Two Hypotheses

OK, basic review is over. Shit’s gonna get real. Here I give a short recounting of the primary reason I came to doubt the dark matter paradigm. This is entirely conventional – my concern about the viability of dark matter is a contradiction within its own context. It had nothing to do with MOND, which I was blissfully ignorant of when I ran head-long into this problem in 1994. Most of the community chooses to remain blissfully ignorant, which I understand: it’s way more comfortable. It is also why the field has remained mired in the ’90s, with all the apparent progress since then being nothing more than the perpetual reinvention of the same square wheel.

To make a completely generic point that does not depend on the specifics of dark matter halo profiles or the details of baryonic assembly, I discuss two basic hypotheses for the distribution of disk galaxy size at a given mass. These broad categories I label SH (Same Halo) and DD (Density begets Density) following McGaugh and de Blok (1998a). In both cases, galaxies of a given baryonic mass are assumed to reside in dark matter halos of a corresponding total mass. Hence, at a given halo mass, the baryonic mass is the same, and variations in galaxy size follow from one of two basic effects:

SH: variations in size follow from variations in the spin of the parent dark matter halo.
DD: variations in surface brightness follow from variations in the density of the dark matter halo.

Recall that at a given luminosity, size and surface brightness are not independent, so variation in one corresponds to variation in the other. Consequently, we have two distinct ideas for why galaxies of the same mass vary in size. In SH, the halo may have the same density profile ρ(r), and it is only variations in angular momentum that dictate variations in the disk size. In DD, variations in the surface brightness of the luminous disk are reflections of variations in the density profile ρ(r) of the dark matter halo. In principle, one could have a combination of both effects, but we will keep them separate for this discussion, and note that mixing them defeats the virtues of each without curing their ills.

The SH hypothesis traces back to at least Fall and Efstathiou (1980). The notion is simple: variations in the size of disks correspond to variations in the angular momentum of their host dark matter halos. The mass destined to become a dark matter halo initially expands with the rest of the universe, reaching some maximum radius before collapsing to form a gravitationally bound object. At the point of maximum expansion, the nascent dark matter halos torque one another, inducing a small but non-zero net spin in each, quantified by the dimensionless spin parameter λ (Peebles, 1969). One then imagines that as a disk forms within a dark matter halo, it collapses until it is centrifugally supported: λ → 1 from some initially small value (typically λ ≈ 0.05, Barnes & Efstathiou, 1987, with some modest distribution about this median value). The spin parameter thus determines the collapse factor and the extent of the disk: low spin halos harbor compact, high surface brightness disks while high spin halos produce extended, low surface brightness disks.

The distribution of primordial spins is fairly narrow, and does not correlate with environment (Barnes & Efstathiou, 1987). The narrow distribution was invoked as an explanation for Freeman’s Law: the small variation in spins from halo to halo resulted in a narrow distribution of disk central surface brightness (van der Kruit, 1987). This association, while apparently natural, proved to be incorrect: when one goes through the mathematics to transform spin into scale length, even a narrow distribution of initial spins predicts a broad distribution in surface brightness (Dalcanton, Spergel, & Summers, 1997; McGaugh and de Blok, 1998a). Indeed, it predicts too broad a distribution: to prevent the formation of galaxies much higher in surface brightness than observed, one must invoke a stability criterion (Dalcanton, Spergel, & Summers, 1997; McGaugh and de Blok, 1998a) that precludes the existence of very high surface brightness disks. While it is physically quite reasonable that such a criterion should exist (Ostriker and Peebles, 1973), the observed surface density threshold does not emerge naturally, and must be inserted by hand. It is an auxiliary hypothesis invoked to preserve SH. Once done, size variations and the trend of average size with mass work out in reasonable quantitative detail (e.g., Mo et al., 1998).

Angular momentum conservation must hold for an isolated galaxy, but the assumption made in SH is stronger: baryons conserve their share of the angular momentum independently of the dark matter. It is considered a virtue that this simple assumption leads to disk sizes that are about right. However, this assumption is not well justified. Baryons and dark matter are free to exchange angular momentum with each other, and are seen to do so in simulations that track both components (e.g., Book et al., 2011; Combes, 2013; Klypin et al., 2002). There is no guarantee that this exchange is equitable, and in general it is not: as baryons collapse to form a small galaxy within a large dark matter halo, they tend to lose angular momentum to the dark matter. This is a one-way street that runs in the wrong direction, with the final destination uncomfortably invisible with most of the angular momentum sequestered in the unobservable dark matter. Worse still, if we impose rigorous angular momentum conservation among the baryons, the result is a disk with a completely unrealistic surface density profile (van den Bosch, 2001a). It then becomes necessary to pick and choose which baryons manage to assemble into the disk and which are expelled or otherwise excluded, thereby solving one problem by creating another.

Early work on LSB disk galaxies led to a rather different picture. Compared to the previously known population of HSB galaxies around which our theories had been built, the LSB galaxy population has a younger mean stellar age (de Blok & van der Hulst, 1998; McGaugh and Bothun, 1994), a lower content of heavy elements (McGaugh, 1994), and a systematically higher gas fraction (McGaugh and de Blok, 1997; Schombert et al., 1997). These properties suggested that LSB galaxies evolve more gradually than their higher surface brightness brethren: they convert their gas into stars over a much longer timescale (McGaugh et al., 2017). The obvious culprit for this difference is surface density: lower surface brightness galaxies have less gravity, hence less ability to gather their diffuse interstellar medium into dense clumps that could form stars (Gerritsen and de Blok, 1999; Mihos et al., 1999). It seemed reasonable to ascribe the low surface density of the baryons to a correspondingly low density of their parent dark matter halos.

One way to think about a region in the early universe that will eventually collapse to form a galaxy is as a so-called top-hat over-density. The mass density Ω_m → 1 at early times, irrespective of its current value, so a spherical region (the top-hat) that is somewhat over-dense early on may locally exceed the critical density. We may then consider this finite region as its own little closed universe, and follow its evolution with the Friedmann equations with Ω > 1. The top-hat will initially expand along with the rest of the universe, but will eventually reach a maximum radius and recollapse. When that happens depends on the density. The greater the over-density, the sooner the top-hat will recollapse. Conversely, a lesser over-density will take longer to reach maximum expansion before recollapsing.

Everything about LSB galaxies suggested that they were lower density, late-forming systems. It therefore seemed quite natural to imagine a distribution of over-densities and corresponding collapse times for top-hats of similar mass, and to associate LSB galaxy with the lesser over-densities (Dekel and Silk, 1986; McGaugh, 1992). More recently, some essential aspects of this idea have been revived under the monicker of “assembly bias” (e.g. Zehavi et al., 2018).

The work that informed the DD hypothesis was based largely on photometric and spectroscopic observations of LSB galaxies: their size and surface brightness, color, chemical abundance, and gas content. DD made two obvious predictions that had not yet been tested at that juncture. First, late-forming halos should reside preferentially in low density environments. This is a generic consequence of Gaussian initial conditions: big peaks defined on small (e.g., galaxy) scales are more likely to be found in big peaks defined on large (e.g., cluster) scales, and vice-versa. Second, the density of the dark matter halo of an LSB galaxy should be lower than that of an equal mass halo containing and HSB galaxy. This predicts a clear signature in their rotation speeds, which should be lower for lower density.

The prediction for the spatial distribution of LSB galaxies was tested by Bothun et al. (1993) and Mo et al. (1994). The test showed the expected effect: LSB galaxies were less strongly clustered than HSB galaxies. They are clustered: both galaxy populations follow the same large scale structure, but HSB galaxies adhere more strongly to it. In terms of the correlation function, the LSB sample available at the time had about half the amplitude r₀ as comparison HSB samples (Mo et al., 1994). The effect was even more pronounced on the smallest scales (<2 Mpc: Bothun et al., 1993), leading Mo et al. (1994) to construct a model that successfully explained both small and large scale aspects of the spatial distribution of LSB galaxies simply by associating them with dark matter halos that lacked close interactions with other halos. This was strong corroboration of the DD hypothesis.

One way to test the prediction of DD that LSB galaxies should rotate more slowly than HSB galaxies was to use the Tully-Fisher relation (Tully and Fisher, 1977) as a point of reference. Originally identified as an empirical relation between optical luminosity and the observed line-width of single-dish 21 cm observations, more fundamentally it turns out to be a relation between the baryonic mass of a galaxy (stars plus gas) and its flat rotation speed the Baryonic Tully-Fisher relation (BTFR: McGaugh et al., 2000). This relation is a simple power law of the form

M_b = AV_f⁴ (equation 1)

with A ≈ 50 M_⊙ km⁻⁴ s⁴ (McGaugh, 2005).

Aaronson et al. (1979) provided a straightforward interpretation for a relation of this form. A test particle orbiting a mass M at a distance R will have a circular speed V

V² = GM/R (equation 2)

where G is Newton’s constant. If we square this, a relation like the Tully-Fisher relation follows:

V⁴ = (GM/R)² &propto; MΣ (equation 3)

where we have introduced the surface mass density Σ = M/R². The Tully-Fisher relation M ∝ V⁴ is recovered if Σ is constant, exactly as expected from Freeman’s Law (Freeman, 1970).

LSB galaxies, by definition, have central surface brightnesses (and corresponding stellar surface densities Σ₀) that are less than the Freeman value. Consequently, DD predicts, through equation (3), that LSB galaxies should shift systematically off the Tully-Fisher relation: lower Σ means lower velocity. The predicted effect is not subtle² (Fig. 4). For the range of surface brightness that had become available, the predicted shift should have stood out like the proverbial sore thumb. It did not (Hoffman et al., 1996; McGaugh and de Blok, 1998a; Sprayberry et al., 1995; Zwaan et al., 1995). This had an immediate impact on galaxy formation theory: compare Dalcanton et al. (1995, who predict a shift in Tully-Fisher with surface brightness) with Dalcanton et al. (1997b, who do not).

Fig. 4. The Baryonic Tully-Fisher relation and residuals. The top panel shows the flat rotation velocity of galaxies in the SPARC database (Lelli et al., 2016a) as a function of the baryonic mass (stars plus gas). The sample is restricted to those objects for which both quantities are measured to better than 20% accuracy. The bottom panel shows velocity residuals around the solid line in the top panel as a function of the central surface density of the stellar disks. Variations in the stellar surface density predict variations in velocity along the dashed line. These would translate to shifts illustrated by the dotted lines in the top panel, with each dotted line representing a shift of a factor of ten in surface density. The predicted dependence on surface density is not observed (Courteau & Rix, 1999; McGaugh and de Blok, 1998a; Sprayberry et al., 1995; Zwaan et al., 1995).

