes to Jim Schombert, who rolled up his proverbial sleeves and did all the hard work. Federico Lelli and I mostly just played the mother-of-all-jerks referees (I’ve had plenty of role models) by asking about every annoying detail. To make a very long story short, none of the items under our control matter at a level we care about, each making < 1 km/s/Mpc difference to the final answer.

In principle, the Baryonic Tully-Fisher relation (BTFR) helps over the usual luminosity-based version by including the gas, which extends application of the relation to lower mass galaxies that can be quite gas rich. Ignoring this component results in a mess that can only be avoided by restricting attention to bright galaxies. But including it introduces an extra parameter. One has to adopt a stellar mass-to-light ratio to put the stars and the gas on the same footing. I always figured that would make things worse – and for a long time, it did. That is no longer the case. So long as we treat the calibration sample that defines the BTFR and the sample used to measure the Hubble constant self-consistently, plausible choices for the mass-to-light ratio return the same answer for H₀. It’s all relative – the calibration changes with different choices, but the application to more distant galaxies changes in the same way. Same for the treatment of molecular gas and metallicity. It all comes out in the wash. Our relative distance scale is very precise. Putting an absolute number on it simply requires a lot of calibrating galaxies with accurate, independently measured distances.

Here is the absolute calibration of the BTFR that we obtain:

btf_cep_trgb — The Baryonic Tully-Fisher relation calibrated with 50 galaxies with direct distance determinations from either the Tip of the Red Giant Branch method (23) or Cepheids (27).

In constructing this calibrated BTFR, we have relied on distance measurements made or compiled by the Extragalactic Distance Database, which represents the cumulative efforts of Tully and many others to map out the local universe in great detail. We have also benefited from the work of Ponomareva et al, which provides new calibrator galaxies not already in our SPARC sample. Critically, they also measure the flat velocity from rotation curves, which is a huge improvement in accuracy over the more readily available linewidths commonly employed in Tully-Fisher work, but is expensive to obtain so remains the primary observational limitation on this procedure.

Still, we’re in pretty good shape. We now have 50 galaxies with well measured distances as well as the necessary ingredients to construct the BTFR: extended, resolved rotation curves, HI fluxes to measure the gas mass, and Spitzer near-IR data to estimate the stellar mass. This is a huge sample for which to have all of these data simultaneously. Measuring distances to individual galaxies remains challenging and time-consuming hard work that has been done by others. We are not about to second-guess their results, but we can note that they are sensible and remarkably consistent.

There are two primary methods by which the distances we use have been measured. One is Cepheids – the same type of variable stars that Hubble used to measure the distance to spiral nebulae to demonstrate their extragalactic nature. The other is the tip of the red giant branch (TRGB) method, which takes advantage of the brightest red giants having nearly the same luminosity. The sample is split nearly 50/50: there are 27 galaxies with a Cepheid distance measurement, and 23 with the TRGB. The two methods (different colored points in the figure) give the same calibration, within the errors, as do the two samples (circles vs. diamonds). There have been plenty of mistakes in the distance scale historically, so this consistency is important. There are many places where things could go wrong: differences between ourselves and Ponomareva, differences between Cepheids and the TRGB as distance indicators, mistakes in the application of either method to individual galaxies… so many opportunities to go wrong, and yet everything is consistent.

Having followed the distance scale problem my entire career, I cannot express how deeply impressive it is that all these different measurements paint a consistent picture. This is a credit to a large community of astronomers who have worked diligently on this problem for what seems like aeons. There is a temptation to dismiss distance scale work as having been wrong in the past, so it can be again. Of course that is true, but it is also true that matters have improved considerably. Forty years ago, it was not surprising when a distance indicator turned out to be wrong, and distances changed by a factor of two. That stopped twenty years ago, thanks in large part to the Hubble Space Telescope, a key goal of which had been to nail down the distance scale. That mission seems largely to have been accomplished, with small differences persisting only at the level that one expects from experimental error. One cannot, for example, make a change to the Cepheid calibration without creating a tension with the TRGB data, or vice-versa: both have to change in concert by the same amount in the same direction. That is unlikely to the point of wishful thinking.

Having nailed down the absolute calibration of the BTFR for galaxies with well-measured distances, we can apply it to other galaxies for which we know the redshift but not the distance. There are nearly 100 suitable galaxies available in the SPARC database. Consistency between them and the calibrator galaxies requires

H₀ = 75.1 +/- 2.3 (stat) +/- 1.5 (sys) km/s/Mpc.

This is consistent with the result for the standard luminosity-linewidth version of the Tully-Fisher relation reported by Kourkchi et al. Note also that our statistical (random/experimental) error is larger, but our systematic error is smaller. That’s because we have a much smaller number of galaxies. The method is, in principle, more precise (mostly because rotation curves are more accurate than linewidhts), so there is still a lot to be gained by collecting more data.

Our measurement is also consistent with many other “local” measurements of the distance scale,

but not with “global” measurements. See the nice discussion by Telescoper and the paper from which it comes. A Hubble constant in the 70s is the answer that we’ve consistently gotten for the past 20 years by a wide variety of distinct methods, including direct measurements that are not dependent on lower rungs of the distance ladder, like gravitational lensing and megamasers. These are repeatable experiments. In contrast, as I’ve pointed out before, it is the “global” CMB-fitted value of the Hubble parameter that has steadily diverged from the concordance region that originally established LCDM.

So, where does this leave us? In the past, it was easy to dismiss a tension of this sort as due to some systematic error, because that happened all the time – in the 20th century. That’s not so true anymore. It looks to me like the tension is real.

The halo mass function

I haven’t written much here of late. This is mostly because I have been busy, but also because I have been actively refraining from venting about some of the sillier things being said in the scientific literature. I went into science to get away from the human proclivity for what is nowadays called “fake news,” but we scientists are human too, and are not immune from the same self-deception one sees so frequently exercised in other venues.

So let’s talk about something positive. Current grad student Pengfei Li recently published a paper on the halo mass function. What is that and why should we care?

One of the fundamental predictions of the current cosmological paradigm, ΛCDM, is that dark matter clumps into halos. Cosmological parameters are known with sufficient precision that we have a very good idea of how many of these halos there ought to be. Their number per unit volume as a function of mass (so many big halos, so many more small halos) is called the halo mass function.

