When you have eliminated the impossible…

In previous posts, I briefly described some of the results that provoked a crisis of faith in the mid-1990s. Up until that point, I was an ardent believer in the cold dark matter paradigm. But it no longer made sense as an explanation for galaxy dynamics. It didn’t just not make sense, it seemed strewn with self-contradictions, all of which persist to this day.

Amidst this crisis of faith, there came a chance meeting in Middle-Earth: Moti Milgrom visited Cambridge, where I was a postdoc at the time, and gave a talk. I almost didn’t go to this talk because it had modified gravity in the title and who wanted to waste their time listening to that nonsense? I had yet to make any connection between the self-contradictions the data posed for dark matter and something as dire as an entirely different paradigm.

Despite my misgivings, I did go to Milgrom’s talk. Not knowing that I was there or what I worked on, he casually remarked on some specific predictions for low surface brightness galaxies. These sounded like what I was seeing, in particular the things that were most troublesome for the dark matter interpretation. I became interested.

Long story short, it is a case in which, had MOND not already existed, we would have had to invent it. As Sherlock Holmes famously put it

When you have eliminated the impossible, whatever remains, however improbable, must be the truth.
Sir Arthur Conan Doyle

Modified Newtonian Dynamics

There is one and only one theory that predicted in advance the observations described above: the Modified Newtonian Dynamics (MOND) introduced by Milgrom (1983a,b,c). MOND is an extension of Newtonian theory (Milgrom, 2020). It is not a generally covariant theory, so is not, by itself, a complete replacement for General Relativity. Nevertheless, it makes unique, testable predictions within its regime of applicability (McGaugh, 2020).

The basic idea of MOND is that the force law is modified at an acceleration scale, a0. For large accelerations, g ≫ a₀, everything is normal and Newtonian: g = g_N, where g_N is the acceleration predicted by the observed luminous mass distribution obtained by solving the Poisson equation. At low accelerations, the effective acceleration tends towards the limit

g → √(a₀g_N) for g ≪ a₀ (5)

(Bekenstein & Milgrom, 1984; Milgrom, 1983c). This limit is called the deep MOND regime in contrast to the Newtonian regime at high accelerations. The two regimes are smoothly connected by an interpolation function μ(g/a₀) that is not specified (Milgrom, 1983c).

The motivation to make an acceleration-based modification is to explain flat rotation curves (Bosma, 1981; Rubin et al., 1978) that also gives a steep Tully-Fisher relation similar to that which is observed (Aaronson et al., 1979). A test particle in a circular orbit around a point mass M_p in the deep MOND regime (eq. (5)) will experience a centripetal acceleration

V_c²/R = √(a₀GM_p/R²). (6)

Note that the term for the radius R cancels out, so eq. (6) reduces to

V_c⁴ = a₀GM_p (7)

which the reader will recognize as the Baryonic Tully-Fisher relation

M_b = A V_f⁴ (8)

with A = ζ/(a₀G) where ζ is a geometrical factor of order unity.

This simple math explains the flatness of rotation curves. This is not a prediction; it was an input that motivated the theory, as it motivated dark matter. Unlike dark matter, in which rotation curves might rise or fall, the rotation curves of isolated galaxies must tend towards asymptotic flatness.

MOND also explains the Tully-Fisher relation. Indeed, there are several distinct aspects to this prediction. That the relation exists at all is a strong prediction. Fundamentally, the Baryonic Tully-Fisher Relation (BTFR) is a relation between the baryonic mass of a galaxy and its flat rotation speed. There is no dark matter involved: V_f is not a property of a dark matter halo, but of the galaxy itself.

One MOND prediction is the slope of the BTFR: the power law scaling M ~ V^x has x = 4 exactly. While the infrared data of Aaronson et al. (1979) suggested such a slope, the exact value was not well constrained at that time. It was not until later that Tully-Fisher was empirically recognized as a relation driven by baryonic mass (McGaugh et al., 2000), as anticipated by MOND. Moreover, the slope is only four when a good measurement of the flat rotation velocity is available (Verheijen, 2001; McGaugh, 2005, 2012); common proxies like the line-width only crudely approximate the result and typically return shallower slopes (e.g., Zaritsky et al., 2014), as do samples of limited dynamic range (e.g., Pizagno et al., 2007). The latter are common in the literature: selection effects strongly favor bright galaxies, and the majority of published Tully-Fisher relations are dominated by high mass galaxies (M_∗ > 10¹⁰ M_⊙). Consequently, the behavior of the Baryonic Tully-Fisher relation remains somewhat controversial to this day (e.g., Mancera Piña et al., 2019; Ogle et al., 2019). This appears to be entirely a matter of data quality (McGaugh et al., 2019). The slope of the relation is indistinguishable from 4 when a modicum of quality control is imposed (Lelli et al., 2016b; McGaugh, 2005, 2012; Schombert et al., 2020; Stark et al., 2009; Trachternach et al., 2009). Indeed, only a slope of four successfully predicted the rotation speeds of low mass galaxies (Giovanelli et al., 2013; McGaugh, 2011).

Another aspect of the Tully-Fisher relation is its normalization. This is set by fundamental constants: Newton’s constant, G, and the acceleration scale of MOND, a0. For ζ = 0.8, A = 50 M_⊙ km⁻⁴ s⁴. However, there is no theory that predicts the value of a0, which has to be set by the data. Moreover, this scale is distance-dependent, so the precise value of a0 varies with adjustments to the distance scale. For this reason, in part, the initial estimate of a₀ = 2 × 10⁻¹⁰ m s⁻² of (Milgrom, 1983a) was a bit high. Begeman et al. (1991) used the best data then available to obtain a₀ = 1.2 × 10⁻¹⁰ m s⁻². The value of Milgrom’s acceleration constant has not varied meaningfully since then (Famaey and McGaugh, 2012; Li et al., 2018; McGaugh, 2011; McGaugh et al., 2016; Sanders and McGaugh, 2002). This is a consistency check, but not a genuine⁷ prediction.

An important consequence of MOND is that the Tully-Fisher relation is absolute: it should have no dependence on size or surface brightness (Milgrom, 1983a). The mass of baryons is the only thing that sets the flat amplitude of the rotation speed. It matters not at all how those baryons are distributed. MOND was the only theory to correctly predict this in advance of the observation (McGaugh and de Blok, 1998b). The fine-tuning problem that we face conventionally is imposed by this otherwise unanticipated result.

The absolute nature of the Tully-Fisher relation in MOND further predicts that it has no physical residuals whatsoever. That is to say, scatter around the relation can only be caused by observational errors and scatter in the mass-to-light ratios of the stars. The latter is an irreducible unknown: we measure the luminosity produced by the stars in a galaxy, but what we need to know is the mass of those stars. The conversion between them can never be perfect, and inevitably introduces some scatter into the relation. Nevertheless, we can make our best effort to account for known sources of scatter. Between scatter expected from observational uncertainties and that induced by variations in the mass-to-light ratio, the best data are consistent with the prediction of zero intrinsic scatter (McGaugh, 2005, 2012; Lelli et al., 2016b, 2019). Of course, it is impossible to measure zero, but it is possible to set an upper limit on the intrinsic scatter that is very tight by extragalactic standards (<6% Lelli et al., 2019). This leaves very little room for variations beyond the inevitable impact of the stellar mass-to-light ratio. The scatter is no longer entirely accounted for when lower quality data are considered (McGaugh, 2012), but this is expected in astronomy: lower quality data inevitably admit systematic uncertainties that are not readily accounted for in the error budget.

Milgrom (1983a) made a number of other specific predictions. In MOND, the acceleration expected for kinematics follows from the surface density of baryons. Consequently, low surface brightness means low acceleration. Interpreted in terms of conventional dynamics, the prediction is that the ratio of dynamical mass to light, M_dyn/L should increase as surface brightness decreases. This happens both globally — LSB galaxies appear to be more dark matter dominated than HSB galaxies (see Fig. 4(b) of McGaugh and de Blok, 1998a), and locally — the need for dark matter sets in at smaller radii in LSB galaxies than in HSB galaxies (Figs. 3 and 14 of McGaugh and de Blok, 1998b; Famaey and McGaugh, 2012, respectively).

One may also test this prediction by plotting the rotation curves of galaxies binned by surface brightness: acceleration should scale with surface brightness. It does (Figs. 4 and 16 of McGaugh and de Blok, 1998b; Famaey and McGaugh, 2012, respectively). This observation has been confirmed by near-infrared data. The systematic variation of color coded surface brightness is already obvious with optical data, as in Fig. 15 of Famaey and McGaugh (2012), but these suffer some scatter from variations in the stellar mass-to-light ratio. These practically vanish with near-infrared data, which provide such a good tracer of the surface mass density of stars that the equivalent plot is a near-perfect rainbow (Fig. 3 of both McGaugh et al., 2019; McGaugh, 2020). The data strongly corroborate the prediction of MOND that acceleration follows from baryonic surface density.

The central density relation (Fig. 6, Lelli et al., 2016c) was also predicted by MOND (Milgrom, 2016). Both the shape and the amplitude of the correlation are correct. Moreover, the surface density Σ_† at which the data bend follows directly from the acceleration scale of MOND: a₀ = GΣ_†. This surface density also corresponds to the stability limit for disks (Brada & Milgrom, 1999; Milgrom, 1989). The scale we had to insert by hand in dark matter models is a consequence of MOND.