Instead of the systematic variation of velocity with surface brightness expected at fixed mass, there was none. Indeed, there is no hint of a second parameter dependence. The relation is incredibly tight by the standards of extragalactic astronomy (Lelli et al., 2016b): baryonic mass and the flat rotation speed are practically interchangeable.

The above derivation is overly simplistic. The radius at which we should make a measurement is ill-defined, and the surface density is dynamical: it includes both stars and dark matter. Moreover, galaxies are not spherical cows: one needs to solve the Poisson equation for the observed disk geometry of LTGs, and account for the varying radial contributions of luminous and dark matter. While this can be made to sound intimidating, the numerical computations are straightforward and rigorous (e.g., Begeman et al., 1991; Casertano & Shostak, 1980; Lelli et al., 2016a). It still boils down to the same sort of relation (modulo geometrical factors of order unity), but with two mass distributions: one for the baryons M_b(R), and one for the dark matter M_DM(R). Though the dark matter is more massive, it is also more extended. Consequently, both components can contribute non-negligibly to the rotation over the observed range of radii:

V²(R) = GM/R = G(M_b/R + M_DM/R), (equation 4)

(4)where for clarity we have omitted^* geometrical factors. The only absolute requirement is that the baryonic contribution should begin to decline once the majority of baryonic mass is encompassed. It is when rotation curves persist in remaining flat past this point that we infer the need for dark matter.

A recurrent problem in testing galaxy formation theories is that they seldom make ironclad predictions; I attempt a brief summary in Table 1. SH represents a broad class of theories with many variants. By construction, the dark matter halos of galaxies of similar stellar mass are similar. If we associate the flat rotation velocity with halo mass, then galaxies of the same mass have the same circular velocity, and the problem posed by Tully-Fisher is automatically satisfied.

Table 1. Predictions of DD and SH for LSB galaxies.

Observation	DD	SH
Evolutionary rate	+	+
Size distribution	+	+
Clustering	+	X
Tully-Fisher relation	X	?
Central density relation	+	X

While it is common to associate the flat rotation speed with the dark matter halo, this is a half-truth: the observed velocity is a combination of baryonic and dark components (eq. (4)). It is thus a rather curious coincidence that rotation curves are as flat as they are: the Keplerian decline of the baryonic contribution must be precisely balanced by an increasing contribution from the dark matter halo. This fine-tuning problem was dubbed the “disk-halo conspiracy” (Bahcall & Casertano, 1985; van Albada & Sancisi, 1986). The solution offered for the disk-halo conspiracy was that the formation of the baryonic disk has an effect on the distribution of the dark matter. As the disk settles, the dark matter halo respond through a process commonly referred to as adiabatic compression that brings the peak velocities of disk and dark components into alignment (Blumenthal et al., 1986). Some rearrangement of the dark matter halo in response to the change of the gravitational potential caused by the settling of the disk is inevitable, so this seemed a plausible explanation.

The observation that LSB galaxies obey the Tully-Fisher relation greatly compounds the fine-tuning (McGaugh and de Blok, 1998a; Zwaan et al., 1995). The amount of adiabatic compression depends on the surface density of stars (Sellwood and McGaugh, 2005b): HSB galaxies experience greater compression than LSB galaxies. This should enhance the predicted shift between the two in Tully-Fisher. Instead, the amplitude of the flat rotation speed remains unperturbed.

The generic failings of dark matter models was discussed at length by McGaugh and de Blok (1998a). The same problems have been encountered by others. For example, Fig. 5 shows model galaxies formed in a dark matter halo with identical total mass and density profile but with different spin parameters (van den Bosch, 2001b). Variations in the assembly and cooling history were also considered, but these make little difference and are not relevant here. The point is that smaller (larger) spin parameters lead to more (less) compact disks that contribute more (less) to the total rotation, exactly as anticipated from variations in the term M_b/R in equation (4). The nominal variation is readily detectable, and stands out prominently in the Tully-Fisher diagram (Fig. 5). This is exactly the same fine-tuning problem that was pointed out by Zwaan et al. (1995) and McGaugh and de Blok (1998a).

What I describe as a fine-tuning problem is not portrayed as such by van den Bosch (2000) and van den Bosch and Dalcanton (2000), who argued that the data could be readily accommodated in the dark matter picture. The difference is between accommodating the data once known, and predicting it a priori. The dark matter picture is extraordinarily flexible: one is free to distribute the dark matter as needed to fit any data that evinces a non-negative mass discrepancy, even data that are wrong (de Blok & McGaugh, 1998). It is another matter entirely to construct a realistic model a priori; in my experience it is quite easy to construct models with plausible-seeming parameters that bear little resemblance to real galaxies (e.g., the low-spin case in Fig. 5). A similar conundrum is encountered when constructing models that can explain the long tidal tails observed in merging and interacting galaxies: models with realistic rotation curves do not produce realistic tidal tails, and vice-versa (Dubinski et al., 1999). The data occupy a very narrow sliver of the enormous volume of parameter space available to dark matter models, a situation that seems rather contrived.

Fig. 5. Model galaxy rotation curves and the Tully-Fisher relation. Rotation curves (left panel) for model galaxies of the same mass but different spin parameters λ from van den Bosch (2001b, see his Fig. 3). Models with lower spin have more compact stellar disks that contribute more to the rotation curve (V² = GM/R; R being smaller for the same M). These models are shown as square points on the Baryonic Tully-Fisher relation (right) along with data for real galaxies (grey circles: Lelli et al., 2016b) and a fit thereto (dashed line). Differences in the cooling history result in modest variation in the baryonic mass at fixed halo mass as reflected in the vertical scatter of the models. This is within the scatter of the data, but variation due to the spin parameter is not.

Both DD and SH predict residuals from Tully-Fisher that are not observed. I consider this to be an unrecoverable failure for DD, which was my hypothesis (McGaugh, 1992), so I worked hard to salvage it. I could not. For SH, Tully-Fisher might be recovered in the limit of dark matter domination, which requires further consideration.

I will save the further consideration for a future post, as that can take infinite words (there are literally thousands of ApJ papers on the subject). The real problem that rotation curve data pose generically for the dark matter interpretation is the fine-tuning required between baryonic and dark matter components – the balancing act explicit in the equations above. This, by itself, constitutes a practical falsification of the dark matter paradigm.

Without going into interesting but ultimately meaningless details (maybe next time), the only way to avoid this conclusion is to choose to be unconcerned with fine-tuning. If you choose to say fine-tuning isn’t a problem, then it isn’t a problem. Worse, many scientists don’t seem to understand that they’ve even made this choice: it is baked into their assumptions. There is no risk of questioning those assumptions if one never stops to think about them, much less worry that there might be something wrong with them.

Much of the field seems to have sunk into a form of scientific nihilism. The attitude I frequently encounter when I raise this issue boils down to “Don’t care! Everything will magically work out! LA LA LA!”

*Strictly speaking, eq. (4) only holds for spherical mass distributions. I make this simplification here to emphasize the fact that both mass and radius matter. This essential scaling persists for any geometry: the argument holds in complete generality.

Common ground

In order to agree on an interpretation, we first have to agree on the facts. Even when we agree on the facts, the available set of facts may admit multiple interpretations. This was an obvious and widely accepted truth early in my career*. Since then, the field has decayed into a haphazardly conceived set of unquestionable absolutes that are based on a large but well-curated subset of facts that gratuitously ignores any subset of facts that are inconvenient.

Sadly, we seem to have entered a post-truth period in which facts are drowned out by propaganda. I went into science to get away from people who place faith before facts, and comfortable fictions ahead of uncomfortable truths. Unfortunately, a lot of those people seem to have followed me here. This manifests as people who quote what are essentially pro-dark matter talking points at me like I don’t understand LCDM, when all it really does is reveal that they are posers** who picked up on some common myths about the field without actually reading the relevant journal articles.

Indeed, a recent experience taught me a new psychology term: identity protective cognition. Identity protective cognition is the tendency for people in a group to selectively credit or dismiss evidence in patterns that reflect the beliefs that predominate in their group. When it comes to dark matter, the group happens to be a scientific one, but the psychology is the same: I’ve seen people twist themselves into logical knots to protect their belief in dark matter from being subject to critical examination. They do it without even recognizing that this is what they’re doing. I guess this is a human foible we cannot escape.

I’ve addressed these issues before, but here I’m going to start a series of posts on what I think some of the essential but underappreciated facts are. This is based on a talk that I gave at a conference on the philosophy of science in 2019, back when we had conferences, and published in Studies in History and Philosophy of Science. I paid the exorbitant open access fee (the journal changed its name – and publication policy – during the publication process), so you can read the whole thing all at once if you are eager. I’ve already written it to be accessible, so mostly I’m going to post it here in what I hope are digestible chunks, and may add further commentary if it seems appropriate.

Cosmic context

Cosmology is the science of the origin and evolution of the universe: the biggest of big pictures. The modern picture of the hot big bang is underpinned by three empirical pillars: an expanding universe (Hubble expansion), Big Bang Nucleosynthesis (BBN: the formation of the light elements through nuclear reactions in the early universe), and the relic radiation field (the Cosmic Microwave Background: CMB) (Harrison, 2000; Peebles, 1993). The discussion here will take this framework for granted.

The three empirical pillars fit beautifully with General Relativity (GR). Making the simplifying assumptions of homogeneity and isotropy, Einstein’s equations can be applied to treat the entire universe as a dynamical entity. As such, it is compelled either to expand or contract. Running the observed expansion backwards in time, one necessarily comes to a hot, dense, early phase. This naturally explains the CMB, which marks the transition from an opaque plasma to a transparent gas (Sunyaev and Zeldovich, 1980; Weiss, 1980). The abundances of the light elements can be explained in detail with BBN provided the universe expands in the first few minutes as predicted by GR when radiation dominates the mass-energy budget of the universe (Boesgaard & Steigman, 1985).

The marvelous consistency of these early universe results with the expectations of GR builds confidence that the hot big bang is the correct general picture for cosmology. It also builds overconfidence that GR is completely sufficient to describe the universe. Maintaining consistency with modern cosmological data is only possible with the addition of two auxiliary hypotheses: dark matter and dark energy. These invisible entities are an absolute requirement of the current version of the most-favored cosmological model, ΛCDM. The very name of this model is born of these dark materials: Λ is Einstein’s cosmological constant, of which ‘dark energy’ is a generalization, and CDM is cold dark matter.