An important test of the paradigm is thus to measure the halo mass function. Does the predicted number match the observed number? This is hard to do, since dark matter halos are invisible! So how do we go about it?

Galaxies are thought to form within dark matter halos. Indeed, that’s kinda the whole point of the ΛCDM galaxy formation paradigm. So by counting galaxies, we should be able to count dark matter halos. Counting galaxies was an obvious task long before we thought there was dark matter, so this should be straightforward: all one needs is the measured galaxy luminosity function – the number density of galaxies as a function of how bright they are, or equivalently, how many stars they are made of (their stellar mass). Unfortunately, this goes tragically wrong.

Galaxy stellar mass function and the predicted halo mass function — Fig. 5 from the review by Bullock & Boylan-Kolchin. The number density of objects is shown as a function of their mass. Colored points are galaxies. The solid line is the predicted number of dark matter halos. The dotted line is what one would expect for galaxies if all the normal matter associated with each dark matter halo turned into stars.

This figure shows a comparison of the observed stellar mass function of galaxies and the predicted halo mass function. It is from a recent review, but it illustrates a problem that goes back as long as I can remember. We extragalactic astronomers spent all of the ’90s obsessing over this problem. [I briefly thought that I had solved this problem, but I was wrong.] The observed luminosity function is nearly flat while the predicted halo mass function is steep. Consequently, there should be lots and lots of faint galaxies for every bright one, but instead there are relatively few. This discrepancy becomes progressively more severe to lower masses, with the predicted number of halos being off by a factor of many thousands for the faintest galaxies. The problem is most severe in the Local Group, where the faintest dwarf galaxies are known. Locally it is called the missing satellite problem, but this is just a special case of a more general problem that pervades the entire universe.

Indeed, the small number of low mass objects is just one part of the problem. There are also too few galaxies at large masses. Even where the observed and predicted numbers come closest, around the scale of the Milky Way, they still miss by a large factor (this being a log-log plot, even small offsets are substantial). If we had assigned “explain the observed galaxy luminosity function” as a homework problem and the students had returned as an answer a line that had the wrong shape at both ends and at no point intersected the data, we would flunk them. This is, in effect, what theorists have been doing for the past thirty years. Rather than entertain the obvious interpretation that the theory is wrong, they offer more elaborate interpretations.

Faced with the choice between changing one’s mind and proving that there is no need to do so, almost everybody gets busy on the proof.
J. K. Galbraith

Theorists persist because this is what CDM predicts, with or without Λ, and we need cold dark matter for independent reasons. If we are unwilling to contemplate that ΛCDM might be wrong, then we are obliged to pound the square peg into the round hole, and bend the halo mass function into the observed luminosity function. This transformation is believed to take place as a result of a variety of complex feedback effects, all of which are real and few of which are likely to have the physical effects that are required to solve this problem. That’s way beyond the scope of this post; all we need to know here is that this is the “physics” behind the transformation that leads to what is currently called Abundance Matching.

Abundance matching boils down to drawing horizontal lines in the above figure, thus matching galaxies with dark matter halos with equal number density (abundance). So, just reading off the graph, a galaxy of stellar mass M_* = 10⁸ M_☉ resides in a dark matter halo of 10¹¹ M_☉, one like the Milky Way with M_* = 5 x 10¹⁰ M_☉ resides in a 10¹² M_☉ halo, and a giant galaxy with M_* = 10¹² M_☉ is the “central” galaxy of a cluster of galaxies with a halo mass of several 10¹⁴ M_☉. And so on. In effect, we abandon the obvious and long-held assumption that the mass in stars should be simply proportional to that in dark matter, and replace it with a rolling fudge factor that maps what we see to what we predict. The rolling fudge factor that follows from abundance matching is called the stellar mass–halo mass relation. Many of the discussions of feedback effects in the literature amount to a post hoc justification for this multiplication of forms of feedback.

This is a lengthy but insufficient introduction to a complicated subject. We wanted to get away from this, and test the halo mass function more directly. We do so by use of the velocity function rather than the stellar mass function.

The velocity function is the number density of galaxies as a function of how fast they rotate. It is less widely used than the luminosity function, because there is less data: one needs to measure the rotation speed, which is harder to obtain than the luminosity. Nevertheless, it has been done, as with this measurement from the HIPASS survey:

Galaxy velocity function — The number density of galaxies as a function of their rotation speed (Zwaan et al. 2010). The bottom panel shows the raw number of galaxies observed; the top panel shows the velocity function after correcting for the volume over which galaxies can be detected. Faint, slow rotators cannot be seen as far away as bright, fast rotators, so the latter are always over-represented in galaxy catalogs.

The idea here is that the flat rotation speed is the hallmark of a dark matter halo, providing a dynamical constraint on its mass. This should make for a cleaner measurement of the halo mass function. This turns out to be true, but it isn’t as clean as we’d like.

Those of you who are paying attention will note that the velocity function Martin Zwaan measured has the same basic morphology as the stellar mass function: approximately flat at low masses, with a steep cut off at high masses. This looks no more like the halo mass function than the galaxy luminosity function did. So how does this help?

To measure the velocity function, one has to use some readily obtained measure of the rotation speed like the line-width of the 21cm line. This, in itself, is not a very good measurement of the halo mass. So what Pengfei did was to fit dark matter halo models to galaxies of the SPARC sample for which we have good rotation curves. Thanks to the work of Federico Lelli, we also have an empirical relation between line-width and the flat rotation velocity. Together, these provide a connection between the line-width and halo mass:

Halo mass-line width relation — The relation Pengfei found between halo mass (M200) and line-width for the NFW (ΛCDM standard) halo model fit to rotation curves from the SPARC galaxy sample.

Once we have the mass-line width relation, we can assign a halo mass to every galaxy in the HIPASS survey and recompute the distribution function. But now we have not the velocity function, but the halo mass function. We’ve skipped the conversion of light to stellar mass to total mass and used the dynamics to skip straight to the halo mass function:

Empirical halo mass function — The halo mass function. The points are the data; these are well fit by a Schechter function (black line; this is commonly used for the galaxy luminosity function). The red line is the prediction of ΛCDM for dark matter halos.

The observed mass function agrees with the predicted one! Test successful! Well, mostly. Let’s think through the various aspects here.