Since MOND is a force law, the entirety of the rotation curve should follow from the baryonic mass distribution. The stellar mass-to-light ratio can modulate the amplitude of the stellar contribution to the rotation curve, but not its shape, which is specified by the observed distribution of light. Consequently, there is rather limited freedom in fitting rotation curves.

Example fits are shown in Fig. 8. The procedure is to construct Newtonian mass models by numerically solving the Poisson equation to determine the gravitational potential that corresponds to the observed baryonic mass distribution. Indeed, it is important to make a rigorous solution of the Poisson equation in order to capture details in the shape of the mass distribution (e.g., the wiggles in Fig. 8). Common analytic approximations like the exponential disk assume these features out of existence. Building proper mass models involves separate observations for the stars, conducted at optical or near-infrared wavelengths, and the gas of the interstellar medium, which is traced by radio wavelength observations. It is sometimes necessary to consider separate mass-to-light ratios for the stellar bulge and disk components, as there can be astrophysical differences between these distinct stellar populations (Baade, 1944). This distinction applies in any theory.

Fig. 8. Example rotation curve fits. MOND fits (heavy solid lines: Li et al., 2018) to the rotation curves of a bright, star-dominated galaxy (UGC 2953, left panel) and a faint, gas-dominated galaxy (DDO 64, right panel). The thin solid lines shows the Newtonian expectation, which is the sum of the atomic gas (dotted lines), stellar disk (dashed lines), and stellar bulge (dash-dotted line; present only in UGC 2953). Note the different scales: UGC 2953 is approximately 400 times more massive than DDO 64.

The gravitational potential of each baryonic component is represented by the circular velocity of a test particle in Fig. 8. The amplitude of the rotation curve of the mass model for each stellar component scales as the square root of its mass-to-light ratio. There is no corresponding mass-to-light ratio for the gas of the interstellar medium as there is a well-understood relation between the observed flux at 21 cm and the mass of hydrogen atoms that emit it (Draine, 2011). Consequently, the line for the gas components in Fig. 8 is practically fixed.

In addition to the mass-to-light ratio, there are two “nuisance” parameters that are sometimes considered in MOND fits: distance and inclination. These are known from independent observations, but of course these have some uncertainty. Consequently, the best MOND fit sometimes occurs for slightly different values of the distance and inclination, within their observational uncertainties (Begeman et al., 1991; de Blok & McGaugh, 1998; Sanders, 1996).

Distance matters because it sets the absolute scale. The further a galaxy, the greater its mass for the same observed flux. The distances to individual galaxies are notoriously difficult to measure. Though usually not important, small changes to the distance can occasionally have powerful effects, especially in gas rich galaxies. Compare, for example, the fit to DDO 154 by Li et al. (2018) to that of Ren et al. (2019).

Inclinations matter because we must correct the observed velocities for the inclination of each galaxy as projected on the sky. The inclination correction is V = V_obs/sin(i), so is small at large inclinations (edge-on) but large at small inclinations (face-on). For this reason, dynamical analyses often impose an inclination limit. This is an issue in any theory, but MOND is particularly sensitive since M ∝ V⁴ so any errors in the inclination are amplified to the fourth power (see Fig. 2 of de Blok & McGaugh, 1998). Worse, inclination estimates can suffer systematic errors (de Blok & McGaugh, 1998; McGaugh, 2012; Verheijen, 2001): a galaxy seen face-on may have an oval distortion that makes it look more inclined than it is, but it can’t be more face-on than face-on.

MOND fits will fail if either the distance or inclination is wrong. Such problems cannot be discerned in fits with dark matter halos, which have ample flexibility to absorb the imparted variance (see Fig. 6 of de Blok & McGaugh, 1998). Consequently, a fit with a dark matter halo will not fail if the distance happens to be wrong; we just won’t notice it.

MOND generally fits rotation curves well (Angus et al, 2012, 2015; Begeman et al., 1991; de Blok & McGaugh, 1998; Famaey and McGaugh, 2012; Gentile et al, 2010, 2011; Haghi et al., 2016; Hees et al., 2016; Kent, 1987; Li et al., 2018; Milgrom, 1988; Sánchez-Salcedo et al., 2013; Sanders, 1996, 2019; Sanders and McGaugh, 2002; Sanders and Verheijen, 1998; Swaters et al., 2010). There are of course exceptions (e.g, NGC 2915 Li et al., 2018). This is to be expected, as there are always some misleading data, especially in astronomy where it is impossible to control for systematic effects in the same manner that is possible in closed laboratories. It is easily forgotten that this type of analysis assumes circular orbits in a static potential, a condition that many spiral galaxies appear to have achieved to a reasonable approximation but which certainly will not hold in all cases.

The best-fit mass-to-light ratios found in MOND rotation curve fits can be checked against independent stellar population models. There is no guarantee that this procedure will return plausible values for the stellar mass-to-light ratio. Nevertheless, MOND fits recover the amplitude that is expected for stellar populations, the expected variation with color, and the band-dependent scatter (e.g., Fig. 28 of Famaey and McGaugh, 2012). Indeed, to a good approximation, the rotation curve can be predicted directly from near-infrared data (McGaugh, 2020; Sanders and Verheijen, 1998) modulo only the inevitable scatter in the mass-to-light ratio. This is a spectacular success of the paradigm that is not shared by dark matter fits (de Blok et al., 2003; de Blok & McGaugh, 1997; Kent, 1987).

Gas rich galaxies provide an even stronger test. When gas dominates the mass budget, the mass-to-light ratio of the stars ceases to have much leverage on the fit. There is no fitting parameter for gas equivalent to the mass-to-light ratio for stars: the gas mass follows directly from the observations. This enables MOND to predict the locations of such galaxies in the Baryonic Tully-Fisher plane (McGaugh, 2011) and essentially their full rotation curves (Sanders, 2019) with no free parameters (McGaugh, 2020).

It should be noted that the acceleration scale a0 is kept fixed when fitting rotation curves. If one allows a0 to vary, both it and the mass-to-light ratio spread over an unphysically large range of values (Li et al., 2018). The two are highly degenerate, causing such fits to be meaningless (Li et al., 2021): the data do not have the power to constrain multiple parameters per galaxy.

Table 2 lists the successful predictions of MOND that are discussed here. A more comprehensive list is given by Famaey and McGaugh (2012) and McGaugh (2020) who also discuss some of the problems posed for dark matter. MOND has had many predictive successes beyond rotation curves (e.g., McGaugh and Milgrom, 2013a,b; McGaugh, 2016) and has inspired successful predictions in cosmology (e.g., Sanders, 1998; McGaugh, 1999, 2000; Sanders, 2001; McGaugh, 2015, 2018). In this context, it makes sense to associate LSB galaxies with low density fluctuations in the initial conditions, thereby recovering the success of DD while its ills are cured by the modified force law. Galaxy formation in general is likely to proceed hierarchically but much more rapidly than in ΛCDM (Sanders, 2001; Stachniewicz and Kutschera, 2001), providing a natural explanation for both the age of stars in elliptical galaxies and allowing for a subsequent settling time for the disks of spiral galaxies (Wittenburg et al., 2020).

Prediction	Observation
Tully-Fisher Relation
Slope = 4	+
No size or surface brightness residuals	+
M_dyn/L depends on surface brightness	+
Central density relation	+
Rotation curve fits	+
Stellar population mass-to-light ratios	+
M_b alone specifies V_f	+

Table 2. Predictions of MOND.

The expert cosmologist may object that there is a great deal more data that must be satisfied. These have been reviewed elsewhere (Bekenstein, 2006; Famaey and McGaugh, 2012; McGaugh, 2015; Sanders and McGaugh, 2002) and are beyond the scope of this discussion. Here I note only that my experience has been that reports of MOND’s falsification are greatly exaggerated. Indeed, it has a great deal more explanatory power for a wider variety of phenomena than is generally appreciated (McGaugh and de Blok, 1998a,b).

The most serious, though certainly not the only, outstanding challenge to MOND is the dynamics of clusters of galaxies (Angus et al., 2008; Sanders and McGaugh, 2002). Contrary to the case in most individual galaxies and some groups of galaxies (Milgrom, 2018, 2019), MOND typically falls short of correcting the mass discrepancy in rich clusters by a factor of ~ 2 in mass. This can be taken as completely fatal, or as a being remarkably close by the standards of astrophysics. Which option one chooses seems to be mostly a matter of confirmation bias: those who are quick to dismiss MOND are happy to spot their own models a factor of two in mass, and even to assert that it is natural to do so (e.g., Ludlow et al., 2017). MOND is hardly alone in suffering problems with clusters of galaxies, which also present problems for ΛCDM (e.g., Angus & McGaugh, 2008; Asencio et al., 2021; Meneghetti et al., 2020).

A common fallacy seems to be that any failing of MOND is automatically considered to be support for ΛCDM. This is seldom the case. More often than not, observations that are problematic for MOND are also problematic for ΛCDM. We do not perceive them as such because we are already convinced that non-baryonic dark matter must exist. From that perspective, any problem encountered by ΛCDM is a mere puzzle that will inevitably be solved, while any problem encountered by MOND is a terminal failure of an irredeemably blasphemous hypothesis. This speaks volumes about human nature but says nothing about how the universe works.

The plain fact is that MOND made many a priori predictions that subsequently came true. This is the essence of the scientific method. LCDM and MOND are largely incommensurate, but whenever I have been able to make a straight comparison, MOND has been the more successful theory. So what am I supposed to say? That it is wrong? Perhaps it is, but that doesn’t make dark matter right. Rather, the predictive successes of MOND must be teaching us something. The field will not progress until these are incorporated into mainstream thinking.