Dark energy does not enter much into the subject of galaxy formation. It mainly helps to set the background cosmology in which galaxies form, and plays some role in the timing of structure formation. This discussion will not delve into such details, and I note only that it was surprising and profoundly disturbing that we had to reintroduce (e.g., Efstathiou et al., 1990; Ostriker and Steinhardt, 1995; Perlmutter et al., 1999; Riess et al., 1998; Yoshii and Peterson, 1995) Einstein’s so-called ‘greatest blunder.’

Dark matter, on the other hand, plays an intimate and essential role in galaxy formation. The term ‘dark matter’ is dangerously crude, as it can reasonably be used to mean anything that is not seen. In the cosmic context, there are at least two forms of unseen mass: normal matter that happens not to glow in a way that is easily seen — not all ordinary material need be associated with visible stars — and non-baryonic cold dark matter. It is the latter form of unseen mass that is thought to dominate the mass budget of the universe and play a critical role in galaxy formation.

Cold Dark Matter

Cold dark matter is some form of slow moving, non-relativistic (‘cold’) particulate mass that is not composed of normal matter (baryons). Baryons are the family of particles that include protons and neutrons. As such, they compose the bulk of the mass of normal matter, and it has become conventional to use this term to distinguish between normal, baryonic matter and the non-baryonic dark matter.

The distinction between baryonic and non-baryonic dark matter is no small thing. Non-baryonic dark matter must be a new particle that resides in a new ‘dark sector’ that is completely distinct from the usual stable of elementary particles. We do not just need some new particle, we need one (or many) that reside in some sector beyond the framework of the stubbornly successful Standard Model of particle physics. Whatever the solution to the mass discrepancy problem turns out to be, it requires new physics.

The cosmic dark matter must be non-baryonic for two basic reasons. First, the mass density of the universe measured gravitationally (Ω_m ≈ 0.3, e.g., Faber and Gallagher, 1979; Davis et al., 1980, 1992) clearly exceeds the mass density in baryons as constrained by BBN (Ω_b ≈ 0.05, e.g., Walker et al., 1991). There is something gravitating that is not ordinary matter: Ω_m > Ω_b.

The second reason follows from the absence of large fluctuations in the CMB (Peebles and Yu, 1970; Silk, 1968; Sunyaev and Zeldovich, 1980). The CMB is extraordinarily uniform in temperature across the sky, varying by only ~ 1 part in 10⁵ (Smoot et al., 1992). These small temperature variations correspond to variations in density. Gravity is an attractive force; it will make the rich grow richer. Small density excesses will tend to attract more mass, making them larger, attracting more mass, and leading to the formation of large scale structures, including galaxies. But gravity is also a weak force: this process takes a long time. In the long but finite age of the universe, gravity plus known baryonic matter does not suffice to go from the initially smooth, highly uniform state of the early universe to the highly clumpy, structured state of the local universe (Peebles, 1993). The solution is to boost the process with an additional component of mass — the cold dark matter — that gravitates without interacting with the photons, thus getting a head start on the growth of structure while not aggravating the amplitude of temperature fluctuations in the CMB.

Taken separately, one might argue away the need for dark matter. Taken together, these two distinct arguments convinced nearly everyone, including myself, of the absolute need for non-baryonic dark matter. Consequently, CDM became established as the leading paradigm during the 1980s (Peebles, 1984; Steigman and Turner, 1985). The paradigm has snowballed since that time, the common attitude among cosmologists being that CDM has to exist.

From an astronomical perspective, the CDM could be any slow-moving, massive object that does not interact with photons nor participate in BBN. The range of possibilities is at once limitless yet highly constrained. Neutrons would suffice if they were stable in vacuum, but they are not. Primordial black holes are a logical possibility, but if made of normal matter, they must somehow form in the first second after the Big Bang to not impair BBN. At this juncture, microlensing experiments have excluded most plausible mass ranges that primordial black holes could occupy (Mediavilla et al., 2017). It is easy to invent hypothetical dark matter candidates, but difficult for them to remain viable.

From a particle physics perspective, the favored candidate is a Weakly Interacting Massive Particle (WIMP: Peebles, 1984; Steigman and Turner, 1985). WIMPs are expected to be the lightest stable supersymmetric partner particle that resides in the hypothetical supersymmetric sector (Martin, 1998). The WIMP has been the odds-on favorite for so long that it is often used synonymously with the more generic term ‘dark matter.’ It is the hypothesized particle that launched a thousand experiments. Experimental searches for WIMPs have matured over the past several decades, making extraordinary progress in not detecting dark matter (Aprile et al., 2018). Virtually all of the parameter space in which WIMPs had been predicted to reside (Trotta et al., 2008) is now excluded. Worse, the existence of the supersymmetric sector itself, once seemingly a sure thing, remains entirely hypothetical, and appears at this juncture to be a beautiful idea that nature declined to implement.

In sum, we must have cold dark matter for both galaxies and cosmology, but we have as yet no clue to what it is.

* There is a trope that late in their careers, great scientists come to the opinion that everything worth discovering has been discovered, because they themselves already did everything worth doing. That is not a concern I have – I know we haven’t discovered all there is to discover. Yet I see no prospect for advancing our fundamental understanding simply because there aren’t enough of us pulling in the right direction. Most of the community is busy barking up the wrong tree, and refuses to be distracted from their focus on the invisible squirrel that isn’t there.

** Many of these people are the product of the toxic culture that Simon White warned us about. They wave the sausage of galaxy formation and feedback like a magic wand that excuses all faults while being proudly ignorant of how the sausage was made. Bitch, please. I was there when that sausage was made. I helped make the damn sausage. I know what went into it, and I recognize when it tastes wrong.

Are there credible deviations from the baryonic Tully-Fisher relation?

There is a rule of thumb in scientific publication that if a title is posed a question, the answer is no.

It sucks being so far ahead of the field that I get to watch people repeat the mistakes I made (or almost made) and warned against long ago. There have been persistent claims of deviations of one sort or another from the Baryonic Tully-Fisher relation (BTFR). So far, these have all been obviously wrong, for reasons we’ve discussed before. It all boils down to data quality. The credibility of data is important, especially in astronomy.

Here is a plot of the BTFR for all the data I have ready at hand, both for gas rich galaxies and the SPARC sample:

*Baryonic mass (stars plus gas) as a function of the rotation speed measured at the outermost detected radius.*

A relation is clear in the plot above, but it’s a mess. There’s lots of scatter, especially at low mass. There is also a systematic tendency for low mass galaxies to fall to the left of the main relation, appearing to rotate too slowly for their mass.

There is no quality control in the plot above. I have thrown all the mud at the wall. Let’s now do some quality control. The plotted quantities are the baryonic mass and the flat rotation speed. We haven’t actually measured the flat rotation speed in all these cases. For some, we’ve simply taken the last measured point. This was an issue we explicitly pointed out in Stark et al (2009):

Fig. 1 from Stark et al (2009): Examples of rotation curves (Swaters et al. 2009) that do and do not satisfy the flatness criterion. The rotation curve of UGC 4173 (top) rises continuously and does not meet the flatness criterion. UGC 5721 (center) is an ideal case with clear flattening of the rotational velocity. UGC 4499 marginally satisfies the flatness criterion.

If we include a galaxy like UGC 4173, we expect it will be offset to the low velocity side because we haven’t measured the flat rotation speed. We’ve merely taken that last point and hoped it is close enough. Sometimes it is, depending on your tolerance for systematic errors. But the plain fact is that we haven’t measured the flat rotation speed in this case. We don’t even know if it has one; it is only empirical experience with other examples that lead us to expect it to flatten if we manage to observe further out.

For our purpose here, it is as if we hadn’t measured this galaxy at all. So let’s not pretend like we have, and restrict the plot to galaxies for which the flat velocity is measured:

*The same as the first plot, restricted to galaxies for which the flat rotation speed has been measured.*

The scatter in the BTFR decreases dramatically when we exclude the galaxies for which we haven’t measured the relevant quantities. This is a simple matter of data quality. We’re no longer pretending to have measured a quantity that we haven’t measured.

There are still some outliers as there are still things that can go wrong. Inclinations are a challenge for some galaxies, as are distances determinations. Remember that Tully-Fisher was first employed as a distance indicator. If we look at the plot above from that perspective, the outliers have obviously been assigned the wrong distance, and we would assign a new one by putting them on the relation. That, in a nutshell, is how astronomical distance indicators work.

If we restrict the data to those with accurate measurements, we get

*Same as the plot above, restricted to galaxies for which the quantities measured on both axes have been measured to an accuracy of 20% or better.*

Now the outliers are gone. They were outliers because they had crappy data. This is completely unsurprising. Some astronomical data are always crappy. You plot crap against crap, you get crap. If, on the other hand, you look at the high quality data, you get a high quality correlation. Even then, you can never be sure that you’ve excluded all the crap, as there are often unknown unknowns – systematic errors you don’t know about and can’t control for.

We have done the exercise of varying the tolerance limits on data quality many times. We have shown that the scatter varies as expected with data quality. If we consider high quality data, we find a small scatter in the BTFR. If we consider low quality data, we get to plot more points, but the scatter goes up. You can see this by eye above. We can quantify this, and have. The amount of scatter varies as expected with the size of the uncertainties. Bigger errors, bigger scatter. Smaller errors, smaller scatter. This shouldn’t be hard to understand.

So why do people – many of them good scientists – keep screwing this up?

There are several answers. One is that measuring the flat rotation speed is hard. We have only done it for a couple hundred galaxies. This seems like a tiny number in the era of the Sloan Digitial Sky Survey, which enables any newbie to assemble a sample of tens of thousands of galaxies… with photometric data. It doesn’t provide any kinematic data. Measuring the stellar mass with the photometric data doesn’t do one bit of good for this problem if you don’t have the kinematic axis to plot against. Consequently, it doesn’t matter how big such a sample is.

Other measurements often provide a proxy measurement that seems like it ought to be close enough to use. If not the flat rotation speed, maybe you have a line width or a maximum speed or V_2.2 or the hybrid S_0.5 or some other metric. That’s fine, so long as you recognize you’re plotting something different so should expect to get something different – not the BTFR. Again, we’ve shown that the flat rotation speed is the measure that minimizes the scatter; if you utilize some other measure you’re gonna get more scatter. That may be useful for some purposes, but it only tells you about what you measured. It doesn’t tell you anything about the scatter in the BTFR constructed with the flat rotation speed if you didn’t measure the flat rotation speed.