First, the normalization is about right. It does not have the offset seen in the first figure. As it should not – we’ve gone straight to the halo mass in this exercise, and not used the luminosity as an intermediary proxy. So that is a genuine success. It didn’t have to work out this well, and would not do so in a very different cosmology (like SCDM).

Second, it breaks down at high mass. The data shows the usual Schechter cut-off at high mass, while the predicted number of dark matter halos continues as an unabated power law. This might be OK if high mass dark matter halos contain little neutral hydrogen. If this is the case, they will be invisible to HIPASS, the 21cm survey on which this is based. One expects this, to a certain extent: the most massive galaxies tend to be gas-poor ellipticals. That helps, but only by shifting the turn-down to slightly higher mass. It is still there, so the discrepancy is not entirely cured. At some point, we’re talking about large dark matter halos that are groups or even rich clusters of galaxies, not individual galaxies. Still, those have HI in them, so it is not like they’re invisible. Worse, examining detailed simulations that include feedback effects, there do seem to be more predicted high-mass halos that should have been detected than actually are. This is a potential missing gas-rich galaxy problem at the high mass end where galaxies are easy to detect. However, the simulations currently available to us do not provide the information we need to clearly make this determination. They don’t look right, so far as we can tell, but it isn’t clear enough to make a definitive statement.

Finally, the faint-end slope is about right. That’s amazing. The problem we’ve struggled with for decades is that the observed slope is too flat. Here a steep slope just falls out. It agrees with the ΛCDM down to the lowest mass bin. If there is a missing satellite-type problem here, it is at lower masses than we probe.

That sounds great, and it is. But before we get too excited, I hope you noticed that the velocity function from the same survey is flat like the luminosity function. So why is the halo mass function steep?

When we fit rotation curves, we impose various priors. That’s statistics talk for a way of keeping parameters within reasonable bounds. For example, we have a pretty good idea of what the mass-to-light ratio of a stellar population should be. We can therefore impose as a prior that the fit return something within the bounds of reason.

One of the priors we imposed on the rotation curve fits was that they be consistent with the stellar mass-halo mass relation. Abundance matching is now part and parcel of ΛCDM, so it made sense to apply it as a prior. The total mass of a dark matter halo is an entirely notional quantity; rotation curves (and other tracers) pretty much never extend far enough to measure this. So abundance matching is great for imposing sense on a parameter that is otherwise ill-constrained. In this case, it means that what is driving the slope of the halo mass function is a prior that builds-in the right slope. That’s not wrong, but neither is it an independent test. So while the observationally constrained halo mass function is consistent with the predictions of ΛCDM; we have not corroborated the prediction with independent data. What we really need at low mass is some way to constrain the total mass of small galaxies out to much larger radii that currently available. That will keep us busy for some time to come.

A personal recollection of how we learned to stop worrying and love the Lambda

There is a tendency when teaching science to oversimplify its history for the sake of getting on with the science. How it came to be isn’t necessary to learn it. But to do science requires a proper understanding of the process by which it came to be.

The story taught to cosmology students seems to have become: we didn’t believe in the cosmological constant (Λ), then in 1998 the Type Ia supernovae (SN) monitoring campaigns detected accelerated expansion, then all of a sudden we did believe in Λ. The actual history was, of course, rather more involved – to the point where this oversimplification verges on disingenuous. There were many observational indications of Λ that were essential in paving the way.

Modern cosmology starts in the early 20th century with the recognition that the universe should be expanding or contracting – a theoretical inevitability of General Relativity that Einstein initially tried to dodge by inventing the cosmological constant – and is expanding in fact, as observationally established by Hubble and Slipher and many others since. The Big Bang was largely considered settled truth after the discovery of the existence of the cosmic microwave background (CMB) in 1964.

The CMB held a puzzle, as it quickly was shown to be too smooth. The early universe was both isotropic and homogeneous. Too homogeneous. We couldn’t detect the density variations that could grow into galaxies and other immense structures. Though such density variations are now well measured as temperature fluctuations that are statistically well described by the acoustic power spectrum, the starting point was that these fluctuations were a disappointing no-show. We should have been able to see them much sooner, unless something really weird was going on…

That something weird was non-baryonic cold dark matter (CDM). For structure to grow, it needed the helping hand of the gravity of some unseen substance. Normal matter matter did not suffice. The most elegant cosmology, the Einstein-de Sitter universe, had a mass density Ω_m= 1. But the measured abundances of the light elements were only consistent with the calculations of big bang nucleosynthesis if normal matter amounted to only 5% of Ω_m = 1. This, plus the need to grow structure, led to the weird but seemingly unavoidable inference that the universe must be full of invisible dark matter. This dark matter needed to be some slow moving, massive particle that does not interact with light nor reside within the menagerie of particles present in the Standard Model of Particle Physics.

CDM and early universe Inflation were established in the 1980s. Inflation gave a mechanism that drove the mass density to exactly one (elegant!), and CDM gave us hope for enough mass to get to that value. Together, they gave us the Standard CDM (SCDM) paradigm with Ω_m = 1.000 and H₀ = 50 km/s/Mpc.

elrondwasthere — I was there when SCDM failed.

It is hard to overstate the ferver with which the SCDM paradigm was believed. Inflation required that the mass density be exactly one; Ω_m < 1 was inconceivable. For an Einstein-de Sitter universe to be old enough to contain the oldest stars, the Hubble constant had to be the lower of the two (50 or 100) commonly discussed at that time. That meant that H₀ > 50 was Right Out. We didn’t even discuss Λ. Λ was Unmentionable. Unclean.

SCDM was Known, Khaleesi.

Λ had attained unmentionable status in part because of its origin as Einstein’s greatest blunder, and in part through its association with the debunked Steady State model. But serious mention of it creeps back into the literature by 1990. The first time I personally heard Λ mentioned as a serious scientific possibility was by Yoshii at a conference in 1993. Yoshii based his argument on a classic cosmological test, N(m) – the number of galaxies as a function of how faint they appeared. The deeper you look, the more you see, in a way that depends on the intrinsic luminosity of galaxies, and how they fill space. Look deep enough, and you begin to trace the geometry of the cosmos.

At this time, one of the serious problems confronting the field was the faint blue galaxies problem. There were so many faint galaxies on the sky, it was incredibly difficult to explain them all. Yoshii made a simple argument. To get so many galaxies, we needed a big volume. The only way to do that in the context of the Robertson-Walker metric that describes the geometry of the universe is if we have a large cosmological constant, Λ. He was arguing for ΛCDM five years before the SN results.

gold_hat_portrayed_by_alfonso_bedoya — Lambda? We don’t need no stinking Lambda!