Cosmic whack-a-mole

The fine-tuning problem encountered by dark matter models that I talked about last time is generic. The knee-jerk reaction of most workers seems to be “let’s build a more sophisticated model.” That’s reasonable – if there is any hope of recovery. The attitude is that dark matter has to be right so something has to work out. This fails to even contemplate the existential challenge that the fine-tuning problem imposes.

Perhaps I am wrong to be pessimistic, but my concern is well informed by years upon years trying to avoid this conclusion. Most of the claims I have seen to the contrary are just specialized versions of the generic models I had already built: they contain the same failings, but these go unrecognized because the presumption is that something has to work out, so people are often quick to declare “close enough!”

In my experience, fixing one thing in a model often breaks something else. It becomes a game of cosmic whack-a-mole. If you succeed in suppressing the scatter in one relation, it pops out somewhere else. A model that seems like it passes the test you built it to pass flunks as soon as you confront it with another test.

Let’s consider a few examples.

Squeezing the toothpaste tube

Our efforts to evade one fine-tuning problem often lead to another. This has been my general experience in many efforts to construct viable dark matter models. It is like squeezing a tube of toothpaste: every time we smooth out the problems in one part of the tube, we simply squeeze them into a different part. There are many published claims to solve this problem or that, but they frequently fail to acknowledge (or notice) that the purported solution to one problem creates another.

One example is provided by Courteau and Rix (1999). They invoke dark matter domination to explain the lack of residuals in the Tully-Fisher relation. In this limit, M_b/R ≪ M_DM/R and the baryons leave no mark on the rotation curve. This can reconcile the model with the Tully-Fisher relation, but it makes a strong prediction. It is not just the flat rotation speed that is the same for galaxies of the same mass, but the entirety of the rotation curve, V(R) at all radii. The stars are just convenient tracers of the dark matter halo in this limit; the dynamics are entirely dominated by the dark matter. The hypothesized solution fixes the problem that is addressed, but creates another problem that is not addressed, in this case the observed variation in rotation curve shape.

The limit of complete dark matter domination is not consistent with the shapes of rotation curves. Galaxies of the same baryonic mass have the same flat outer velocity (Tully-Fisher), but the shapes of their rotation curves vary systematically with surface brightness (de Blok & McGaugh, 1996; Tully and Verheijen, 1997; McGaugh and de Blok, 1998a,b; Swaters et al., 2009, 2012; Lelli et al., 2013, 2016c). High surface brightness galaxies have steeply rising rotation curves while LSB galaxies have slowly rising rotation curves (Fig. 6). This systematic dependence of the inner rotation curve shape on the baryon distribution excludes the SH hypothesis in the limit of dark matter domination: the distribution of the baryons clearly has an impact on the dynamics.

Fig. 6. Rotation curve shapes and surface density. The left panel shows the rotation curves of two galaxies, one HSB (NGC 2403, open circles) and one LSB (UGC 128, filled circles) (de Blok & McGaugh, 1996; Verheijen and de Blok, 1999; Kuzio de Naray et al., 2008). These galaxies have very nearly the same baryonic mass (~ 10¹⁰ M_⊙), and asymptote to approximately the same flat rotation speed (~ 130 km s⁻¹). Consequently, they are indistinguishable in the Tully-Fisher plane (Fig. 4). However, the inner shapes of the rotation curves are readily distinguishable: the HSB galaxy has a steeply rising rotation curve while the LSB galaxy has a more gradual rise. This is a general phenomenon, as illustrated by the central density relation (right panel: Lelli et al., 2016c) where each point is one galaxy; NGC 2403 and UGC 128 are highlighted as open points. The central dynamical mass surface density (Σ_dyn) measured by the rate of rise of the rotation curve (Toomre, 1963) correlates with the central surface density of the stars (Σ₀) measured by their surface brightness. The line shows 1:1 correspondence: no dark matter is required near the centers of HSB galaxies. The need for dark matter appears below 1000 M_⊙ pc⁻² and grows systematically greater to lower surface brightness. This is the origin of the statement that LSB galaxies are dark matter dominated.

A more recent example of this toothpaste tube problem for SH-type models is provided by the EAGLE simulations (Schaye et al., 2015). These are claimed (Ludlow et al., 2017) to explain one aspect of the observations, the radial acceleration relation (McGaugh et al., 2016), but fail to explain another, the central density relation (Lelli et al., 2016c) seen in Fig. 6. This was called the ‘diversity’ problem by Oman et al. (2015), who note that the rotation velocity at a specific, small radius (2 kpc) varies considerably from galaxy to galaxy observationally (Fig. 6), while simulated galaxies show essentially no variation, with only a small amount of scatter. This diversity problem is exactly the same problem that was pointed out before [compare Fig. 5 of Oman et al. (2015) to Fig. 14 of McGaugh and de Blok (1998a)].

There is no single, universally accepted standard galaxy formation model, but a common touchstone is provided by Mo et al. (1998). Their base model has a constant ratio of luminous to dark mass m_d [their assumption (i)], which provides a reasonable description of the sizes of galaxies as a function of mass or rotation speed (Fig. 7). However, this model predicts the wrong slope (3 rather than 4) for the Tully-Fisher relation. This is easily remedied by making the luminous mass fraction proportional to the rotation speed (m_d ∝ V_f), which then provides an adequate fit to the Tully-Fisher⁴ relation. This has the undesirable effect of destroying the consistency of the size-mass relation. We can have one or the other, but not both.

Fig. 7. Galaxy size (as measured by the exponential disk scale length, left) and mass (right) as a function of rotation velocity. The latter is the Baryonic Tully-Fisher relation; the data are the same as in Fig. 4. The solid lines are Mo et al. (1998) models with constant m_d (their equations 12 and 16). This is in reasonable agreement with the size-speed relation but not the BTFR. The latter may be fit by adopting a variable m_d ∝ V_f (dashed lines), but this ruins agreement with the size-speed relation. This is typical of dark matter models in which fixing one thing breaks another.

This failure of the Mo et al. (1998) model provides another example of the toothpaste tube problem. By fixing one problem, we create another. The only way forward is to consider more complex models with additional degrees of freedom.

Feedback

It has become conventional to invoke ‘feedback’ to address the various problems that afflict galaxy formation theory (Bullock & Boylan-Kolchin, 2017; De Baerdemaker and Boyd, 2020). It goes by other monikers as well, variously being called ‘gastrophysics’⁵ for gas phase astrophysics, or simply ‘baryonic physics’ for any process that might intervene between the relatively simple (and calculable) physics of collisionless cold dark matter and messy observational reality (which is entirely illuminated by the baryons). This proliferation of terminology obfuscates the boundaries of the subject and precludes a comprehensive discussion.

Feedback is not a single process, but rather a family of distinct processes. The common feature of different forms of feedback is the deposition of energy from compact sources into the surrounding gas of the interstellar medium. This can, at least in principle, heat gas and drive large-scale winds, either preventing gas from cooling and forming too many stars, or ejecting it from a galaxy outright. This in turn might affect the distribution of dark matter, though the effect is weak: one must move a lot of baryons for their gravity to impact the dark matter distribution.

There are many kinds of feedback, and many devils in the details. Massive, short-lived stars produce copious amounts of ultraviolet radiation that heats and ionizes the surrounding gas and erodes interstellar dust. These stars also produce strong winds through much of their short (~ 10 Myr) lives, and ultimately explode as Type II supernovae. These three mechanisms each act in a distinct way on different time scales. That’s just the feedback associated with massive stars; there are many other mechanisms (e.g., Type Ia supernovae are distinct from Type II supernovae, and Active Galactic Nuclei are a completely different beast entirely). The situation is extremely complicated. While the various forms of stellar feedback are readily apparent on the small scales of stars, it is far from obvious that they have the desired impact on the much larger scales of entire galaxies.

For any one kind of feedback, there can be many substantially different implementations in galaxy formation simulations. Independent numerical codes do not generally return compatible results for identical initial conditions (Scannapieco et al., 2012): there is no consensus on how feedback works. Among the many different computational implementations of feedback, at most one can be correct.

Most galaxy formation codes do not resolve the scale of single stars where stellar feedback occurs. They rely on some empirically calibrated, analytic approximation to model this ‘sub-grid physics’ — which is to say, they don’t simulate feedback at all. Rather, they simulate the accumulation of gas in one resolution element, then follow some prescription for what happens inside that unresolved box. This provides ample opportunity for disputes over the implementation and effects of feedback. For example, feedback is often cited as a way to address the cusp-core problem — or not, depending on the implementation (e.g., Benítez-Llambay et al., 2019; Bose et al., 2019; Di Cintio et al., 2014; Governato et al., 2012; Madau et al., 2014; Read et al., 2019). High resolution simulations (Bland-Hawthorn et al., 2015) indicate that the gas of the interstellar medium is less affected by feedback effects than assumed by typical sub-grid prescriptions: most of the energy is funneled through the lowest density gas — the course of least resistance — and is lost to the intergalactic medium without much impacting the galaxy in which it originates.

From the perspective of the philosophy of science, feedback is an auxiliary hypothesis invoked to patch up theories of galaxy formation. Indeed, since there are many distinct flavors of feedback that are invoked to carry out a variety of different tasks, feedback is really a suite of auxiliary hypotheses. This violates parsimony to an extreme and brutal degree.

This concern for parsimony is not specific to any particular feedback scheme; it is not just a matter of which feedback prescription is best. The entire approach is to invoke as many free parameters as necessary to solve any and all problems that might be encountered. There is little doubt that such models can be constructed to match the data, even data that bear little resemblance to the obvious predictions of the paradigm (McGaugh and de Blok, 1998a; Mo et al., 1998). So the concern is not whether ΛCDM galaxy formation models can explain the data; it is that they can’t not.