Another possibility is that there exist galaxies that fall off the BTFR that we haven’t observed yet. It is a big universe, after all. This is a known unknown unknown: we know that we don’t know if there are non-conforming galaxies. If the relation is indeed absolute, then we never can find any, but never can we know that they don’t exist, only that we haven’t yet found any credible examples.

I’ve addressed the possibility of nonconforming galaxies elsewhere, so all I’ll say here is that I have spent my entire career seeking out the extremes in galaxy properties. Many times I have specifically sought out galaxies that should deviate from the BTFR for some clear reason, only to be surprised when they fall bang on the BTFR. Over and over and over again. It makes me wonder how Vera Rubin felt when her observations kept turning up flat rotation curves. Shouldn’t happen, but it does – over and over and over again. So far, I haven’t found any credible deviations from the BTFR, nor have I seen credible cases provided by others – just repeated failures of quality control.

Finally, an underlying issue is often – not always, but often – an obsession with salvaging the dark matter paradigm. That’s hard to do if you acknowledge that the observed BTFR – its slope, normalization, lack of scale length residuals, negligible intrinsic scatter; indeed, the very quantities that define it, were anticipated and explicitly predicted by MOND and only MOND. It is easy to confirm the dark matter paradigm if you never acknowledge this to be a problem. Often, people redefine the terms of the issue in some manner that is more tractable from the perspective of dark matter. From that perspective, neither the “cold” baryonic mass nor the flat rotation speed have any special meaning, so why even consider them? That is the road to MONDness.

A script for every observational test

Science progresses through hypothesis testing. The primary mechanism for distinguishing between hypotheses is predictive power. The hypothesis that can predict new phenomena is “better.” This is especially true for surprising, a priori predictions: it matters more when the new phenomena was not expected in the context of an existing paradigm.

I’ve seen this happen many times now. MOND has had many predictive successes. As a theory, it has been exposed to potential falsification, and passed many tests. These have often been in the form of phenomena that had not been anticipated in any other way, and were initially received as strange to the point of seeming impossible. It is exactly the situation envisioned in Putnam’s “no miracles” argument: it is unlikely to the point of absurdity that a wholly false theory should succeed in making so many predictions of such diversity and precision.

MOND has many doubters, which I can understand. What I don’t get is the ignorance I so often encounter among them. To me, the statement that MOND has had many unexpected predictions come true is a simple statement of experiential fact. I suspect it will be received by some as a falsehood. It shouldn’t be, so if you don’t know what I’m talking about, you should try reading the relevant literature. What papers about MOND have you actually read?

Ignorance is not a strong basis for making scientific judgements. Before I criticize something, I make sure I know what I’m talking about. That’s rarely true of the complaints I hear against MOND. There are legitimate ones, to be sure, but for the most part I hear assertions like

MOND is guaranteed to fit rotation curves.
It fits rotation curves but does nothing else.
It is just a fitting tool with no predictive power.

These are myths, plain and simple. They are easily debunked, and were long ago. Yet I hear them repeated often by people who think they know better, one as recently as last week. Serious people who expect to be taken seriously as scientists, and yet they repeat known falsehoods as if they were established fact. Is there a recycling bin of debunked myths that gets passed around? I guess it is easy to believe a baseless rumor when it conforms to your confirmation bias: no need for fact-checking!

Aside from straight-up reality denial, another approach is to claim that dark matter predicts exactly the same thing, whatever it is. I’ve seen this happen so often, I know how the script always goes:

• We make a new observation X that is surprising.
• We test the hypothesis, and report the result: “Gee, MOND predicted this strange effect, and we see evidence of it in the data.”
• Inevitable Question: What does LCDM predict?
• Answer: Not that.
• Q: But what does it predict?
• A: It doesn’t really make a clear prediction on this subject, so we have to build some kind of model to even begin to address this question. In the most obvious models one can construct, it predicts Y. Y is not the same as X.
• Q: What about more complicated models?
• A: One can construct more complicated models, but they are not unique. They don’t make a prediction so much as provide a menu of options from which we may select the results that suit us. The obvious danger is that it becomes possible to do anything, and we have nothing more than an epicycle theory of infinite possibilities. If we restrict ourselves to considering the details of serious models that have only been partially fine-tuned over the course of the development of the field, then there are still a lot of possibilities. Some of them come closer to reality than others but still don’t really do the right thing for the following reasons…[here follows 25 pages of minutia in the ApJ considering every up/down left/right stand on your head and squint possibility that still winds up looking more like Y than like X.] You certainly couldn’t predict X this way, as MOND did a priori.
• Q: That’s too long to read. Dr. Z says it works, so he must be right since we already know that LCDM is correct.

The thing is, Dr. Z did not predict X ahead of time. MOND did. Maybe Dr. Z’s explanation in terms of dark matter makes sense. Often it does not, but even if it does, so what? Why should I be more impressed with a theory that only explains things after they’re observed when another predicted them a priori?

There are lots of Dr. Z’s. No matter how carefully one goes through the minutia, no matter how clearly one demonstrates that X cannot work in a purely conventional CDM context, there is always someone who says it does. That’s what people want to hear, so that’s what they choose to believe. Way easier that way. Or, as it has been noted before

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 (1965)

Super spirals on the Tully-Fisher relation

A surprising and ultimately career-altering result that I encountered while in my first postdoc was that low surface brightness galaxies fell precisely on the Tully-Fisher relation. This surprising result led me to test the limits of the relation in every conceivable way. Are there galaxies that fall off it? How far is it applicable? Often, that has meant pushing the boundaries of known galaxies to ever lower surface brightness, higher gas fraction, and lower mass where galaxies are hard to find because of unavoidable selection biases in galaxy surveys: dim galaxies are hard to see.

I made a summary plot in 2017 to illustrate what we had learned to that point. There is a clear break in the stellar mass Tully-Fisher relation (left panel) that results from neglecting the mass of interstellar gas that becomes increasingly important in lower mass galaxies. The break goes away when you add in the gas mass (right panel). The relation between baryonic mass and rotation speed is continuous down to Leo P, a tiny galaxy just outside the Local Group comparable in mass to a globular cluster and the current record holder for the slowest known rotating galaxy at a mere 15 km/s.

The stellar mass (left) and baryonic (right) Tully-Fisher relations constructed in 2017 from SPARC data and gas rich galaxies. Dark blue points are star dominated galaxies; light blue points are galaxies with more mass in gas than in stars. The data are restricted to galaxies with distance measurements accurate to 20% or better; *see McGaugh et al. (2019)* for a discussion of the effects of different quality criteria. The line has a slope of 4 and is identical in both panels for comparison.

At the high mass end, galaxies aren’t hard to see, but they do become progressively rare: there is an exponential cut off in the intrinsic numbers of galaxies at the high mass end. So it is interesting to see how far up in mass we can go. Ogle et al. set out to do that, looking over a huge volume to identify a number of very massive galaxies, including what they dubbed “super spirals.” These extend the Tully-Fisher relation to higher masses.

*The Tully-Fisher relation extended to very massive “super” spirals (blue points) by Ogle et al. (2019).*

Most of the super spirals lie on the top end of the Tully-Fisher relation. However, a half dozen of the most massive cases fall off to the right. Could this be a break in the relation? So it was claimed at the time, but looking at the data, I wasn’t convinced. It looked to me like they were not always getting out to the flat part of the rotation curve, instead measuring the maximum rotation speed.

Bright galaxies tend to have rapidly rising rotation curves that peak early then fall before flattening out. For very bright galaxies – and super spirals are by definition the brightest spirals – the amplitude of the decline can be substantial, several tens of km/s. So if one measures the maximum speed instead of the flat portion of the curve, points will fall to the right of the relation. I decided not to lose any sleep over it, and wait for better data.

Better data have now been provided by Di Teodoro et al. Here is an example from their paper. The morphology of the rotation curve is typical of what we see in massive spiral galaxies. The maximum rotation speed exceeds 300 km/s, but falls to 275 km/s where it flattens out.

*A super spiral (left) and its rotation curve (right) from Di Teodoro et al.*

Adding the updated data to the plot, we see that the super spirals now fall on the Tully-Fisher relation, with no hint of a break. There are a couple of outliers, but those are trees. The relation is the forest.

*The super spiral (red points) stellar mass (left) and baryonic (right) Tully-Fisher relations as updated by Di Teodoro et al. (2021).*

That’s a good plot, but it stops at 10⁸ solar masses, so I couldn’t resist adding the super spirals to my plot from 2017. I’ve also included the dwarfs I discussed in the last post. Together, we see that the baryonic Tully-Fisher relation is continuous over six decades in mass – a factor of million from the smallest to the largest galaxies.

The plot from above updated to include the super spirals (red points) at high mass and Local Group dwarfs (gray squares) at low mass. The SPARC data (blue points) have also been updated with new stellar population mass-to-light ratio estimates that make their bulge components a bit more massive, and with scaling relations for metallicity and molecular gas. The super spirals have been treated in the same way, and adjusted to a matching distance scale (H₀ = 73 km/s/Mpc). There is some overlap between the super spirals and the most massive galaxies in SPARC; here the data are in excellent agreement. The super spirals extend to higher mass by a factor of two.

The strength of this correlation continues to amaze me. This never happens in extragalactic astronomy, where correlations are typically weak and have lots of intrinsic scatter. The opposite is true here. This must be telling us something.

The obvious thing that this is telling us is MOND. The initial report that super spirals fell off of the Tully-Fisher relation was widely hailed as a disproof of MOND. I’ve seen this movie many times, so I am not surprised that the answer changed in this fashion. It happens over and over again. Even less surprising is that there is no retraction, no self-examination of whether maybe we jumped to the wrong conclusion.

I get it. I couldn’t believe it myself, to start. I struggled for many years to explain the data conventionally in terms of dark matter. Worked my ass off trying to save the paradigm. Try as I might, nothing worked. Since then, many people have claimed to explain what I could not, but so far all I have seen are variations on models that I had already rejected as obviously unworkable. They either make unsubstantiated assumptions, building a tautology, or simply claim more than they demonstrate. As long as you say what people want to hear, you will be held to a very low standard. If you say what they don’t want to hear, what they are conditioned not to believe, then no standard of proof is high enough.