Yoshii was shouted down. NO! Galaxies evolve! We don’t need no stinking Λ! In retrospect, Yoshii & Peterson (1995) looks like a good detection of Λ. Perhaps Yoshii & Peterson also deserve a Nobel prize?

Indeed, there were many hints that Λ (or at least low Ω_m) was needed, e.g., the baryon catastrophe in clusters, the power spectrum of IRAS galaxies, the early appearance of bound structures, the statistics of gravitational lenses, and so on. Certainly by the mid-90s it was clear that we were not going to make it to Ω_m = 1. Inflation was threatened – it requires Ω_m = 1 – or at least a flat geometry: Ω_m+Ω_Λ = 1.

SCDM was in crisis.

A very influential 1995 paper by Ostriker & Steinhardt did a lot to launch ΛCDM. I was impressed by the breadth of data Ostriker & Steinhardt discussed, all of which demanded low Ω_m. I thought the case for Λ was less compelling, as it hinged on the age problem in a way that might also have been solved, at that time, by simply having an open universe (low Ω_m with no Λ). This would ruin Inflation, but I wasn’t bothered by that. I expect they were. Regardless, they definitely made that case for ΛCDM three years before the supernovae results. Their arguments were accepted by almost everyone who was paying attention, including myself. I heard Ostriker give a talk around this time during which he was asked “what cosmology are you assuming?” to which he replied “the right one.” Called the “concordance” cosmology by Ostriker & Steinhardt, ΛCDM had already achieved the status of most-favored cosmology by the mid-90s.

omhannotated — A simplified version of the diagram of Ostriker & Steinhardt (1995) illustrating just a few of the constraints they discussed. Direct measurements of the expansion rate, mass density, and ages of the oldest stars excluded SCDM, instead converging on a narrow window – what we now call ΛCDM.

Ostriker & Steinhardt neglected to mention an important prediction of Λ: not only should the universe expand, but that expansion rate should accelerate! In 1995, that sounded completely absurd. People had looked for such an effect, and claimed not to see it. So I wrote a brief note pointing out the predicted acceleration of the expansion rate. I meant it in a bad way: how crazy would it be if the expansion of the universe was accelerating?! This was an obvious and inevitable consequence of ΛCDM that was largely being swept under the rug at that time.

I mean[t], surely we could live with Ω_m < 1 but no Λ. Can’t we all just get along? Not really, as it turned out. I remember Mike Turner pushing the SN people very hard in Aspen in 1997 to Admit Λ. He had an obvious bias: as an Inflationary cosmologist, he had spent the previous decade castigating observers for repeatedly finding Ω_m < 1. That’s too little mass, you fools! Inflation demands Ω_m = 1.000! Look harder!

By 1997, Turner had, like many cosmologists, finally wrapped his head around the fact that we weren’t going to find enough mass for Ω_m = 1. This was a huge problem for Inflation. The only possible solution, albeit an ugly one, was if Λ made up the difference. So there he was at Aspen, pressuring the people who observed supernova to Admit Λ. One, in particular, was Richard Ellis, a great and accomplished astronomer who had led the charge in shouting down Yoshii. They didn’t yet have enough data to Admit Λ. Not.Yet.

By 1998, there were many more high redshift SNIa. Enough to see Λ. This time, after the long series of results only partially described above, we were intellectually prepared to accept it – unlike in 1993. Had the SN experiments been conducted five years earlier, and obtained exactly the same result, they would not have been awarded the Nobel prize. They would instead have been dismissed as a trick of astrophysics: the universe evolves, metallicity was lower at earlier times, that made SN then different from now, they evolve and so cannot be used as standard candles. This sounds silly now, as we’ve figured out how to calibrate for intrinsic variations in the luminosities of Type Ia SN, but that is absolutely how we would have reacted in 1993, and no amount of improvements in the method would have convinced us. This is exactly what we did with faint galaxy counts: galaxies evolve; you can’t hope to understand that well enough to constrain cosmology. Do you ever hear them cited as evidence for Λ?

Great as the supernovae experiments to measure the metric genuinely were, they were not a discovery so much as a confirmation of what cosmologists had already decided to believe. There was no singular discovery that changed the way we all thought. There was a steady drip, drip, drip of results pointing towards Λ all through the ’90s – the age problem in which the oldest stars appeared to be older than the universe in which they reside, the early appearance of massive clusters and galaxies, the power spectrum of galaxies from redshift surveys that preceded Sloan, the statistics of gravitational lenses, and the repeated measurement of 1/4 < Ω_m < 1/3 in a large variety of independent ways – just to name a few. By the mid-90’s, SCDM was dead. We just refused to bury it until we could accept ΛCDM as a replacement. That was what the Type Ia SN results really provided: a fresh and dramatic reason to accept the accelerated expansion that we’d already come to terms with privately but had kept hidden in the closet.

Note that the acoustic power spectrum of temperature fluctuations in the cosmic microwave background (as opposed to the mere existence of the highly uniform CMB) plays no role in this history. That’s because temperature fluctuations hadn’t yet been measured beyond their rudimentary detection by COBE. COBE demonstrated that temperature fluctuations did indeed exist (finally!) as they must, but precious little beyond that. Eventually, after the settling of much dust, COBE was recognized as one of many reasons why Ω_m ≠ 1, but it was neither the most clear nor most convincing reason at that time. Now, in the 21st century, the acoustic power spectrum provides a great way to constrain what all the parameters of ΛCDM have to be, but it was a bit player in its development. The water there was carried by traditional observational cosmology using general purpose optical telescopes in a great variety of different ways, combined with a deep astrophysical understanding of how stars, galaxies, quasars and the whole menagerie of objects found in the sky work. All the vast knowledge incorporated in textbooks like those by Harrison, by Peebles, and by Peacock – knowledge that often seems to be lacking in scientists trained in the post-WMAP era.