One could go on at much greater length about feedback and its impact on galaxy formation. This is pointless. It is a form of magical thinking to expect that the combined effects of numerous complicated feedback effects are going to always add up to looking like MOND in each and every galaxy. It is also the working presumption of an entire field of modern science.

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.

Galaxy Formation – a few basics

Galaxies are gravitationally bound condensations of stars and gas in a mostly empty, expanding universe. The tens of billions of solar masses of baryonic material that comprise the stars and gas of the Milky Way now reside mostly within a radius of 20 kpc. At the average density of the universe, the equivalent mass fills a spherical volume with a comoving radius a bit in excess of 1 Mpc. This is a large factor by which a protogalaxy must collapse, starting from the very smooth (~ 1 part in 10⁵) initial condition at z = 1090 observed in the CMB (Planck Collaboration et al., 2018). Dark matter — in particular, non-baryonic cold dark matter — plays an essential role in speeding this process along.

The mass-energy of the early universe is initially dominated by the radiation field. The baryons are held in thrall to the photons until the expansion of the universe turns the tables and matter becomes dominant. Exactly when this happens depends on the mass density (Peebles, 1980); for our purposes it suffices to realize that the baryonic components of galaxies can not begin to form until well after the time of the CMB. However, since CDM does not interact with photons, it is not subject to this limitation. The dark matter can begin to form structures — dark matter halos — that form the scaffolding of future structure. Essential to the ΛCDM galaxy formation paradigm is that the dark matter halos form first, seeding the subsequent formation of luminous galaxies by providing the potential wells into which baryons can condense once free from the radiation field.

The theoretical expectation for how dark matter halos form is well understood at this juncture. Numerical simulations of cold dark matter — mass that interacts only through gravity in an expanding universe — show that quasi-spherical dark matter halos form with a characteristic ‘NFW’ (e.g., Navarro et al., 1997) density profile. These have a ‘cuspy’ inner density profile in which the density of dark matter increases towards the center approximately ¹ as a power law, ρ(r → 0) ~ r⁻¹. At larger radii, the density profile falls of as ρ(r → ∞) ~ r⁻³. The centers of these halos are the density peaks around which galaxies can form.

The galaxies that we observe are composed of stars and gas: normal baryonic matter. The theoretical expectation for how baryons behave during galaxy formation is not well understood (Scannapieco et al., 2012). This results in a tremendous and long-standing disconnect between theory and observation. We can, however, stipulate a few requirements as to what needs to happen. Dark matter halos must form first; the baryons fall into these halos afterwards. Dark matter halos are observed to extend well beyond the outer edges of visible galaxies, so baryons must condense to the centers of dark matter halos. This condensation may proceed through both the hierarchical merging of protogalactic fragments (a process that has a proclivity to form ETGs) and the more gentle accretion of gas into rotating disks (a requirement to form LTGs). In either case, some fraction of the baryons form the observed, luminous component of a galaxy at the center of a CDM halo. This condensation of baryons necessarily affects the dark matter gravitationally, with the net effect of dragging some of it towards the center (Blumenthal et al., 1986; Dubinski, 1994; Gnedin et al., 2004; Sellwood and McGaugh, 2005a), thus compressing the dark matter halo from its initial condition as indicated by dark matter-only simulations like those of Navarro et al. (1997). These processes must all occur, but do not by themselves suffice to explain real galaxies.

Galaxies formed in models that consider only the inevitable effects described above suffer many serious defects. They tend to be too massive (Abadi et al., 2003; Benson et al., 2003), too small (the angular momentum catastrophe: Katz, 1992; Steinmetz, 1999; D’Onghia et al., 2006), have systematically too large bulge-to-disk ratios (the bulgeless galaxy problem: D’Onghia and Burkert, 2004; Kormendy et al., 2010), have dark matter halos with too much mass at small radii (the cusp-core problem: Moore et al., 1999b; Kuzio de Naray et al., 2008, 2009; de Blok, 2010; Kuzio de Naray and McGaugh, 2014), and have the wrong over-all mass function (the over-cooling problem, e.g., Benson, 2010), also known locally as the missing satellite problem (Klypin et al., 1999; Moore et al., 1999a). This long list of problems have kept the field of galaxy formation a lively one: there is no risk of it becoming a victim its own success through the appearance of one clearly-correct standard model.

Historical threads of development

Like last time, this is a minimalist outline of the basics that are relevant to our discussion. A proper history of this field would be much longer. Indeed, I rather doubt it would be possible to write a coherent text on the subject, which means different things to different scientists.

Entering the 1980s, options for galaxy formation were frequently portrayed as a dichotomy between monolithic galaxy formation (Eggen et al., 1962) and the merger of protogalactic fragments (Searle and Zinn, 1978). The basic idea of monolithic galaxy formation is that the initial ~ 1 Mpc cloud of gas that would form the Milky Way experienced dissipational collapse in one smooth, adiabatic process. This is effective at forming the disk, with only a tiny bit of star formation occurring during the collapse phase to provide the stars of the ancient, metal-poor stellar halo. In contrast, the Galaxy could have been built up by the merger of smaller protogalactic fragments, each with their own life as smaller galaxies prior to merging. The latter is more natural to the emergence of structure from the initial conditions observed in the CMB, where small lumps condense more readily than large ones. Indeed, this effectively forms the basis of the modern picture of hierarchical galaxy formation (Efstathiou et al., 1988).

Hierarchical galaxy formation is effective at forming bulges and pressure-supported ETGs, but is anathema to the formation of orderly disks. Dynamically cold disks are fragile and prefer to be left alone: the high rate of merging in the hierarchical ΛCDM model tends to destroy the dynamically cold state in which most spirals are observed to exist (Abadi et al., 2003; Peebles, 2020; Toth and Ostriker, 1992). Consequently, there have been some rather different ideas about galaxy formation: if one starts from the initial conditions imposed by the CMB, hierarchical galaxy formation is inevitable. If instead one works backwards from the observed state of galaxy disks, the smooth settling of gaseous disks in relatively isolated monoliths seems more plausible.

In addition to different theoretical notions, our picture of the galaxy population was woefully incomplete. An influential study by Freeman (1970) found that 28 of three dozen spirals shared very nearly the same central surface brightness. This was generalized into a belief that all spirals had the same (high) surface brightness, and came to be known as Freeman’s Law. Ultimately this proved to be a selection effect, as pointed out early by Disney (1976) and Allen and Shu (1979). However, it was not until much later (McGaugh et al., 1995a) that this became widely recognized. In the mean time, the prevailing assumption was that Freeman’s Law held true (e.g., van der Kruit, 1987) and all spirals had practically the same surface brightness. In particular, it was the central surface brightness of the disk component of spiral galaxies that was thought to be universal, while bulges and ETGs varied in surface brightness. Variation in the disk component of LTGs was thought to be restricted to variations in size, which led to variations in luminosity at fixed surface brightness.

Consequently, most theoretical effort was concentrated on the bright objects in the high-mass (M_∗ > 10¹⁰ M_⊙) clump in Fig. 2. Some low mass dwarf galaxies were known to exist, but were considered to be insignificant because they contained little mass. Low surface brightness galaxies violated Freeman’s Law, so were widely presumed not to exist, or be at most a rare curiosity (Bosma & Freeman, 1993). A happy consequence of this unfortunate state of affairs was that as observations of diffuse LSB galaxies were made, they forced then-current ideas about galaxy formation into a regime that they had not anticipated, and which many could not accommodate.

The similarity and difference between high surface brightness (HSB) and LSB galaxies is illustrated by Fig. 3. Both are rotationally supported, late type disk galaxies. Both show spiral structure, though it is more prominent in the HSB. More importantly, both systems are of comparable linear diameter. They exist roughly at opposite ends of a horizontal line in Fig. 2. Their differing stellar masses stem from the surface density of their stars rather than their linear extent — exactly the opposite of what had been inferred from Freeman’s Law. Any model of galaxy formation and evolution must account for the distribution of size (or surface brightness) at a given mass as well as the number density of galaxies as a function of mass. Both aspects of the galaxy population remain problematic to this day.

Fig. 3. High and low surface brightness galaxies. NGC 7757 (left) and UGC 1230 (right) are examples of high and low surface brightness galaxies, respectively. These galaxies are about the same distance away and span roughly the same physical diameter. The chief difference is in the surface brightness, which follows from the separation between stars (McGaugh et al., 1995b). Note that the intensity scale of these images is not identical; the contrast has been increased for the LSB galaxy so that it appears as more than a smudge.

Throughout my thesis work, my spouse joked that my LSB galaxy images looked like bug splots on the telescope. You can see more of them here. And a few more here. And lots more on Jim Schombert’s web pages, here and here and here.

Primer on Galaxy Properties

When we look up at the sky, we see stars. Stars are the building blocks of galaxies; we can see the stellar disk of the galaxy in which we live as the vault of the Milky Way arching across the sky. When we look beyond the Milky Way, we see galaxies. Just as stars are the building blocks of galaxies, galaxies are the building blocks of the universe. One can no more hope to understand cosmology without understanding galaxies than one can hope to understand galaxies without understanding stars.

Here I give a very brief primer on basic galaxy properties. This is a subject on which entire textbooks are written, so what I say here is necessarily very incomplete. It is a bare minimum to go on for the ensuing discussion.