MOND was the only theory to predict the observed behavior a priori. There are no free parameters in the plots above. We measure the mass and the rotation speed. The data fall on the predicted line. Dark matter models did not predict this, and can at best hope to provide a convoluted, retroactive explanation. Why should I be impressed by that?

Divergence

I read somewhere – I don’t think it was Kuhn himself, but someone analyzing Kuhn – that there came a point in the history of science where there was a divergence between scientists, with different scientists disagreeing about what counts as a theory, what counts as a test of a theory, what even counts as evidence. We have reached that point with the mass discrepancy problem.

For many years, I worried that if the field ever caught up with me, it would zoom past. That hasn’t happened. Instead, it has diverged towards a place that I barely recognize as science. It looks more like the Matrix – a simulation – that is increasingly sophisticated yet self-contained, making only parsimonious contact with observational reality and unable to make predictions that apply to real objects. Scaling relations and statistical properties, sure. Actual galaxies with NGC numbers, not so much. That, to me, is not science.

I have found it increasingly difficult to communicate across the gap built on presumptions buried so deep that they cannot be questioned. One obvious one is the existence of dark matter. This has been fueled by cosmologists who take it for granted and particle physicists eager to discover it who repeat “we know dark matter exists*; we just need to find it” like a religious mantra. This is now ingrained so deeply that it has become difficult to convey even the simple concept that what we call “dark matter” is really just evidence of a discrepancy: we do not know whether it is literally some kind of invisible mass, or a breakdown of the equations that lead us to infer invisible mass.

I try to look at all sides of a problem. I can say nice things about dark matter (and cosmology); I can point out problems with it. I can say nice things about MOND; I can point out problems with it. The more common approach is to presume that any failing of MOND is an automatic win for dark matter. This is a simple-minded logical fallacy: just because MOND gets something wrong doesn’t mean dark matter gets it right. Indeed, my experience has been that cases that don’t make any sense in MOND don’t make any sense in terms of dark matter either. Nevertheless, this attitude persists.

*I made this flowchart as a joke in 2012, but it persists in being an uncomfortably fair depiction of how many people who work on dark matter approach the problem.*

I don’t know what is right, but I’m pretty sure this attitude is wrong. Indeed, it empowers a form of magical thinking: dark matter has to be correct, so any data that appear to contradict it are either wrong, or can be explained with feedback. Indeed, the usual trajectory has been denial first (that can’t be true!) and explanation later (we knew it all along!) This attitude is an existential threat to the scientific method, and I am despondent in part because I worry we are slipping into a post-scientific reality, where even scientists are little more than priests of a cold, dark religion.

*If we’re sure dark matter exists, it is not obvious that we need to be doing expensive experiments to find it.

Why bother?

Divergence

Reality check

Before we can agree on the interpretation of a set of facts, we have to agree on what those facts are. Even if we agree on the facts, we can differ about their interpretation. It is OK to disagree, and anyone who practices astrophysics is going to be wrong from time to time. It is the inevitable risk we take in trying to understand a universe that is vast beyond human comprehension. Heck, some people have made successful careers out of being wrong. This is OK, so long as we recognize and correct our mistakes. That’s a painful process, and there is an urge in human nature to deny such things, to pretend they never happened, or to assert that what was wrong was right all along.

This happens a lot, and it leads to a lot of weirdness. Beyond the many people in the field whom I already know personally, I tend to meet two kinds of scientists. There are those (usually other astronomers and astrophysicists) who might be familiar with my work on low surface brightness galaxies or galaxy evolution or stellar populations or the gas content of galaxies or the oxygen abundances of extragalactic HII regions or the Tully-Fisher relation or the cusp-core problem or faint blue galaxies or big bang nucleosynthesis or high redshift structure formation or joint constraints on cosmological parameters. These people behave like normal human beings. Then there are those (usually particle physicists) who have only heard of me in the context of MOND. These people often do not behave like normal human beings. They conflate me as a person with a theory that is Milgrom’s. They seem to believe that both are evil and must be destroyed. My presence, even the mere mention of my name, easily destabilizes their surprisingly fragile grasp on sanity.

One of the things that scientists-gone-crazy do is project their insecurities about the dark matter paradigm onto me. People who barely know me frequently attribute to me motivations that I neither have nor recognize. They presume that I have some anti-cosmology, anti-DM, pro-MOND agenda, and are remarkably comfortably about asserting to me what it is that I believe. What they never explain, or apparently bother to consider, is why I would be so obtuse? What is my motivation? I certainly don’t enjoy having the same argument over and over again with their ilk, which is the only thing it seems to get me.

The only agenda I have is a pro-science agenda. I want to know how the universe works.

This agenda is not theory-specific. In addition to lots of other astrophysics, I have worked on both dark matter and MOND. I will continue to work on both until we have a better understanding of how the universe works. Right now we’re very far away from obtaining that goal. Anyone who tells you otherwise is fooling themselves – usually by dint of ignoring inconvenient aspects of the evidence. Everyone is susceptible to cognitive dissonance. Scientists are no exception – I struggle with it all the time. What disturbs me is the number of scientists who apparently do not. The field is being overrun with posers who lack the self-awareness to question their own assumptions and biases.

So, I feel like I’m repeating myself here, but let me state my bias. Oh wait. I already did. That’s why it felt like repetition. It is.

The following bit of this post is adapted from an old web page I wrote well over a decade ago. I’ve lost track of exactly when – the file has been through many changes in computer systems, and unix only records the last edit date. For the linked page, that’s 2016, when I added a few comments. The original is much older, and was written while I was at the University of Maryland. Judging from the html style, it was probably early to mid-’00s. Of course, the sentiment is much older, as it shouldn’t need to be said at all.

I will make a few updates as seem appropriate, so check the link if you want to see the changes. I will add new material at the end.

Long standing remarks on intellectual honesty

The debate about MOND often degenerates into something that falls well short of the sober, objective discussion that is suppose to characterize scientific debates. One can tell when voices are raised and baseless ad hominem accusations made. I have, with disturbing frequency, found myself accused of partisanship and intellectual dishonesty, usually by people who are as fair and balanced as Fox News.

Let me state with absolute clarity that intellectual honesty is a bedrock principle of mine. My attitude is summed up well by the quote

When a man lies, he murders some part of the world.
Paul Gerhardt

I first heard this spoken by the character Merlin in the movie Excalibur (1981 version). Others may have heard it in a song by Metallica. As best I can tell, it is originally attributable to the 17th century cleric Paul Gerhardt.

This is a great quote for science, as the intent is clear. We don’t get to pick and choose our facts. Outright lying about them is antithetical to science.

I would extend this to ignoring facts. One should not only be honest, but also as complete as possible. It does not suffice to be truthful while leaving unpleasant or unpopular facts unsaid. This is lying by omission.

I “grew up” believing in dark matter. Specifically, Cold Dark Matter, presumably a WIMP. I didn’t think MOND was wrong so much as I didn’t think about it at all. Barely heard of it; not worth the bother. So I was shocked – and angered – when it its predictions came true in my data for low surface brightness galaxies. So I understand when my colleagues have the same reaction.

Nevertheless, Milgrom got the prediction right. I had a prediction, it was wrong. There were other conventional predictions, they were also wrong. Indeed, dark matter based theories generically have a very hard time explaining these data. In a Bayesian sense, given the prior that we live in a ΛCDM universe, the probability that MONDian phenomenology would be observed is practically zero. Yet it is. (This is very well established, and has been for some time.)

So – confronted with an unpopular theory that nevertheless had some important predictions come true, I reported that fact. I could have ignored it, pretended it didn’t happen, covered my eyes and shouted LA LA LA NOT LISTENING. With the benefit of hindsight, that certainly would have been the savvy career move. But it would also be ignoring a fact, and tantamount to a lie.

In short, though it was painful and protracted, I changed my mind. Isn’t that what the scientific method says we’re suppose to do when confronted with experimental evidence?

That was my experience. When confronted with evidence that contradicted my preexisting world view, I was deeply troubled. I tried to reject it. I did an enormous amount of fact-checking. The people who presume I must be wrong have not had this experience, and haven’t bothered to do any fact-checking. Why bother when you already are sure of the answer?

Willful Ignorance

I understand being skeptical about MOND. I understand being more comfortable with dark matter. That’s where I started from myself, so as I said above, I can empathize with people who come to the problem this way. This is a perfectly reasonable place to start.

For me, that was over a quarter century ago. I can understand there being some time lag. That is not what is going on. There has been ample time to process and assimilate this information. Instead, most physicists have chosen to remain ignorant. Worse, many persist in spreading what can only be described as misinformation. I don’t think they are liars; rather, it seems that they believe their own bullshit.

To give an example of disinformation, I still hear said things like “MOND fits rotation curves but nothing else.” This is not true. The first thing I did was check into exactly that. Years of fact-checking went into McGaugh & de Blok (1998), and I’ve done plenty more since. It came as a great surprise to me that MOND explained the vast majority of the data as well or better than dark matter. Not everything, to be sure, but lots more than “just” rotation curves. Yet this old falsehood still gets repeated as if it were not a misconception that was put to rest in the previous century. We’re stuck in the dark ages by choice.

It is not a defensible choice. There is no excuse to remain ignorant of MOND at this juncture in the progress of astrophysics. It is incredibly biased to point to its failings without contending with its many predictive successes. It is tragi-comically absurd to assume that dark matter provides a better explanation when it cannot make the same predictions in advance. MOND may not be correct in every particular, and makes no pretense to be a complete theory of everything. But it is demonstrably less wrong than dark matter when it comes to predicting the dynamics of systems in the low acceleration regime. Pretending like this means nothing is tantamount to ignoring essential facts.

Even a lie of omission murders a part of the world.

25 years a heretic

People seem to like to do retrospectives at year’s end. I take a longer view, but the end of 2020 seems like a fitting time to do that. Below is the text of a paper I wrote in 1995 with collaborators at the Kapteyn Institute of the University of Groningen. The last edit date is from December of that year, so this text (in plain TeX, not LaTeX!) is now a quarter century old. I am just going to cut & paste it as-was; I even managed to recover the original figures and translate them into something web-friendly (postscript to jpeg). This is exactly how it was.

This was my first attempt to express in the scientific literature my concerns for the viability of the dark matter paradigm, and my puzzlement that the only theory to get any genuine predictions right was MOND. It was the hardest admission in my career that this could be even a remote possibility. Nevertheless, intellectual honesty demanded that I report it. To fail to do so would be an act of reality denial antithetical to the foundational principles of science.