Despite being a late arrival, the CMB power spectrum measured in 2000 by Boomerang and 2003 by WMAP did one important new thing to corroborate the ΛCDM picture. The supernovae data didn’t detect accelerated expansion so much as exclude the deceleration we had nominally expected. The data were also roughly consistent with a coasting universe (neither accelerating nor decelerating); the case for acceleration only became clear when we assumed that the geometry of the universe was flat (Ω_m+Ω_Λ = 1). That didn’t have to work out, so it was a great success of the paradigm when the location of the first peak of the power spectrum appeared in exactly the right place for a flat FLRW geometry.

The consistency of these data have given ΛCDM an air of invincibility among cosmologists. But a modern reconstruction of the Ostriker & Steinhardt diagram leaves zero room remaining – hence the tension between H₀ = 73 measured directly and H₀ = 67 from multiparameter CMB fits.

omhannotated_cmb — Constraints from the acoustic power spectrum of the CMB overplotted on the direct measurements from the plot above. Initially in great consistency with those measurement, the best fit CMB values have steadily wandered away from the most-favored region of parameter space that established ΛCDM in the first place. This is most apparent in the tension with H₀.

In cosmology, we are accustomed to having to find our way through apparently conflicting data. The difference between an expansion rate of 67 and 73 seems trivial given that the field was long riven – in living memory – by the dispute between 50 and 100. This gives rise to the expectation that the current difference is just a matter of some subtle systematic error somewhere. That may well be correct. But it is also conceivable that FLRW is inadequate to describe the universe, and we have been driven to the objectively bizarre parameters of ΛCDM because it happens to be the best approximation that can be obtained to what is really going on when we insist on approximating it with FLRW.

Though a logical possibility, that last sentence will likely drive many cosmologists to reach for their torches and pitchforks. Before killing the messenger, we should remember that we once endowed SCDM with the same absolute certainty we now attribute to ΛCDM. I was there, 3,000 internet years ago, when SCDM failed. There is nothing so sacred in ΛCDM that it can’t suffer the same fate, as has every single cosmology ever devised by humanity.

Today, we still lack definitive knowledge of either dark matter or dark energy. These add up to 95% of the mass-energy of the universe according to ΛCDM. These dark materials must exist.

It is Known, Khaleesi.

Hypothesis testing with gas rich galaxies

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, The Dark Matter Problem: A Historical Perspective and Deconstructing Cosmology.

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:

DDO154_MOND_180805695 — The MOND fit to DDO 154 from Ren et al. (2018). The black points are the rotation curve data, the green line is the Newtonian expectation for the baryons, and the red line is their MOND fit.

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.:

DDO154_RAR_Li2018 — The MOND fit to DDO 154 from Li et al. (2018) using the same data as above, as tabulated in SPARC.

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.

DDO154_SIDM_180805695 — The SIDM fit to DDO 154 from Ren et al.

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.

NGC2403UGC128trickMOND — An attempt to fit MOND to a hybrid galaxy with the rotation curve of NGC 2403 and the baryon distribution of UGC 128. The mass-to-light ratio is driven to unphysical values (6 in solar units), but an acceptable fit is not obtained.

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.

NGC2403UGC128trickDM — A fit to the hybrid galaxy with a cored (pseudo-isothermal) dark matter halo. A satisfactory fit is readily obtained.

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:

rcresid_mondfits — Residuals of MOND fits from Famaey & McGaugh (2012).

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.

DDO154mond_i66 — The rotation curve of DDO 154 from THINGS (points with error bars). The Newtonian expectation for stars is the green line; the gas is the blue line. The red line is the MOND prediction. Not that the gas greatly outweighs the stars beyond 1.5 kpc; the stellar mass-to-light ratio has extremely little leverage in this MOND fit.

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.

DDO154mond_i61 — MOND fit to the THINGS data for DDO 154 with the inclination adjusted to the value found by Li et al. (2018).

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.

star_trek_tv_spock_3_copy_-_h_2018 — This outcome is illogical.

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?

It Must Be So. But which Must?

In the last post, I noted some of the sociological overtones underpinning attitudes about dark matter and modified gravity theories. I didn’t get as far as the more scientifically interesting part, which illustrates a common form of reasoning in physics.

About modified gravity theories, Bertone & Tait state

“the only way these theories can be reconciled with observations is by effectively, and very precisely, mimicking the behavior of cold dark matter on cosmological scales.”

Leaving aside just which observations need to be mimicked so precisely (I expect they mean power spectrum; perhaps they consider this to be so obvious that it need not be stated), this kind of reasoning is both common and powerful – and frequently correct. Indeed, this is exactly the attitude I expressed in my review a few years ago for the Canadian Journal of Physics, quoted in the image above. I get it. There are lots of positive things to be said for the standard cosmology.

This upshot of this reasoning is, in effect, that “cosmology works so well that non-baryonic dark matter must exist.” I have sympathy for this attitude, but I also remember many examples in the history of cosmology where it has gone badly wrong. There was a time, not so long ago, that the matter density had to be the critical value, and the Hubble constant had to be 50 km/s/Mpc. By and large, it is the same community that insisted on those falsehoods with great intensity that continues to insist on conventionally conceived cold dark matter with similarly fundamentalist insistence.

I think it is an overstatement to say that the successes of cosmology (as we presently perceive them) prove the existence of dark matter. A more conservative statement is that the ΛCDM cosmology is correct if, and only if, dark matter exists. But does it? That’s a separate question, which is why laboratory searches are so important – including null results. It was, after all, the null result of Michelson & Morley that ultimately put an end to the previous version of an invisible aetherial medium, and sparked a revolution in physics.

Here I point out that the same reasoning asserted by Bertone & Tait as a slam dunk in favor of dark matter can just as accurately be asserted in favor of MOND. To directly paraphrase the above statement:

“the only way ΛCDM can be reconciled with observations is by effectively, and very precisely, mimicking the behavior of MOND on galactic scales.”

This is a terrible problem for dark matter. Even if it were true, as is often asserted, that MOND only fits rotation curves, this would still be tantamount to a falsification of dark matter by the same reasoning applied by Bertone & Tait.

Lets look at just one example, NGC 1560:

ngc1560mond — The rotation curve of NGC 1560 (points) together with the Newtonian expectation (black line) and the MOND fit (blue line). Data from Begeman et al. (1991) and Gentile et al. (2010).