Galaxy Properties

Cosmology entered the modern era when Hubble (1929) resolved the debate over the nature of spiral nebulae by measuring the distance to Andromeda, establishing that vast stellar systems — galaxies — exist external to and coequal with the Milky Way. Galaxies are the primary type of object observed when we look beyond the confines of our own Milky Way: they are the building blocks of the universe. Consequently, galaxies and cosmology are intertwined: it is impossible to understand one without the other.

Here I sketch a few essential facts about the properties of galaxies. This is far from a comprehensive list (see, for example Binney & Tremaine, 1987) and serves only to provide a minimum framework for the subsequent discussion. The properties of galaxies are often cast in terms of morphological type, starting with Hubble’s tuning fork diagram. The primary distinction is between Early Type Galaxies (ETGs) and Late Type Galaxies (LTGs), which is a matter of basic structure. ETGs, also known as elliptical galaxies, are three dimensional, ellipsoidal systems that are pressure supported: there is more kinetic energy in random motions than in circular motions, a condition described as dynamically hot. The orbits of stars are generally eccentric and oriented randomly with respect to one another, filling out the ellipsoidal shape seen in projection on the sky. LTGs, including spiral and irregular galaxies, are thin, quasi-two dimensional, rotationally supported disks. The majority of their stars orbit in the same plane in the same direction on low eccentricity orbits. The lion’s share of kinetic energy is invested in circular motion, with only small random motions, a condition described as dynamically cold. Examples of early and late type galaxies are shown in Fig. 1.

Fig. 1. Galaxy morphology. These examples shown an early type elliptical galaxy (NGC 3379, left), and two late type disk galaxies: a face-on spiral (NGC 628, top right), and an edge-on disk galaxy (NGC 891, bottom right). Elliptical galaxies are quasi-spherical, pressure supported stellar systems that tend to have predominantly old stellar populations, usually lacking young stars or much in the way of the cold interstellar gas from which they might form. In contrast, late type galaxies (spirals and irregulars) are thin, rotationally supported disks. They typically contain a mix of stellar ages and cold interstellar gas from which new stars continue to form. Interstellar dust is also present, being most obvious in the edge-on case. Images from Palomar Observatory, Caltech.

Finer distinctions in morphology can be made within the broad classes of early and late type galaxies, but the basic structural and kinematic differences suffice here. The disordered motion of ETGs is a natural consequence of violent relaxation (Lynden-Bell, 1967) in which a stellar system reaches a state of dynamical equilibrium from a chaotic initial state. This can proceed relatively quickly from a number of conceivable initial conditions, and is a rather natural consequence of the hierarchical merging of sub-clumps expected from the Gaussian initial conditions indicated by observations of the CMB (White, 1996). In contrast, the orderly rotation of dynamically cold LTGs requires a gentle settling of gas into a rotationally supported disk. It is essential that disk formation occur in the gaseous phase, as gas can dissipate and settle to the preferred plane specified by the net angular momentum of the system. Once stars form, their orbits retain a memory of their initial state for a period typically much greater than the age of the universe (Binney & Tremaine, 1987). Consequently, the bulk of the stars in the spiral disk must have formed there after the gas settled.

In addition to the dichotomy in structure, ETGs and LTGs also differ in their evolutionary history. ETGs tend to be ‘red and dead,’ which is to say, dominated by old stars. They typically lack much in the way of recent star formation, and are often devoid of the cold interstellar gas from which new stars can form. Most of their star formation happened in the early universe, and may have involved the merger of multiple protogalactic fragments. Irrespective of these details, massive ETGs appeared early in the universe (Steinhardt et al., 2016), and for the most part seem to have evolved passively since (Franck and McGaugh, 2017).

Again in contrast, LTGs have on-going star formation in interstellar media replete with cold atomic and molecular gas. They exhibit a wide range in stellar ages, from newly formed stars to ancient stars dating to near the beginning of time. Old stars seem to be omnipresent, famously occupying globular clusters but also present in the general disk population. This implies that the gaseous disk settled fairly early, though accretion may continue over a long timescale (van den Bergh, 1962; Henry and Worthey, 1999). Old stars persist in the same orbital plane as young stars (Binney & Merrifield, 1998), which precludes much subsequent merger activity, as the chaos of merging distorts orbits. Disks can be over-heated (Toth and Ostriker, 1992) and transformed by interactions between galaxies (Toomre and Toomre, 1972), even turning into elliptical galaxies during major mergers (Barnes & Hernquist, 1992).

Aside from its morphology, an obvious property of a galaxy is its mass. Galaxies exist over a large range of mass, with a type-dependent characteristic stellar mass of 5 × 10¹⁰ M_⊙ for disk dominated systems (the Milky Way is very close to this mass: Bland-Hawthorn & Gerhard, 2016) and 10¹¹ M_⊙ for elliptical galaxies (Moffett et al., 2016). Above this characteristic mass, the number density of galaxies declines sharply, though individual galaxies exceeding a few 10¹¹ M_⊙ certainly exist. The number density of galaxies increases gradually to lower masses, with no known minimum. The gradual increase in numbers does not compensate for the decrease in mass: integrating over the distribution, one finds that most of the stellar mass is in bright galaxies close to the characteristic mass.

Galaxies have a characteristic size and surface brightness. The same amount of stellar mass can be concentrated in a high surface brightness (HSB) galaxies, or spread over a much larger area in a low surface brightness (LSB) galaxy. For the purposes of this discussion, it suffices to assume that the observed luminosity is proportional to the mass of stars that produces the light. Similarly, the surface brightness measures the surface density of stars. Of the three observable quantities of luminosity, size, and surface brightness, only two are independent: the luminosity is the product of the surface brightness and the area over which it extends. The area scales as the square of the linear size.

The distribution of size and mass of galaxies is shown in Fig. 2. This figure spans the range from tiny dwarf irregular galaxies containing ‘only’ a few hundred thousand stars to giant spirals composed of hundreds of billions of stars with half-light radii ranging from hundreds of parsecs to tens of kpc. The upper boundaries represent real, physical limits on the sizes and masses of galaxies. Bright objects are easy to see; if still higher mass galaxies were common, they would be readily detected and cataloged. In contrast, the lower boundaries are set by the limits of observational sensitivity (“selection effects”): galaxies that are physically small or low in surface brightness are difficult to detect and are systematically under-represented in galaxy catalogs (Allen & Shu, 1979; Disney, 1976; McGaugh et al., 1995a).

Fig. 2. Galaxy size and mass. The radius that contains half of the light is plotted against the stellar mass. Galaxies exist over many decades in mass, and exhibit a considerable variation in size at a given mass. Early and late type galaxies are demarcated with different symbols, as noted. Lines illustrate tracks of constant stellar surface density. The data for ETGs are from the compilation of Dabringhausen and Fellhauer (2016) augmented by dwarf Spheroidal (dSph) galaxies in the Local Group compiled by Lelli et al. (2017). Ultra-diffuse galaxies (UDGs: van Dokkum et al., 2015; Mihos et al., 2015, × and +, respectively) have unsettled kinematic classifications at present, but most seem likely to be pressure supported ETGs. The bulk of the data for LTGs is from the SPARC database (Lelli et al., 2016a), augmented by cases that are noteworthy for their extremity in mass or surface brightness (Brunker et al., 2019; Dalcanton, Spergel, Gunn, et al., 1997; de Blok et al., 1995; McGaugh and Bothun, 1994; Mihos et al., 2018; Rhode et al., 2013; Schombert et al., 2011). The gas content of these star-forming systems adds a third axis, illustrated crudely here by whether an LTG is made more of stars or gas (filled and open symbols, respectively).

Individual galaxies can be early type or late type, high mass or low mass, large or small in linear extent, high or low surface brightness, gas poor or gas rich. No one of these properties is completely predictive of the others: the correlations that do exist tend to have lots of intrinsic scatter. The primary exception to this appears to involve the kinematics. Massive galaxies are fast rotators; low mass galaxies are slow rotators. This Tully-Fisher relation (Tully and Fisher, 1977) is one of the strongest correlations in extragalactic astronomy (Lelli et al., 2016b). It is thus necessary to simultaneously explain both the chaotic diversity of galaxy properties and the orderly nature of their kinematics (McGaugh et al., 2019).

Galaxies do not exist in isolation. Rather than being randomly distributed throughout the universe, they tend to cluster together: the best place to find a galaxy is in the proximity of another galaxy (Rubin, 1954). A common way to quantify the clustering of galaxies is the two-point correlation function ξ(r) (Peebles, 1980). This measures the excess probability of finding a galaxy within a distance r of a reference galaxy relative to a random distribution. The observed correlation function is well approximated as a power law whose slope and normalization varies with galaxy population. ETGs are more clustered than LTGs, having a longer correlation length: r₀ ≈ 9 Mpc for red galaxies vs. ~ 5 Mpc for blue galaxies (Zehavi et al., 2011). Here we will find this quantity to be of interest for comparing the distribution of high and low surface brightness galaxies.

Galaxies are sometimes called island universes. That is partly a hangover from pre-Hubble times during which it was widely believed that the Milky Way contained everything: it was one giant island universe embedded in an indefinite but otherwise empty void. We know that’s not true now – there are lots of stellar systems of similar size to the Milky Way – but they often seem to stand alone even if they are clustered in non-random ways.

For example, here is the spiral galaxy NGC 7757, an island unto itself.