It was never published. There were three referees. Initially, one was positive, one was negative, and one insisted that rotation curves weren’t flat. There was one iteration; this is the resubmitted version in which the concerns of the second referee were addressed to his apparent satisfaction by making the third figure a lot more complicated. The third referee persisted that none of this was valid because rotation curves weren’t flat. Seems like he had a problem with something beyond the scope of this paper, but the net result was rejection.

One valid concern that ran through the refereeing process from all sides was “what about everything else?” This is a good question that couldn’t fit into a short letter like this. Thanks to the support of Vera Rubin and a Carnegie Fellowship, I spent the next couple of years looking into everything else. The results were published in 1998 in a series of three long papers: one on dark matter, one on MOND, and one making detailed fits.

This had started from a very different place intellectually with my efforts to write a paper on galaxy formation that would have been similar to contemporaneous papers like Dalcanton, Spergel, & Summers and Mo, Mao, & White. This would have followed from my thesis and from work with Houjun Mo, who was an office mate when we were postdocs at the IoA in Cambridge. (The ideas discussed in Mo, McGaugh, & Bothun have been reborn recently in the galaxy formation literature under the moniker of “assembly bias.”) But I had realized by then that my ideas – and those in the papers cited – were wrong. So I didn’t write a paper that I knew to be wrong. I wrote this one instead.

Nothing substantive has changed since. Reading it afresh, I’m amazed how many of the arguments over the past quarter century were anticipated here. As a scientific community, we are stuck in a rut, and seem to prefer to spin the wheels to dig ourselves in deeper than consider the plain if difficult path out.

Testing hypotheses of dark matter and alternative gravity with low surface density galaxies

The missing mass problem remains one of the most vexing in astrophysics. Observations clearly indicate either the presence of a tremendous amount of as yet unidentified dark matter^1,2, or the need to modify the law of gravity^3-7. These hypotheses make vastly different predictions as a function of density. Observations of the rotation curves of galaxies of much lower surface brightness than previously studied therefore provide a powerful test for discriminating between them. The dark matter hypothesis requires a surprisingly strong relation between the surface brightness and mass to light ratio⁸, placing stringent constraints on theories of galaxy formation and evolution. Alternatively, the observed behaviour is predicted⁴ by one of the hypothesised alterations of gravity known as modified Newtonian dynamics^3,5 (MOND).

Spiral galaxies are observed to have asymptotically flat [i.e., V(R) ~ constant for large R] rotation curves that extend well beyond their optical edges. This trend continues for as far (many, sometimes > 10 galaxy scale lengths) as can be probed by gaseous tracers^1,2 or by the orbits of satellite galaxies⁹. Outside a galaxy’s optical radius, the gravitational acceleration is a_N = GM/R² = V²/R so one expects V(R) ~ R^-1/2. This Keplerian behaviour is not observed in galaxies.

One approach to this problem is to increase M in the outer parts of galaxies in order to provide the extra gravitational acceleration necessary to keep the rotation curves flat. Indeed, this is the only option within the framework of Newtonian gravity since both V and R are directly measured. The additional mass must be invisible, dominant, and extend well beyond the optical edge of the galaxies.

Postulating the existence of this large amount of dark matter which reveals itself only by its gravitational effects is a radical hypothesis. Yet the kinematic data force it upon us, so much so that the existence of dark matter is generally accepted. Enormous effort has gone into attempting to theoretically predict its nature and experimentally verify its existence, but to date there exists no convincing detection of any hypothesised dark matter candidate, and many plausible candidates have been ruled out¹⁰.

Another possible solution is to alter the fundamental equation a_N = GM/R². Our faith in this simple equation is very well founded on extensive experimental tests of Newtonian gravity. Since it is so fundamental, altering it is an even more radical hypothesis than invoking the existence of large amounts of dark matter of completely unknown constituent components. However, a radical solution is required either way, so both possibilities must be considered and tested.

A phenomenological theory specifically introduced to address the problem of the flat rotation curves is MOND³. It has no other motivation and so far there is no firm physical basis for the theory. It provides no satisfactory cosmology, having yet to be reconciled with General Relativity. However, with the introduction of one new fundamental constant (an acceleration a₀), it is empirically quite successful in fitting galaxy rotation curves^11-14. It hypothesises that for accelerations a < a₀ = 1.2 x 10^-10 m s^-2, the effective acceleration is given by a_eff = (a_N a₀)^1/2. This simple prescription works well with essentially only one free parameter per galaxy, the stellar mass to light ratio, which is subject to independent constraint by stellar evolution theory. More importantly, MOND makes predictions which are distinct and testable. One specific prediction⁴ is that the asymptotic (flat) value of the rotation velocity, V_a, is V_a = (GMa₀)^1/4. Note that V_a does not depend on R, but only on M in the regime of small accelerations (a < a₀).

In contrast, Newtonian gravity depends on both M and R. Replacing R with a mass surface density variable S = M(R)/R², the Newtonian prediction becomes M S ~ V_a⁴ which contrasts with the MOND prediction M ~ V_a⁴. These relations are the theoretical basis in each case for the observed luminosity-linewidth relation L ~ V_a⁴ (better known as the Tully-Fisher¹⁵ relation. Note that the observed value of the exponent is bandpass dependent, but does obtain the theoretical value of 4 in the near infrared¹⁶ which is considered the best indicator of the stellar mass. The systematic variation with bandpass is a very small effect compared to the difference between the two gravitational theories, and must be attributed to dust or stars under either theory.) To transform from theory to observation one requires the mass to light ratio Y: Y = M/L = S/s, where s is the surface brightness. Note that in the purely Newtonian case, M and L are very different functions of R, so Y is itself a strong function of R. We define Y to be the mass to light ratio within the optical radius R_*, as this is the only radius which can be measured by observation. The global mass to light ratio would be very different (since M ~ R for R > R_*, the total masses of dark haloes are not measurable), but the particular choice of definition does not affect the relevant functional dependences is all that matters. The predictions become Y²sL ~ V_a⁴ for Newtonian gravity^8,16 and YL ~ V_a⁴ for MOND⁴.

The only sensible¹⁷ null hypothesis that can be constructed is that the mass to light ratio be roughly constant from galaxy to galaxy. Clearly distinct predictions thus emerge if galaxies of different surface brightnesses s are examined. In the Newtonian case there should be a family of parallel Tully-Fisher relations for each surface brightness. In the case of MOND, all galaxies should follow the same Tully-Fisher relation irrespective of surface brightness.

Recently it has been shown that extreme objects such as low surface brightness galaxies^8,18 (those with central surface brightnesses fainter than s₀ = 23 B mag./[] corresponding 40 L_☉ pc^-2) obey the same Tully-Fisher relation as do the high surface brightness galaxies (typically with s₀ = 21.65 B mag./[] or 140 L_☉ pc^-2) which originally¹⁵ defined it. Fig. 1 shows the luminosity-linewidth plane for galaxies ranging over a factor of 40 in surface brightness. Regardless of surface brightness, galaxies fall on the same Tully-Fisher relation.

The luminosity-linewidth (Tully-Fisher) relation for spiral galaxies over a large range in surface brightness. The B-band relation is shown; the same result is obtained in all bands^8,18. Absolute magnitudes are measured from apparent magnitudes assuming H₀ = 75 km/s/Mpc. Rotation velocities V_a are directly proportional to observed 21 cm linewidths (measured as the full width at 20% of maximum) W₂₀ corrected for inclination [sin^-1(i)]. Open symbols are an independent sample which defines⁴² the Tully-Fisher relation (solid line). The dotted lines show the expected shift of the Tully-Fisher relation for each step in surface brightness away from the canonical value s₀ = 21.5 if the mass to light ratio remains constant. Low surface brightness galaxies are plotted as solid symbols, binned by surface brightness: red triangles: 22 < s₀ < 23; green squares: 23 < s₀ < 24; blue circles: s₀ > 24. One galaxy with two independent measurements is connected by a line. This gives an indication of the typical uncertainty which is sufficient to explain nearly all the scatter. Contrary to the clear expectation of a readily detectable shift as indicated by the dotted lines, galaxies fall on the same Tully-Fisher relation regardless of surface brightness, as predicted by MOND.

MOND predicts this behaviour in spite of the very different surface densities of low surface brightness galaxies. In order to understand this observational fact in the framework of standard Newtonian gravity requires a subtle relation⁸ between surface brightness and the mass to light ratio to keep the product sY² constant. If we retain normal gravity and the dark matter hypothesis, this result is unavoidable, and the null hypothesis of similar mass to light ratios (which, together with an assumed constancy of surface brightness, is usually invoked to explain the Tully-Fisher relation) is strongly rejected. Instead, the current epoch surface brightness is tightly correlated with the properties of the dark matter halo, placing strict constraints on models of galaxy formation and evolution.

The mass to light ratios computed for both cases are shown as a function of surface brightness in Fig. 2. Fig. 2 is based solely on galaxies with full rotation curves^19,20 and surface photometry, so V_a and R_* are directly measured. The correlation in the Newtonian case is very clear (Fig. 2a), confirming our inference⁸ from the Tully-Fisher relation. Such tight correlations are very rare in extragalactic astronomy, and the Y-s relation is probably the real cause of an inferred Y-L relation. The latter is much weaker because surface brightness and luminosity are only weakly correlated^21-24.

The mass to light ratio Y (in M_☉/L_☉) determined with (a) Newtonian dynamics and (b) MOND, plotted as a function of central surface brightness. The mass determination for Newtonian dynamics is M = V² R_*/G and for MOND is M = V⁴/(G a₀). We have adopted as a consistent definition of the optical radius R_* four scale lengths of the exponential optical disc. This is where discs tend to have edges, and contains essentially all the light^21,22. The definition of R_* makes a tremendous difference to the absolute value of the mass to light ratio in the Newtonian case, but makes no difference at all to the functional relation will be present regardless of the precise definition. These mass measurements are more sensitive to the inclination corrections than is the Tully-Fisher relation since there is a sin^-2(i) term in the Newtonian case and one of sin^-4(i) for MOND. It is thus very important that the inclination be accurately measured, and we have retained only galaxies which have adequate inclination determinations — error bars are plotted for a nominal uncertainty of 6 degrees. The sensitivity to inclination manifests itself as an increase in the scatter from (a) to (b). The derived mass is also very sensitive to the measured value of the asymptotic velocity itself, so we have used only those galaxies for which this can be taken directly from a full rotation curve^19,20,42. We do not employ profile widths; the velocity measurements here are independent of those in Fig. 1. In both cases, we have subtracted off the known atomic gas mass^19,20,42, so what remains is essentially only the stars and any dark matter that may exist. A very strong correlation (regression coefficient = 0.85) is apparent in (a): this is the mass to light ratio — surface brightness conspiracy. The slope is consistent (within the errors) with the theoretical expectation s ~ Y^-2 derived from the Tully-Fisher relation⁸. At the highest surface brightnesses, the mass to light ratio is similar to that expected for the stellar population. At the faintest surface brightnesses, it has increased by a factor of nearly ten, indicating increasing dark matter domination within the optical disc as surface brightness decreases or a very systematic change in the stellar population, or both. In (b), the mass to light ratio scatters about a constant value of 2. This mean value, and the lack of a trend, is what is expected for stellar populations^17,21-24.