MOND fits the details of this rotation curve in excruciating detail. It provides just the right amount of boost over the Newtonian expectation, which varies from galaxy to galaxy. Features in the baryon distribution are reflected in the rotation curve. That is required in MOND, but makes no sense in dark matter, where the excess velocity over the Newtonian expectation is attributed to a dynamically hot, dominant, quasi-spherical dark matter halo. Such entities cannot support the features commonly seen in thin, dynamically cold disks. Even if they could, there is no reason that features in the dominant dark matter halo should align with those in the disk: a sphere isn’t a disk. In short, it is impossible to explain this with dark matter – to the extent that anything is ever impossible for the invisible.

NGC 1560 is a famous case because it has such an obvious feature. It is common to dismiss this as some non-equilibrium fluke that should simply be ignored. That is always a dodgy path to tread, but might be OK if it were only this galaxy. But similar effects are seen over and over again, to the point that they earned an empirical moniker: Renzo’s Rule. Renzo’s rule is known to every serious student of rotation curves, but has not informed the development of most dark matter theory. Ignoring this information is like leaving money on the table.

MOND fits not just NGC 1560, but very nearly* every galaxy we measure. It does so with excruciatingly little freedom. The only physical fit parameter is the stellar mass-to-light ratio. The gas fraction of NGC 1560 is 75%, so M*/L plays little role. We understand enough about stellar populations to have an idea what to expect; MOND fits return mass-to-light ratios that compare well with the normalization, color dependence, and band-pass dependent scatter expected from stellar population synthesis models.

MLBV_MOND — The mass-to-light ratio from MOND fits (points) in the blue (left panel) and near-infrared (right panel) pass-bands plotted against galaxy color (blue to the left, red to the right). From the perspective of stellar populations, one expects more scatter and a steeper color dependence in the blue band, as observed. The lines are stellar population models from Bell et al. (2003). These are completely independent, and have not been fit to the data in any way. One could hardly hope for better astrophysical agreement.

One can also fit rotation curve data with dark matter halos. These require a minimum of three parameters to the one of MOND. In addition to M*/L, one also needs at least two parameters to describe the dark matter halo of each galaxy – typically some characteristic mass and radius. In practice, one finds that such fits are horribly degenerate: one can not cleanly constrain all three parameters, much less recover a sensible distribution of M*/L. One cannot construct the plot above simply by asking the data what it wants as one can with MOND.

The “disk-halo degeneracy” in dark matter halo fits to rotation curves has been much discussed in the literature. Obsessed over, dismissed, revived, and ultimately ignored without satisfactory understanding. Well, duh. This approach uses three parameters per galaxy when it takes only one to describe the data. Degeneracy between the excess fit parameters is inevitable.

From a probabilistic perspective, there is a huge volume of viable parameter space that could (and should) be occupied by galaxies composed of dark matter halos plus luminous galaxies. Two identical dark matter halos might host very different luminous galaxies, so would have rotation curves that differed with the baryonic component. Two similar looking galaxies might reside in rather different dark matter halos, again having rotation curves that differ.

The probabilistic volume in MOND is much smaller. Absolutely tiny by comparison. There is exactly one and only one thing each rotation curve can do: what the particular distribution of baryons in each galaxy says it should do. This is what we observe in Nature.

The only way ΛCDM can be reconciled with observations is by effectively, and very precisely, mimicking the behavior of MOND on galactic scales. There is a vast volume of parameter space that the rotation curves of galaxies could, in principle, inhabit. The naive expectation was exponential disks in NFW halos. Real galaxies don’t look like that. They look like MOND. Magically, out of the vast parameter space available to galaxies in the dark matter picture, they only ever pick the tiny sub-volume that very precisely mimics MOND.

The ratio of probabilities is huge. So many dark matter models are possible (and have been mooted over the years) that it is indefinably huge. The odds of observing MOND-like phenomenology in a ΛCDM universe is practically zero. This amounts to a practical falsification of dark matter.

I’ve never said dark matter is falsified, because I don’t think it is a falsifiable concept. It is like epicycles – you can always fudge it in some way. But at a practical level, it was falsified a long time ago.

That is not to say MOND has to be right. That would be falling into the same logical trap that says ΛCDM has to be right. Obviously, both have virtues that must be incorporated into whatever the final answer may be. There are some efforts in this direction, but by and large this is not how science is being conducted at present. The standard script is to privilege those data that conform most closely to our confirmation bias, and pour scorn on any contradictory narrative.

In my assessment, the probability of ultimate success through ignoring inconvenient data is practically zero. Unfortunately, that is the course upon which much of the field is currently set.

*There are of course exceptions: no data are perfect, so even the right theory will get it wrong once in a while. The goof rate for MOND fits is about what I expect: rare, but more frequent for lower quality data. Misfits are sufficiently rare that to obsess over them is to refuse to see the forest for a few outlying trees.

Here’s a residual plot of MOND fits. See the peak at right? That’s the forest. See the tiny tail to one side? That’s an outlying tree.

Dwarf Satellite Galaxies. III. The dwarfs of Andromeda

Like the Milky Way, our nearest giant neighbor, Andromeda (aka M31), has several dozen dwarf satellite galaxies. A few of these were known and had measured velocity dispersions at the time of my work with Joe Wolf, as discussed previously. Also like the Milky Way, the number of known objects has grown rapidly in recent years – thanks in this case largely to the PAndAS survey.

PAndAS imaged the area around M31 and M33, finding many individual red giant stars. These trace out the debris from interactions and mergers as small dwarfs are disrupted and consumed by their giant host. They also pointed up the existence of previously unknown dwarf satellites.

M31fromPANDAS_ McC2012_EPJ_19_01003 — The PAndAS survey field. Dwarf satellites are circled.

As the PAndAS survey started reporting the discovery of new dwarf satellites around Andromeda, it occurred to me that this provided the opportunity to make genuine a priori predictions. These are the gold standard of the scientific method. We could use the observed luminosity and size of the newly discovered dwarfs to predict their velocity dispersions.

I tried to do this for both ΛCDM and MOND. I will not discuss the ΛCDM case much, because it can’t really be done. But it is worth understanding why this is.

In ΛCDM, the velocity dispersion is determined by the dark matter halo. This has only a tenuous connection to the observed stars, so just knowing how big and bright a dwarf is doesn’t provide much predictive power about the halo. This can be seen from this figure by Tollerud et al (2011):

Tollerud2011_ml_scatter — Virial mass of the dark matter halo as a function of galaxy luminosity. Dwarfs satellites reside in the wide colored band of low luminosities.