*NGC 7757 from the digitized sky survey (© 1994, Association of Universities for Research in Astronomy, Inc).*

NGC 7757 is a high surface brightness spiral. It is easy to spot amongst the foreground stars of the Milky Way. In contrast, there are strong selection effects against low surface brightness galaxies, like UGC 1230:

UGC 1230 from the digitized sky survey *(© 1994, Association of Universities for Research in Astronomy, Inc).*

The LSB galaxy is rather harder to spot. Even when noticed, it doesn’t seem as important as the HSB galaxy. This, in a nutshell, is the history of selection effects in galaxy surveys, which are inevitably biased towards the biggest and the brightest. Advances in detectors (especially the CCD revolution of the 1980s) helped open our eyes to the existence of these LSB galaxies, and allowed us to measure their physical properties. Doing so provided a stringent test of galaxy formation theories, which have scrambled to catch up ever since.

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.

Galaxy models in compressed halos

The last post was basically an introduction to this one, which is about the recent work of Pengfei Li. In order to test a theory, we need to establish its prior. What do we expect?

The prior for fully formed galaxies after 13 billion years of accretion and evolution is not an easy problem. The dark matter halos need to form first, with the baryonic component assembling afterwards. We know from dark matter-only structure formation simulations that the initial condition (A) of the dark matter halo should resemble an NFW halo, and from observations that the end product of baryonic assembly needs to look like a real galaxy (Z). How the universe gets from A to Z is a whole alphabet of complications.

The simplest thing we can do is ignore B-Y and combine a model galaxy with a model dark matter halo. The simplest model for a spiral galaxy is an exponential disk. True to its name, the azimuthally averaged stellar surface density falls off exponentially from a central value over some scale length. This is a tolerable approximation of the stellar disks of spiral galaxies, ignoring their central bulges and their gas content. It is an inadequate yet surprisingly decent starting point for describing gravitationally bound collections of hundreds of billions of stars with just two parameters.

So a basic galaxy model is an exponential disk in an NFW dark matter halo. This is they type of model I discussed in the last post, the kind I was considering two decades ago, and the kind of model still frequently considered. It is an obvious starting point. However, we know that this starting point is not adequate. On the baryonic side, we should model all the major mass components: bulge, disk, and gas. On the halo side, we need to understand how the initial halo depends on its assembly history and how it is modified by the formation of the luminous galaxy within it. The common approach to do all that is to run a giant cosmological simulation and watch what happens. That’s great, provided we know how to model all the essential physics. The action of gravity in an expanding universe we can compute well enough, but we do not enjoy the same ability to calculate the various non-gravitational effects of baryons.

Rather than blindly accept the outcome of simulations that have become so complicated that no one really seems to understand them, it helps to break the problem down into its basic steps. There is a lot going on, but what we’re concerned about here boils down to a tug of war between two competing effects: adiabatic compression tends to concentrate the dark matter, while feedback tends to redistribute it outwards.

Adiabatic compression refers to the response of the dark matter halo to infalling baryons. Though this name stuck, the process isn’t necessarily adiabatic, and the A-word word tends to blind people to a generic and inevitable physical process. As baryons condense into the centers of dark matter halos, the gravitational potential is non-stationary. The distribution of dark matter has to respond to this redistribution of mass: the infall of dissipating baryons drags some dark matter in with them, so we expect dark matter halos to become more centrally concentrated. The most common approach to computing this effect is to assume the process is adiabatic (hence the name). This means a gentle settling that is gradual enough to be time-reversible: you can imagine running the movie backwards, unlike a sudden, violent event like a car crash. It needn’t be rigorously adiabatic, but the compressive response of the halo is inevitable. Indeed, forming a thin, dynamically cold, well-organized rotating disk in a preferred plane – i.e., a spiral galaxy – pretty much requires a period during which the adiabatic assumption is a decent approximation. There is a history of screwing up even this much, but Jerry Sellwood showed that it could be done correctly and that when one does so, it reproduces the results of more expensive numerical simulations. This provides a method to go beyond a simple exponential disk in an NFW halo: we can compute what happens to an NFW halo in response to an observed mass distribution.

After infall and compression, baryons form stars that produce energy in the form of radiation, stellar winds, and the blast waves of supernova explosions. These are sources of energy that complicate what until now has been a straightforward calculation of gravitational dynamics. With sufficient coupling to the surrounding gas, these energy sources might be converted into enough kinetic energy to alter the equilibrium mass distribution and the corresponding gravitational potential. I say might because we don’t really know how this works, and it is a lot more complicated than I’ve made it sound. So let’s not go there, and instead just calculate the part we do know how to calculate. What happens from the inevitable adiabatic compression in the limit of zero feedback?

We have calculated this for a grid of model galaxies that matches the observed distribution or real galaxies. This is important; it often happens that people do not explore a realistic parameter space. Here is a plot of size against stellar mass:

The size of galaxy disks as measured by the exponential scale length as a function of stellar mass. Grey points are real galaxies; red circles are model galaxies with parameters chosen to cover the same parameter space. This, and all plots, from Li et al. (2022).

Note that at a given stellar mass, there is a wide range of sizes. This is an essential aspect of galaxy properties; one has to explain size variations as well as the trend with mass. This obvious point has been frequently forgotten and rediscovered in the literature.

The two parameter plot above only suffices to approximate the stellar disks of spiral and irregular galaxies. Real galaxies have bulges and interstellar gas. We include these in our models so that they cover the same distribution as real galaxies in terms of bulge mass, size, and gas fraction. We then assign a dark matter halo to each model galaxy using an abundance matching relation (the stellar mass tells us the halo mass) and adopt the cosmologically appropriate halo mass-concentration relation. These specify the initial condition of the NFW halo in which each model galaxy is presumed to reside.

At this point, it is worth remarking that there are a variety of abundance matching relations in the literature. Some of these give tragically bad predictions for the kinematics. I won’t delve into this here, but do want to note that in what follows, we have adopted the most favorable abundance matching relation, which turns out to be that of Kravstov et al. (2018). Note that this means that we are already engaged in a kind of fine-tuning by cherry-picking the most favorable relation.

Before considering adiabatic compression, let’s see what happens if we simply add our model galaxies to NFW halos. This is the same exercise we did last time with exponential disks; now we’re including bulges and gas:

Galaxy models in the RAR plane. Models are color coded by their stellar surface density. The dotted line is 1:1 (Newton with no dark matter or other funny business). The black line is the fit to the observed RAR.

This looks pretty good, at least at a first glance. Most of the models fall nearly on top of each other. This isn’t entirely true, as the most massive models overpredict the RAR. This is a generic consequence of the bend in abundance matching relations. This bend is mildest in the Kravtsov relation, which is what makes it “best” here – other relations, like the commonly cited one of Behroozi, predict a lot more high-acceleration models. One sees only a hint of that here.

The scatter is respectably small, mostly solving the problem I initially encountered in the nineties. Despite predicting a narrow relation, the models do have a finite scatter that is a bit more than we observe. This isn’t too tragic, so maybe we can work with it. These models also miss the low acceleration end of the relation by a modest but appreciable amount. This seems more significant, as we found the same thing for pure exponential models: it is hard to make this part of the problem go away.

Including bulges in the models extends them to high accelerations. This would seem to explain a region of the RAR that pure exponential models do not address. Bulges are high surface density, star dominated regions, so they fall on the 1:1 part of the RAR at high accelerations.

And then there are the hooks. These are obvious in the plot above. They occur in low and intermediate mass galaxies that lack a significant bulge component. A pure exponential disk has a peak acceleration at finite radius, but an NFW halo has its peak at zero radius. So if you imagine following a given model line inwards in radius, it goes up in acceleration until it reaches the maximum for the disk along the x-axis. The baryonic component of the acceleration then starts to decline while that due to the NFW halo continues to rise. The model doubles back to lower baryonic acceleration while continuing to higher total acceleration, making the little hook shape. This deviation from the RAR is not commonly observed; indeed, these hooks are the signature of the cusp-core problem in the RAR plane.

Results so far are mixed. With the “right” choice of abundance matching relation, we are well ahead of where we were at the turn of the century, but some real problems remain. We have yet to compute the necessary adiabatic contraction, so hopefully doing that right will result in further improvement. So let’s make a rigorous calculation of the compression that would result from forming a galaxy of the stipulated parameters.

*Galaxy models in the RAR plane after compression.*

Adiabatic compression makes things worse. There is a tiny improvement at low accelerations, but the most pronounced effects are at small radii where accelerations are large. Compression makes cuspy halos cuspier, making the hooks more pronounced. Worse, the strong concentration of starlight that is a bulge inevitably leads to strong compression. These models don’t approach the 1:1 line at high acceleration, and never can: higher acceleration means higher stellar surface density means greater compression. One cannot start from an NFW halo and ever reach a state of baryon domination; too much dark matter is always in the mix.

It helps to look at the residual diagram. The RAR is a log-log plot over a large dynamic range; this can hide small but significant deviations. For some reason, people who claim to explain the RAR with dark matter models never seem to show these residuals.

*As above, with the observed RAR divided out. Model galaxies are mostly above the RAR. The cusp-core problem is exacerbated in disks, and bulges never reach the 1:1 line at high accelerations.*

The models built to date don’t have the right shape to explain the RAR, at least when examined closely. Still, I’m pleased: what we’ve done here comes closer than all my many previous efforts, and most of the other efforts that are out there. Still, I wouldn’t claim it as a success. Indeed, the inevitable compressive effects that occur at high surface densities means that we can’t invoke simple offsets to accommodate the data: if a model gets the shape of the RAR right but the normalization wrong, it doesn’t work to simply shift it over.