The Y-s relation is not predicted by any dark matter theory^25,26. It can not be purely an effect of the stellar mass to light ratio, since no other stellar population indicator such as color^21-24 or metallicity^27,28 is so tightly correlated with surface brightness. In principle it could be an effect of the stellar mass fraction, as the gas mass to light ratio follows a relation very similar to that of total mass to light ratio²⁰. We correct for this in Fig. 2 by subtracting the known atomic gas mass so that Y refers only to the stars and any dark matter. We do not correct for molecular gas, as this has never been detected in low surface brightness galaxies to rather sensitive limits³⁰ so the total mass of such gas is unimportant if current estimates³¹ of the variation of the CO to H₂ conversion factor with metallicity are correct. These corrections have no discernible effect at all in Fig. 2 because the dark mass is totally dominant. It is thus very hard to see how any evolutionary effect in the luminous matter can be relevant.

In the case of MOND, the mass to light ratio directly reflects that of the stellar population once the correction for gas mass fraction is made. There is no trend of Y* with surface brightness (Fig. 2b), a more natural result and one which is consistent with our studies of the stellar populations of low surface brightness galaxies^21-23. These suggest that Y* should be roughly constant or slightly declining as surface brightness decreases, with much scatter. The mean value Y* = 2 is also expected from stellar evolutionary theory¹⁷, which always gives a number 0 < Y* < 10 and usually gives 0.5 < Y* < 3 for disk galaxies. This is particularly striking since Y* is the only free parameter allowed to MOND, and the observed mean is very close to that directly observed²⁹ in the Milky Way (1.7 ± 0.5 M_☉/L_☉).

The essence of the problem is illustrated by Fig. 3, which shows the rotation curves of two galaxies of essentially the same luminosity but vastly different surface brightnesses. Though the asymptotic velocities are the same (as required by the Tully-Fisher relation), the rotation curve of the low surface brightness galaxy rises less quickly than that of the high surface brightness galaxy as expected if the mass is distributed like the light. Indeed, the ratio of surface brightnesses is correct to explain the ratio of velocities at small radii if both galaxies have similar mass to light ratios. However, if this continues to be the case as R increases, the low surface brightness galaxy should reach a lower asymptotic velocity simply because R_* must be larger for the same L. That this does not occur is the problem, and poses very significant systematic constraints on the dark matter distribution.

The rotation curves of two galaxies, one of high surface brightness¹¹ (NGC 2403; open circles) and one of low surface brightness¹⁹ (UGC 128; filled circles). The two galaxies have very nearly the same asymptotic velocity, and hence luminosity, as required by the Tully-Fisher relation. However, they have central surface brightnesses which differ by a factor of 13. The lines give the contributions to the rotation curves of the various components. Green: luminous disk. Blue: dark matter halo. Red: luminous disk (stars and gas) with MOND. Solid lines refer to NGC 2403 and dotted lines to UGC 128. The fits for NGC 2403 are taken from ref. 11, for which the stars have Y* = 1.5 *M_☉/L_☉*. For UGC 128, no specific fit is made: the blue and green dotted lines are simply the NGC 2403 fits scaled by the ratio of disk scale lengths h. This provides a remarkably good description of the UGC 128 rotation curve and illustrates one possible manifestation of the fine tuning problem: if disks have similar Y, the halo parameters p₀ and R₀ must scale with the disk parameters s₀ and h while conspiring to keep the product *p₀ R₀*² fixed at any given luminosity. Note also that the halo of NGC 2403 gives an adequate fit to the rotation curve of UGC 128. This is another possible manifestation of the fine tuning problem: all galaxies of the same luminosity have the same halo, with Y systematically varying with s₀ so that Y* goes to zero as s₀ goes to zero. Neither of these is exactly correct because the contribution of the gas can not be set to zero as is mathematically possible with the stars. This causes the resulting fin tuning problems to be even more complex, involving more parameters. Alternatively, the green dotted line is the rotation curve expected by MOND for a galaxy with the observed luminous mass distribution of UGC 128.

Satisfying the Tully-Fisher relation has led to some expectation that haloes all have the same density structure. This simplest possibility is immediately ruled out. In order to obtain L ~ V_a⁴ ~ MS, one might suppose that the mass surface density S is constant from galaxy to galaxy, irrespective of the luminous surface density s. This achieves the correct asymptotic velocity V_a, but requires that the mass distribution, and hence the complete rotation curve, be essentially identical for all galaxies of the same luminosity. This is obviously not the case (Fig. 3), as the rotation curves of lower surface brightness galaxies rise much more gradually than those of higher surface brightness galaxies (also a prediction⁴ of MOND). It might be possible to have approximately constant density haloes if the highest surface brightness disks are maximal and the lowest minimal in their contribution to the inner parts of the rotation curves, but this then requires fine tuning of Y* with this systematically decreasing with surface brightness.

The expected form of the halo mass distribution depends on the dominant form of dark matter. This could exist in three general categories: baryonic (e.g., MACHOs), hot (e.g., neutrinos), and cold exotic particles (e.g., WIMPs). The first two make no specific predictions. Baryonic dark matter candidates are most subject to direct detection, and most plausible candidates have been ruled out¹⁰ with remaining suggestions of necessity sounding increasingly contrived³². Hot dark matter is not relevant to the present problem. Even if neutrinos have a small mass, their velocities considerably exceed the escape velocities of the haloes of low mass galaxies where the problem is most severe. Cosmological simulations involving exotic cold dark matter^33,34 have advanced to the point where predictions are being made about the density structure of haloes. These take the form^33,34 p(R) = p_H/[R(R+R_H)^b] where p_H characterises the halo density and R_H its radius, with b ~ 2 to 3. The characteristic density depends on the mean density of the universe at the collapse epoch, and is generally expected to be greater for lower mass galaxies since these collapse first in such scenarios. This goes in the opposite sense of the observations, which show that low mass and low surface brightness galaxies are less, not more, dense. The observed behaviour is actually expected in scenarios which do not smooth on a particular mass scale and hence allow galaxies of the same mass to collapse at a variety of epochs²⁵, but in this case the Tully-Fisher relation should not be universal. Worse, note that at small R < R_H, p(R) ~ R^-1. It has already been noted^32,35 that such a steep interior density distribution is completely inconsistent with the few (4) analysed observations of dwarf galaxies. Our data^19,20 confirm and considerably extend this conclusion for 24 low surface brightness galaxies over a wide range in luminosity.

The failure of the predicted exotic cold dark matter density distribution either rules out this form of dark matter, indicates some failing in the simulations (in spite of wide-spread consensus), or requires some mechanism to redistribute the mass. Feedback from star formation is usually invoked for the last of these, but this can not work for two reasons. First, an objection in principle: a small mass of stars and gas must have a dramatic impact on the distribution of the dominant dark mass, with which they can only interact gravitationally. More mass redistribution is required in less luminous galaxies since they start out denser but end up more diffuse; of course progressively less baryonic material is available to bring this about as luminosity declines. Second, an empirical objection: in this scenario, galaxies explode and gas is lost. However, progressively fainter and lower surface brightness galaxies, which need to suffer more severe explosions, are actually very gas rich.

Observationally, dark matter haloes are inferred to have density distributions^1,2,11 with constant density cores, p(R) = p₀/[1 + (R/R₀)^g]. Here, p₀ is the core density and R₀ is the core size with g ~ 2 being required to produce flat rotation curves. For g = 2, the rotation curve resulting from this mass distribution is V(R) = V_a [1-(R₀/R) tan^-1({R/R₀)]^1/2 where the asymptotic velocity is V_a = (4πG p₀ R₀²)^1/2. To satisfy the Tully-Fisher relation, V_a, and hence the product p₀ R₀², must be the same for all galaxies of the same luminosity. To decrease the rate of rise of the rotation curves as surface brightness decreases, R₀ must increase. Together, these two require a fine tuning conspiracy to keep the product p₀ R₀² constant while R₀ must vary with the surface brightness at a given luminosity. Luminosity and surface brightness themselves are only weakly correlated, so there exists a wide range in one parameter at any fixed value of the other. Thus the structural properties of the invisible dark matter halo dictate those of the luminous disk, or vice versa. So, s and L give the essential information about the mass distribution without recourse to kinematic information.

A strict s-p₀-R₀ relation is rigorously obeyed only if the haloes are spherical and dominate throughout. This is probably a good approximation for low surface brightness galaxies but may not be for the those of the highest surface brightness. However, a significant non-halo contribution can at best replace one fine tuning problem with another (e.g., surface brightness being strongly correlated with the stellar population mass to light ratio instead of halo core density) and generally causes additional conspiracies.

There are two perspectives for interpreting these relations, with the preferred perspective depending strongly on the philosophical attitude one has towards empirical and theoretical knowledge. One view is that these are real relations which galaxies and their haloes obey. As such, they provide a positive link between models of galaxy formation and evolution and reality.

The other view is that this list of fine tuning requirements makes it rather unattractive to maintain the dark matter hypothesis. MOND provides an empirically more natural explanation for these observations. In addition to the Tully-Fisher relation, MOND correctly predicts the systematics of the shapes of the rotation curves of low surface brightness galaxies^19,20 and fits the specific case of UGC 128 (Fig. 3). Low surface brightness galaxies were stipulated⁴ to be a stringent test of the theory because they should be well into the regime a < a₀. This is now observed to be true, and to the limit of observational accuracy the predictions of MOND are confirmed. The critical acceleration scale a₀ is apparently universal, so there is a single force law acting in galactic disks for which MOND provides the correct description. The cause of this could be either a particular dark matter distribution³⁶ or a real modification of gravity. The former is difficult to arrange, and a single force law strongly supports the latter hypothesis since in principle the dark matter could have any number of distributions which would give rise to a variety of effective force laws. Even if MOND is not correct, it is essential to understand why it so closely describe the observations. Though the data can not exclude Newtonian dynamics, with a working empirical alternative (really an extension) at hand, we would not hesitate to reject as incomplete any less venerable hypothesis.