This graph is obtained by relating the number density of galaxies (an observed quantity) to that of the dark matter halos in which they reside (a theoretical construct). It is highly non-linear, deviating strongly from the one-to-one line we expected early on. There is no reason to expect this particular relation; it is imposed on us by the fact that the observed luminosity function of galaxies is rather flat while the predicted halo mass function is steep. Nowadays, this is usually called the missing satellite problem, but this is a misnomer as it pervades the field.

Addressing the missing satellites problem would be another long post, so lets just accept that the relation between mass and light has to follow something like that illustrated above. If a dwarf galaxy has a luminosity of a million suns, one can read off the graph that it should live in a dark halo with a mass of about 10¹⁰ M_☉. One could use this to predict the velocity dispersion, but not very precisely, because there’s a big range corresponding to that luminosity (the bands in the figure). It could be as much as 10¹¹ M_☉ or as little as 10⁹ M_☉. This corresponds to a wide range of velocity dispersions. This wide range is unavoidable because of the difference in the luminosity function and halo mass function. Small variations in one lead to big variations in the other, and some scatter in dark halo properties is unavoidable.

Consequently, we only have a vague range of expected velocity dispersions in ΛCDM. In practice, we never make this prediction. Instead, we compare the observed velocity dispersion to the luminosity and say “gee, this galaxy has a lot of dark matter” or “hey, this one doesn’t have much dark matter.” There’s no rigorously testable prior.

In MOND, what you see is what you get. The velocity dispersion has to follow from the observed stellar mass. This is straightforward for isolated galaxies: M_* ∝ σ⁴ – this is essentially the equivalent of the Tully-Fisher relation for pressure supported systems. If we can estimate the stellar mass from the observed luminosity, the predicted velocity dispersion follows.

Many dwarf satellites are not isolated in the MONDian sense: they are subject to the external field effect (EFE) from their giant hosts. The over-under for whether the EFE applies is the point when the internal acceleration from all the stars of the dwarf on each other is equal to the external acceleration from orbiting the giant host. The amplitude of the discrepancy in MOND depends on how low the total acceleration is relative to the critical scale a₀. The external field in effect adds some acceleration that wouldn’t otherwise be there, making the discrepancy less than it would be for an isolated object. This means that two otherwise identical dwarfs may be predicted to have different velocity dispersions is they are or are not subject to the EFE. This is a unique prediction of MOND that has no analog in ΛCDM.

It is straightforward to derive the equation to predict velocity dispersions in the extreme limits of isolated (a_ex ≪ a_in < a₀) or EFE dominated (a_in ≪ a_ex < a₀) objects. In reality, there are many objects for which a_in ≈ a_ex, and no simple formula applies. In practice, we apply the formula that more nearly applies, and pray that this approximation is good enough.

There are many other assumptions and approximations that must be made in any theory: that an object is spherical, isotropic, and in dynamical equilibrium. All of these must fail at some level, but it is the last one that is the most serious concern. In the case of the EFE, one must also make the approximation that the object is in equilibrium at the current level of the external field. That is never true, as both the amplitude and the vector of the external field vary as a dwarf orbits its host. But it might be an adequate approximation if this variation is slow. In the case of a circular orbit, only the vector varies. In general the orbits are not known, so we make the instantaneous approximation and once again pray that it is good enough. There is a fairly narrow window between where the EFE becomes important and where we slip into the regime of tidal disruption, but lets plow ahead and see how far we can get, bearing in mind that the EFE is a dynamical variable of which we only have a snapshot.

To predict the velocity dispersion in the isolated case, all we need to know is the luminosity and a stellar mass-to-light ratio. Assuming the dwarfs of Andromeda to be old stellar populations, I adopted a V-band mass-to-light ratio of 2 give or take a factor of 2. That usually dominates the uncertainty, though the error in the distance can sometimes impact the luminosity at a level that impacts the prediction.

To predict the velocity dispersion in the EFE case, we again need the stellar mass, but now also need to know the size of the stellar system and the intensity of the external field to which it is subject. The latter depends on the mass of the host galaxy and the distance from it to the dwarf. This latter quantity is somewhat fraught: it is straightforward to measure the projected distance on the sky, but we need the 3D distance – how far in front or behind each dwarf is as well as its projected distance from the host. This is often a considerable contributor to the error budget. Indeed, some dwarfs may be inferred to be in the EFE regime for the low end of the range of adopted stellar mass-to-light ratio, and the isolated regime for the high end.

In this fashion, we predicted velocity dispersions for the dwarfs of Andromeda. We in this case were Milgrom and myself. I had never collaborated with him before, and prefer to remain independent. But I also wanted to be sure I got the details described above right. Though it wasn’t much work to make the predictions once the preliminaries were established, it was time consuming to collect and vet the data. As we were writing the paper, velocity dispersion measurements started to appear. People like Michelle Collins, Erik Tollerud, and Nicolas Martin were making follow-up observations, and publishing velocity dispersion for the objects we were making predictions for. That was great, but they were too good – they were observing and publishing faster than we could write!

Nevertheless, we managed to make and publish a priori predictions for 10 dwarfs before any observational measurements were published. We also made blind predictions for the other known dwarfs of Andromeda, and checked the predicted velocity dispersions against all measurements that we could find in the literature. Many of these predictions were quickly tested by on-going programs (i.e., people were out to measure velocity dispersions, whether we predicted them or not). Enough data rolled in that we were soon able to write a follow-up paper testing our predictions.

Nailed it. Good data were soon available to test the predictions for 8 of the 10* a priori cases. All 8 were consistent with our predictions. I was particularly struck by the case of And XXVIII, which I had called out as perhaps the best test. It was isolated, so the messiness of the EFE didn’t apply, and the uncertainties were low. Moreover, the predicted velocity dispersion was low – a good deal lower than broadly expected in ΛCDM: 4.3 km/s, with an uncertainty just under 1 km/s. Two independent observations were subsequently reported. One found 4.9 ± 1.6 km/s, the other 6.6 ± 2.1 km/s, both in good agreement within the uncertainties.