So, where does that leave us? Up the proverbial creek? Perhaps. We have yet to consider feedback, which is too complicated to delve into here. Instead, while we haven’t engaged in any specific fine-tuning, we have already engaged in some cherry picking. First, we’ve abandoned the natural proportionality between halo and disk mass, replacing it with abundance matching. This is no small step, as it converts a single-valued parameter of our theory to a rolling function of mass. Abundance matching has become familiar enough that people seemed to be lulled into thinking this is natural. There is nothing natural about it. Regardless of how much fancy jargon we use to justify it, it’s still basically a rolling fudge factor – the scientific equivalent of a lipstick smothered pig.

Abundance matching does, at least, use data that are independent of the kinematics to set the relation between stellar and halo mass, and it does go in the right direction for the RAR. This only gets us into the right ballpark, and only if we cherry-pick the particular abundance matching relation that we use. So we’re well down the path of tuning whether we realize it or not. Invoking feedback is simply another step along this path.

Feedback is usually invoked in the kinematic context to convert cusps into cores. That could help with the hooks. This kind of feedback is widely thought to affect low and intermediate mass galaxies, or galaxies of a particular stellar to halo mass ratio. Opinions vary a bit, but it is generally not thought to have such a strong effect on massive galaxies. And yet, we find that we need some (second?) kind of feedback for them, as we need to move bulges back onto the 1:1 line in the RAR plane. That’s perhaps related to the cusp-core problem, but it’s also different. Getting bulges right requires a fine-tuned amount of feedback to exactly cancel out the effects of compression. A third distinct place where the models need some help is at low accelerations. This is far from the region where feedback is thought to have much effect at all.

I could go on, and perhaps will in a future post. Point is, we’ve been tuning our feedback prescriptions to match observed facts about galaxies, not computing how we think it really works. We don’t know how to do the latter, and there is no guarantee that our approximations do justice to reality. So on the one hand, I don’t doubt that with enough tinkering this process can be made to work in a model. On the other hand, I do question whether this is how the universe really works.

What should we expect for the radial acceleration relation?

In the previous post, I related some of the history of the Radial Acceleration Relation (henceforth RAR). Here I’ll discuss some of my efforts to understand it. I’ve spent more time trying to do this in terms of dark matter than pretty much anything else, but I have not published most of those efforts. As I related briefly in this review, that’s because most of the models I’ve considered are obviously wrong. Just because I have refrained from publishing explanations of the RAR that are manifestly incorrect has not precluded others from doing so.

A theory is only as good as its prior. If a theory makes a clear prediction, preferably ahead of time, then we can test it. If it has not done so ahead of time, that’s still OK, if we can work out what it would have predicted without being guided by the data. A good historical example of this is the explanation of the excess perihelion precession of Mercury provided by General Relativity. The anomaly had been known for decades, but the right answer falls out of the theory without input from the data. A more recent example is our prediction of the velocity dispersions of the dwarf satellites of Andromeda. Some cases were genuine a priori predictions, but even in the cases that weren’t, the prediction is what it is irrespective of the measurement.

Dark matter-based explanations of the RAR do not fall in either category. They have always chased the data and been informed by it. This has been going on for so long that new practitioners have entered field unaware of the extent to which the simulations they inherited had already been informed by the data. They legitimately seem to think that there has been no fine-tuning of the models because they weren’t personally present for every turn of the knob.

So let’s set the way-back machine. I became concerned about fine-tuning problems in the context of galaxy dynamics when I was trying to explain the Tully-Fisher relation of low surface brightness galaxies in the mid-1990s. This was before I was more than dimly aware that MOND existed, much less taken it seriously. Many of us were making earnest efforts to build proper galaxy formation theories at the time (e.g., Mo, McGaugh, & Bothun 1994, Dalcanton, Spergel, & Summers 1997; Mo, Mao, & White 1998 [MMW]; McGaugh & de Blok 1998), though of course these were themselves informed by observations to date. My own paper had started as an effort to exploit the new things we had discovered about low surface brightness galaxies to broaden our conventional theory of galaxy formation, but over the course of several years, turned into a falsification of some of the ideas I had considered in my 1992 thesis. Dalcanton’s model evolved from one that predicted a shift in Tully-Fisher (as mine had) to one that did not (after the data said no). It may never be possible to completely separate theoretical prediction from concurrent data, but it is possible to ask what a theory plausibly predicts. What is the LCDM prior for the RAR?

In order to do this, we need to predict both the baryon distribution (g_bar) and that of the dark matter (g_obs-g_bar). Unfortunately, nobody seems to really agree on what LCDM predicts for galaxies. There seems to be a general consensus that dark matter halos should start out with the NFW form, but opinions vary widely about whether and how this is modified during galaxy formation. The baryonic side of the issue is simply seen as a problem.

That there is no clear prediction is in itself a problem. I distinctly remember expressing my concerns to Martin Rees while I was still a postdoc. He said not to worry; galaxies were such non-linear entities that we shouldn’t be surprised by anything they do. This verbal invocation of a blanket dodge for any conceivable observation did not inspire confidence. Since then, I’ve heard that excuse repeated by others. I have lost count of the number of more serious, genuine, yet completely distinct LCDM predictions I have seen, heard, or made myself. Many dozens, at a minimum; perhaps hundreds at this point. Some seem like they might work but don’t while others don’t even cross the threshold of predicting both axes of the RAR. There is no coherent picture that adds up to an agreed set of falsifiable predictions. Individual models can be excluded, but not the underlying theory.

To give one example, let’s consider the specific model of MMW. I make this choice here for two reasons. One, it is a credible effort by serious workers and has become a touchstone in the field, to the point that a sizeable plurality of practitioners might recognize it as a plausible prior – i.e., the closest thing we can hope to get to a legitimate, testable prior. Two, I recently came across one of my many unpublished attempts to explain the RAR which happens to make use of it. Unix says that the last time I touched these files was nearly 22 years ago, in 2000. The postscript generated then is illegible now, so I have to update the plot:

The prediction of MMW (lines) compared to data (points). Each colored line represents a model galaxy of a given mass. Different lines of the same color represent models with different disk scale lengths, as galaxies of the same mass exist over a range of sizes. Models are only depicted over the range of radii typically observed in real galaxies.

At first glance, this might look OK. The trend is at least in the right direction. This is not a success so much as it is an inevitable consequence of the fact that the observed acceleration includes the contribution of the baryons. The area below the dashed line is excluded, as it is impossible to have g_obs < g_bar. Moreover, since g_obs = g_bar+g_DM, some correlation in this plane is inevitable. Quite a lot, if baryons dominate, as they always seem to do at high accelerations. Not that these models explain the high acceleration part of the RAR, but I’ll leave that detail for later. For now, note that this is a log-log plot. That the models miss the data a little to the eye translates to a large quantitative error. Individual model galaxies sometimes fall too high, sometimes too low: the model predicts considerably more scatter than is observed. The RAR is not predicted to be a narrow relation, but one with lots of scatter with large intrinsic deviations from the mean. That’s the natural prediction of MMW-type models.

I have explored many flavors of [L]CDM models. They generically predicts more scatter in the RAR than is observed. This is the natural expectation, and some fine-tuning has to be done to reduce the scatter to the observed level. The inevitable need for fine-tuning is why I became concerned for the dark matter paradigm, even before I became aware that MOND predicted exactly this. It is also why the observed RAR was considered to be against orthodoxy at the time: everybody’s prior was for a large scatter. It wasn’t just me.

In order to build a model, one has to make some assumptions. The obvious assumption to make, at the time, was a constant ratio of dark matter to baryons. Indeed, for many years, the working assumption was that this was about 10:1, maybe 20:1. This type of assumption is built into the models of MMW, who thought that they worked provided “(i) the masses of disks are a few percent of those of their haloes”. The (i) is there because it is literally their first point, and the assumption that everybody made. We were terrified of dropping this natural assumption, as the obvious danger is that it becomes a rolling fudge factor, assuming any value that is convenient for explaining any given observation.

Unfortunately, it had already become clear by this time from the data that a constant ratio of dark to luminous matter could not work. The earliest I said this on the record is 1996. [That was before LCDM had supplanted SCDM as the most favored cosmology. From that perspective, the low baryon fractions of galaxies seemed natural; it was clusters of galaxies that were weird.] I pointed out the likely failure of (i) to Mo when I first saw a draft of MMW (we had been office mates in Cambridge). I’ve written various papers about it since. The point here is that, from the perspective of the kinematic data, the ratio of dark to luminous mass has to vary. It cannot be a constant as we had all assumed. But it has to vary in a way that doesn’t introduce scatter into relations like the RAR or the Baryonic Tully-Fisher relation, so we have to fine-tune this rolling fudge factor so that it varies with mass but always obtains the same value at the same mass.

A constant ratio of dark to luminous mass wasn’t just a convenient assumption. There is good physical reason to expect that this should be the case. The baryons in galaxies have to cool and dissipate to form a galaxy in the center of a dark matter halo. This takes time, imposing an upper limit on galaxy mass. But the baryons in small galaxies have ample time to cool and condense, so one naively expects that they should all do so. That would have been natural. It would also lead to a steeply increasing luminosity function, which is not observed, leading to the over-cooling and missing satellite problems.