Nevertheless, MOND itself remains incomplete as a theory, being more of a Kepler’s Law for galaxies. It provides only an empirical description of kinematic data. While successful for disk galaxies, it was thought to fail in clusters of galaxies³⁷. Recently it has been recognized that there exist two missing mass problems in galaxy clusters, one of which is now solved³⁸: most of the luminous matter is in X-ray gas, not galaxies. This vastly improves the consistency of MOND with with cluster dynamics³⁹. The problem with the theory remains a reconciliation with Relativity and thereby standard cosmology (which is itself in considerable difficulty^38,40), and a lack of any prediction about gravitational lensing⁴¹. These are theoretical problems which need to be more widely addressed in light of MOND’s empirical success.

ACKNOWLEDGEMENTS. We thank R. Sanders and M. Milgrom for clarifying aspects of a theory with which we were previously unfamiliar. SSM is grateful to the Kapteyn Astronomical Institute for enormous hospitality during visits when much of this work was done. [Note added in 2020: this work was supported by a cooperative grant funded by the EU and would no longer be possible thanks to Brexit.]

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Oh… you don’t want to look in there

This post is a recent conversation with David Garofalo for his blog.

Today we talk to Dr. Stacy McGaugh, Chair of the Astronomy Department at Case Western Reserve University.

David: Hi Stacy. You had set out to disprove MOND and instead found evidence to support it. That sounds like the poster child for how science works. Was praise forthcoming?

Stacy: In the late 1980s and into the 1990s, I set out to try to understand low surface brightness galaxies. These are diffuse systems of stars and gas that rotate like the familiar bright spirals, but whose stars are much more spread out. Why? How did these things come to be? Why were they different from brighter galaxies? How could we explain their properties? These were the problems I started out working on that inadvertently set me on a collision course with MOND.

I did not set out to prove or disprove either MOND or dark matter. I was not really even aware of MOND at that time. I had head of it only on a couple of occasions, but I hadn’t payed any attention, and didn’t really know anything about it. Why would I bother? It was already well established that there had to be dark matter.

I worked to develop our understanding of low surface brightness galaxies in the context of dark matter. Their blue colors, low metallicities, high gas fractions, and overall diffuse nature could be explained if they had formed in dark matter halos that are themselves lower than average density: they occupy the low concentration side of the distribution of dark matter halos at a given mass. I found this interpretation quite satisfactory, so gave me no cause to doubt dark matter to that point.

This picture made two genuine predictions that had yet to be tested. First, low surface brightness galaxies should be less strongly clustered than brighter galaxies. Second, having their mass spread over a larger area, they should shift off of the Tully-Fisher relation defined by denser galaxies. The first prediction came true, and for a period I was jubilant that we had made an important new contribution to out understanding of both galaxies and dark matter. The second prediction failed badly: low surface brightness galaxies adhere to the same Tully-Fisher relation that other galaxies follow.

I tried desperately to understand the failure of the second prediction in terms of dark matter. I tried what seemed like a thousand ways to explain this, but ultimately they were all tautological: I could only explain it if I assumed the answer from the start. The adherence of low surface brightness galaxies to the Tully-Fisher relation poses a serious fine-tuning problem: the distribution of dark matter must be adjusted to exactly counterbalance that of the visible matter so as not to leave any residuals. This makes no sense, and anyone who claims it does is not thinking clearly.

It was in this crisis of comprehension in which I became aware that MOND predicted exactly what I was seeing. No fine-tuning was required. Low surface brightness galaxies followed the same Tully-Fisher relation as other galaxies because the modified force law stipulates that they must. It was only at this point (in the mid-’90s) at which I started to take MOND seriously. If it had got this prediction right, what else did it predict?

I was still convinced that the right answer had to be dark matter. There was, after all, so much evidence for it. So this one prediction must be a fluke; surely it would fail the next test. That was not what happened: MOND passed test after test after test, successfully predicting observations both basic and detailed that dark matter theory got wrong or did not even address. It was only after this experience that I realized that what I thought was evidence for dark matter was really just evidence that something was wrong: the data cannot be explained with ordinary gravity without invisible mass. The data – and here I mean ALL the data – were mostly ambiguous: they did not clearly distinguish whether the problem was with mass we couldn’t see or with the underlying equations from which we inferred the need for dark matter.

So to get back to your original question, yes – this is how science should work. I hadn’t set out to test MOND, but I had inadvertently performed exactly the right experiment for that purpose. MOND had its predictions come true where the predictions of other theories did not: both my own theory and those of others who were working in the context of dark matter. We got it wrong while MOND got it right. That led me to change my mind: I had been wrong to be sure the answer had to be dark matter, and to be so quick to dismiss MOND. Admitting this was the most difficult struggle I ever faced in my career.

David: From the perspective of dark matter, how does one understand MOND’s success?

Stacy: One does not.

That the predictions of MOND should come true in a universe dominated by dark matter makes no sense.

Before I became aware of MOND, I spent lots of time trying to come up with dark matter-based explanations for what I was seeing. It didn’t work. Since then, I have continued to search for a viable explanation with dark matter. I have not been successful. Others have claimed such success, but whenever I look at their work, it always seems that what they assert to be a great success is just a specific elaboration of a model I had already considered and rejected as obviously unworkable. The difference boils down to Occam’s razor. If you give dark matter theory enough free parameters, it can be adjusted to “predict” pretty much anything. But the best we can hope to do with dark matter theory is to retroactively explain what MOND successfully predicted in advance. Why should we be impressed by that?

David: Does MOND fail in clusters?

Stacy: Yes and no: there are multiple tests in clusters. MOND passes some and flunks others – as does dark matter.

The most famous test is the baryon fraction. This should be one in MOND – all the mass is normal baryonic matter. With dark matter, it should be the cosmic ratio of normal to dark matter (about 1:5).

MOND fails this test: it explains most of the discrepancy in clusters, but not all of it. The dark matter picture does somewhat better here, as the baryon fraction is close to the cosmic expectation — at least for the richest clusters of galaxies. In smaller clusters and groups of galaxies, the normal matter content falls short of the cosmic value. So both theories suffer a “missing baryon” problem: MOND in rich clusters; dark matter in everything smaller.

Another test is the mass-temperature relation. Both theories predict a relation between the mass of a cluster and the temperature of the gas it contains, but they predict different slopes for this relation. MOND gets the slope right but the amplitude wrong, leading to the missing baryon problem above. Dark matter gets the amplitude right for the most massive clusters, but gets the slope wrong – which leads to it having a missing baryon problem for systems smaller than the largest clusters.

There are other tests. Clusters continue to merge; the collision velocity of merging clusters is predicted to be higher in MOND than with dark matter. For example, the famous bullet cluster, which is often cited as a contradiction to MOND, has a collision speed that is practically impossible with dark matter: there just isn’t enough time for the two components of the bullet to accelerate up to the observed relative speed if they fall together under the influence of normal gravity and the required amount of dark mass. People have argued over the severity of this perplexing problem, but the high collision speed happens quite naturally in MOND as a consequence of its greater effective force of attraction. So, taken at face value, the bullet cluster both confirms and refutes both theories!

I could go on… one expects clusters to form earlier and become more massive in MOND than in dark matter. There are some indications that this is the case – the highest redshift clusters came as a surprise to conventional structure formation theory – but the relative numbers of clusters as a function of mass seems to agree well with current expectations with dark matter. So clusters are a mixed bag.

More generally, there is a widespread myth that MOND fits rotation curves, but gets nothing else right. This is what I expected to find when I started fact checking, but the opposite is true. MOND explains a huge variety of data well. The presumptive superiority of dark matter is just that – a presumption.

David: At a physics colloquium two decades ago, Vera Rubin described how theorists were willing and eager to explain her data to her. At an astronomy colloquium a few years later, you echoed that sentiment in relation to your data on velocity curves. One concludes that theorists are uniquely insightful and generous people. Is there anyone you would like to thank for putting you straight?

Stacy: So they perceive themselves to be.

MOND has made many successful a priori predictions. This is the golden standard of the scientific method. If there is another explanation for it, I’d like to know what it is.

As your questions supposes, many theorists have offered such explanations. At most one of them can be correct. I have yet to hear a satisfactory explanation.

David: What are MOND people working on these days?

Stacy: Any problem that is interesting in extragalactic astronomy is interesting in the context of MOND. Outstanding questions include planes of satellite dwarf galaxies, clusters of galaxies, the formation of large scale structure, and the microwave background. MOND-specific topics include the precise value of the MOND acceleration constant, predicting the velocity dispersions of dwarf galaxies, and the search for the predicted external field effect, which is a unique signature of MOND.

The phrasing of this question raises a sociological issue. I don’t know what a “MOND person” is. Before now, I have only heard it used as a pejorative.

I am a scientist who has worked on many topics. MOND is just one of them. Does that make me a “MOND person”? I have also worked on dark matter, so am I also a “dark matter person”? Are these mutually exclusive?

I have attended conferences where I have heard people say ‘“MOND people” do this’ or ‘“MOND people” fail to do that.’ Never does the speaker of these words specify who they’re talking about: “MOND people” are a nameless Other. In all cases, I am more familiar with the people and the research they pretend to describe, but in no way do I recognize what they’re talking about. It is just a way of saying “Those People” are Bad.

There are many experts on dark matter in the world. I am one of them. There are rather fewer experts on MOND. I am also one of them. Every one of these “MOND people” is also an expert on dark matter. This situation is not reciprocated: many experts on dark matter are shockingly ignorant about MOND. I was once guilty of that myself, but realized that ignorance is not a sound basis on which to base a scientific judgement.

David: Are you tired of getting these types of questions?

Stacy: Yes and no.

No, in that these are interesting questions about fundamental science. That is always fun to talk about.

Yes, in that I find myself having the same arguments over and over again, usually with scientists who remain trapped in the misconceptions I suffered myself a quarter century ago, but whose minds are closed to ideas that threaten their sacred cows. If dark matter is a real, physical substance, then show me a piece already.

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.

Triton Station

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

Category: Philosophy of Science

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Cosmology, then and now

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