We made further predictions in the second paper as people had continued to discover new dwarfs. These also came true. Here is a summary plot for all of the dwarfs of Andromeda:

AndDwarfswithGoldStars.002 — The velocity dispersions of the dwarf satellites of Andromeda. Each numbered box corresponds to one dwarf (x=1 is for And I and so on). Measured velocity dispersions have a number next to them that is the number of stars on which the measurement is based. MOND predictions are circles: green if isolated, open if the EFE applies. Points appear within each box in the order they appeared in the literature, from left to right. The vast majority of Andromeda’s dwarfs are consistent with MOND (large green circles). Two cases are ambiguous (large yellow circles), having velocity dispersions based only a few stars. Only And V appears to be problematic (large red circle).

MOND works well for And I, And II, And III, And VI, And VII, And IX, And X, And XI, And XII, And XIII, And XIV, And XV, And XVI, And XVII, And XVIII, And XIX, And XX, And XXI, And XXII, And XXIII, And XXIV, And XXV, And XXVIII, And XXIX, And XXXI, And XXXII, and And XXXIII. There is one problematic case: And V. I don’t know what is going on there, but note that systematic errors frequently happen in astronomy. It’d be strange if there weren’t at least one goofy case.

Nevertheless, the failure of And V could be construed as a falsification of MOND. It ought to work in every single case. But recall the discussion of assumptions and uncertainties above. Is falsification really the story these data tell?

We do have experience with various systematic errors. For example, we predicted that the isolated dwarfs spheroidal Cetus should have a velocity dispersion in MOND of 8.2 km/s. There was already a published measurement of 17 ± 2 km/s, so we reported that MOND was wrong in this case by over 3σ. Or at least we started to do so. Right before we submitted that paper, a new measurement appeared: 8.3 ± 1 km/s. This is an example of how the data can sometimes change by rather more than the formal error bars suggest is possible. In this case, I suspect the original observations lacked the spectral resolution to resolve the velocity dispersion. At any rate, the new measurement (8.3 km/s) was somewhat more consistent with our prediction (8.2 km/s).

The same predictions cannot even be made in ΛCDM. The velocity data can always be fit once they are in hand. But there is no agreed method to predict the velocity dispersion of a dwarf from its observed luminosity. As discussed above, this should not even be possible: there is too much scatter in the halo mass-stellar mass relation at these low masses.

An unsung predictive success of MOND absent from the graph above is And IV. When And IV was discovered in the general direction of Andromeda, it was assumed to be a new dwarf satellite – hence the name. Milgrom looked at the velocities reported for this object, and said it had to be a background galaxy. No way it could be a dwarf satellite – at least not in MOND. I see no reason why it couldn’t have been in ΛCDM. It is absent from the graph above, because it was subsequently confirmed to be much farther away (7.2 Mpc vs. 750 kpc for Andromeda).

The box for And XVII is empty because this system is manifestly out of equilibrium. It is more of a stellar stream than a dwarf, appearing as a smear in the PAndAS image rather than as a self-contained dwarf. I do not recall what the story with the other missing object (And VIII) is.

While writing the follow-up paper, I also noticed that there were a number of Andromeda dwarfs that were photometrically indistinguishable: basically the same in terms of size and stellar mass. But some were isolated while others were subject to the EFE. MOND predicts that the EFE cases should have lower velocity dispersion than the isolated equivalents.

AndDwarfswithGoldStars.003 — The velocity dispersions of the dwarfs of Andromeda, highlighting photometrically matched pairs – dwarfs that should be indistinguishable, but aren’t because of the EFE.

And XXVIII (isolated) has a higher velocity dispersion than its near-twin And XVII (EFE). The same effect might be acting in And XVIII (isolated) and And XXV (EFE). This is clear if we accept the higher velocity dispersion measurement for And XVIII, but an independent measurements begs to differ. The former has more stars, so is probably more reliable, but we should be cautious. The effect is not clear in And XVI (isolated) and And XXI (EFE), but the difference in the prediction is small and the uncertainties are large.

An aggressive person might argue that the pairs of dwarfs is a positive detection of the EFE. I don’t think the data for the matched pairs warrant that, at least not yet. On the other hand, the appropriate use of the EFE was essential to all the predictions, not just the matched pairs.

The positive detection of the EFE is important, as it is a unique prediction of MOND. I see no way to tune ΛCDM galaxy simulations to mimic this effect. Of course, there was a very recent time when it seemed impossible for them to mimic the isolated predictions of MOND. They claim to have come a long way in that regard.

But that’s what we’re stuck with: tuning ΛCDM to make it look like MOND. This is why a priori predictions are important. There is ample flexibility to explain just about anything with dark matter. What we can’t seem to do is predict the same things that MOND successfully predicts… predictions that are both quantitative and very specific. We’re not arguing that dwarfs in general live in ~15 or 30 km/s halos, as we must in ΛCDM. In MOND we can say this dwarf will have this velocity dispersion and that dwarf will have that velocity dispersion. We can distinguish between 4.9 and 7.3 km/s. And we can do it over and over and over. I see no way to do the equivalent in ΛCDM, just as I see no way to explain the acoustic power spectrum of the CMB in MOND.

This is not to say there are no problematic cases for MOND. Read, Walker, & Steger have recently highlighted the matched pair of Draco and Carina as an issue. And they are – though here I already have reason to suspect Draco is out of equilibrium, which makes it challenging to analyze. Whether it is actually out of equilibrium or not is a separate question.

I am not thrilled that we are obliged to invoke non-equilibrium effects in both theories. But there is a difference. Brada & Milgrom provided a quantitative criterion to indicate when this was an issue before I ran into the problem. In ΛCDM, the low velocity dispersions of objects like And XIX, XXI, XXV and Crater 2 came as a complete surprise despite having been predicted by MOND. Tidal disruption was only invoked after the fact – and in an ad hoc fashion. There is no way to know in advance which dwarfs are affected, as there is no criterion equivalent to that of Brada. We just say “gee, that’s a low velocity dispersion. Must have been disrupted.” That might be true, but it gives no explanation for why MOND predicted it in the first place – which is to say, it isn’t really an explanation at all.

I still do not understand is why MOND gets any predictions right if ΛCDM is the universe we live in, let alone so many. Shouldn’t happen. Makes no sense.

If this doesn’t confuse you, you are not thinking clearly.

*The other two dwarfs were also measured, but with only 4 stars in one and 6 in the other. These are too few for a meaningful velocity dispersion measurement.

Triton Station

A Blog About the Science and Sociology of Cosmology and Dark Matter

Category: LCDM