Reconciling the observed and predicted mass functions is one of the reasons we invoke feedback. The energy produced by the stars that form in the first gas to condense are an energy source that feeds back into the surrounding gas. This can, in principle, reheat the remaining gas or expel it entirely, thereby precluding it from condensing and forming more stars as in the naive expectation. In principle. In practice, we don’t know how this works, or even if the energy provided by star formation couples to the surrounding gas in a way that does what we need it to do. Simulations do not have the resolution to follow feedback in detail, so instead make some assumptions (“subgrid physics”) about how this might happen, and tune the assumed prescription to fit some aspect of the data. Once this is done, it is possible to make legitimate predictions about other aspects of the data, provided they are unrelated. But we still don’t know if that’s how feedback works, and in no way is it natural. Rather, it is a deus ex machina that we invoke to save us from a glaring problem without really knowing how it works or even if it does. This is basically just theoretical hand-waving in the computational age.

People have been invoking feedback as a panacea for all ills in galaxy formation theory for so long that it has become familiar. Once something becomes familiar, everybody knows it. Since everybody knows that feedback has to play some role, it starts to seem like it was always expected. This is easily confused with being natural.

I could rant about the difficulty of making predictions with feedback afflicted models, but never mind the details. Let’s find some aspect of the data that is independent of the kinematics that we can use to specify the dark to luminous mass ratio. The most obvious candidate is abundance matching, in which the number density of observed galaxies is matched to the predicted number density of dark matter halos. We don’t have to believe feedback-based explanations to apply this, we merely have to accept that there is some mechanism to make the dark to luminous mass ratio variable. Whatever it is that makes this happen had better predict the right thing for both the mass function and the kinematics.

When it comes to the RAR, the application of abundance matching to assign halo masses to observed galaxies works out much better than the natural assumption of a constant ratio. This was first pointed out by Di Cintio & Lelli (2016), which inspired me to consider appropriately modified models. All I had to do was update the relation between stellar and halo mass from a constant ratio to a variable specified by abundance matching. This gives rather better results:

*A model like that from 2000 but updated by assigning halo masses using an abundance matching relation.*

This looks considerably better! The predicted scatter is much lower. How is this accomplished?

Abundance matching results in a non-linear relation bewteen stellar mass and halo mass. For the RAR, the scatter is reduced by narrowing the dynamic range of halo masses relative to the observed stellar masses. There is less variation in g_DM. Empirically, this is what needs to happen – to a crude first approximation, the data are roughly consistent with all galaxies living in the same halo – i.e., no variation in halo mass with stellar mass. This was already known before abundance matching became rife; both the kinematic data and the mass function push us in this direction. There’s nothing natural about any of this; it’s just what we need to do to accommodate the data.

Still, it is tempting to say that we’ve succeeded in explaining the RAR. Indeed, some people have built the same kind of models to claim exactly this. While matters are clearly better, really we’re just less far off. By reducing the dynamic range in halo masses that are occupied by galaxies, the partial contribution of g_DM to the g_obs axis is compressed, and model lines perforce fall closer together. There’s less to distinguish an L* galaxy from a dwarf galaxy in this plane.

Nevertheless, there’s still too much scatter in the models. Harry Desmond made a specific study of this, finding that abundance matching “significantly overpredicts the scatter in the relation and its normalisation at low acceleration”, which is exactly what I’ve been saying. The offset in the normalization at low acceleration is obvious from inspection in the figure above: the models overshoot the low acceleration data. This led Navarro et al. to argue that there was a second acceleration scale, “an effective minimum acceleration probed by kinematic tracers in isolated galaxies” a little above 10^-11 m/s/s. The models do indeed do this, over a modest range in g_bar, and there is some evidence for it in some data. This does not persist in the more reliable data; those shown above are dominated by atomic gas so there isn’t even the systematic uncertainty of the stellar mass-to-light ratio to save us.

The astute observer will notice some pink model lines that fall well above the RAR in the plot above. These are for the most massive galaxies, those with luminosities in excess of L*. Below the knee in the Schechter function, there is a small range of halo masses for a given range of stellar masses. Above the knee, this situation is reversed. Consequently, the nonlinearity of abundance matching works against us instead of for us, and the scatter explodes. One can suppress this with an apt choice of abundance matching relation, but we shouldn’t get to pick and choose which relation we use. It can be made to work only because there remains enough uncertainty in abundance matching to select the “right” one. There is nothing natural about any this.

There are also these little hooks, the kinks at the high acceleration end of the models. I’ve mostly suppressed them here (as did Navarro et al.) but they’re there in the models if one plots to small enough radii. This is the signature of the cusp-core problem in the RAR plane. The hooks occur because the exponential disk model has a maximum acceleration at a finite radius that is a little under one scale length; this marks the maximum value that such a model can reach in g_bar. In contrast, the acceleration g_DM of an NFW halo continues to increase all the way to zero radius. Consequently, the predicted g_obs continues to increase even after g_bar has peaked and starts to decline again. This leads to little hook-shaped loops at the high acceleration end of the models in the RAR plane.

These hooks were going to be the segue to discuss more sophisticated models built by Pengfei Li, but that’s going to be a whole ‘nother post because these are quite enough words for now. So, until next time, don’t invest in bitcoins, Russian oil, or LCDM models that claim to explain the RAR.

A brief history of the Radial Acceleration Relation

In science, all new and startling facts must encounter in sequence the responses
1. It is not true!
2. It is contrary to orthodoxy.
3. We knew it all along.
Louis Agassiz (circa 1861)

This expression exactly depicts the progression of the radial acceleration relation. Some people were ahead of this curve, others are still behind it, but it quite accurately depicts the mass sociology. This is how we react to startling new facts.

For quotation purists, I’m not sure exactly what the original phrasing was. I have paraphrased it to be succinct and have substituted orthodoxy for religion, because even scientists can have orthodoxies: holy cows that must not be slaughtered.

I might even add a precursor stage zero to the list above:

0. It goes unrecognized.

This is to say, that if a new fact is sufficiently startling, we don’t just disbelieve it (stage 1); at first we fail to see it at all. We lack the cognitive framework to even recognize how important it is. An example is provided by the 1941 detection of the microwave background by Andrew McKellar. In retrospect, this is as persuasive as the 1964 detection of Penzias and Wilson to which we usually ascribe the discovery. At the earlier time, there was simply no framework for recognizing what it was that was being detected. It appears to me that P&Z didn’t know what they were looking at either until Peebles explained it to them.

The radial acceleration relation was first posed as the mass discrepancy-acceleration relation. They’re fundamentally the same thing, just plotted in a slightly different way. The mass discrepancy-acceleration relation shows the ratio of total mass to that which is visible. This is basically the ratio of the observed acceleration to that predicted by the observed baryons. This is useful to see how much dark matter is needed, but by construction the axes are not independent, as both measured quantities are used in forming the ratio.

The radial acceleration relation shows independent observations along each axis: observed vs. predicted acceleration. Though measured independently, they are not physically independent, as the baryons contribute some to the total observed acceleration – they do have mass, after all. One can construct a halo acceleration relation by subtracting the baryonic contribution away from the total; in principle the remainders are physically independent. Unfortunately, the axes again become observationally codependent, and the uncertainties blow up, especially in the baryon dominated regime. Which of these depictions is preferable depends a bit on what you’re looking to see; here I just want to note that they are the same information packaged somewhat differently.

To the best of my knowledge, the first mention of the mass discrepancy-acceleration relation in the scientific literature is by Sanders (1990). Its existence is explicit in MOND (Milgrom 1983), but here it is possible to draw a clear line between theory and data. I am only speaking of the empirical relation as it appears in the data, irrespective of anything specific to MOND.

I met Bob Sanders, along with many other talented scientists, in a series of visits to the University of Groningen in the early 1990s. Despite knowing him and having talked to him about rotation curves, I was unaware that he had done this.

Stage 0: It goes unrecognized.

For me, stage one came later in the decade at the culmination of a several years’ campaign to examine the viability of the dark matter paradigm from every available perspective. That’s a long paper, which nevertheless drew considerable praise from many people who actually read it. If you go to the bother of reading it today, you will see the outlines of many issues that are still debated and others that have been forgotten (e.g., the fine-tuning issues).

Around this time (1998), the dynamicists at Rutgers were organizing a meeting on galaxy dynamics, and asked me to be one of the speakers. I couldn’t possibly discuss everything in the paper in the time allotted, so was looking for a way to show the essence of the challenge the data posed. Consequently, I reinvented the wheel, coming up with the mass discrepancy-acceleration relation. Here I show the same data that I had then in the form of the radial acceleration relation:

*The Radial Acceleration Relation from the data in McGaugh (1999). Plot credit: Federico Lelli. (There is a time delay in publication: the 1998 meeting’s proceedings appeared in 1999.)*

I recognize this version of the plot as having been made by Federico Lelli. I’ve made this plot many times, but this is version I came across first, and it is better than mine in that the opacity of the points illustrates where the data are concentrated. I had been working on low surface brightness galaxies; these have low accelerations, so that part of the plot is well populated.

The data show a clear correlation. By today’s standards, it looks crude. Going on what we had then, it was fantastic. Correlations practically never look this good in extragalactic astronomy, and they certainly don’t happen by accident. Low quality data can hide a correlation – uncertainties cause scatter – but they can’t create a correlation where one doesn’t exist.

This result was certainly startling if not as new as I then thought. That’s why I used the title How Galaxies Don’t Form. This was contrary to our expectations, as I had explained in exhaustive detail in the long paper and revisit in a recent review for philosophers and historians of science.

I showed the same result later that year (1998) at a meeting on the campus of the University of Maryland where I was a brand new faculty member. It was a much shorter presentation, so I didn’t have time to justify the context or explain much about the data. Contrary to the reception at Rutgers where I had adequate time to speak, the hostility of the audience to the result was palpable, their stony silence eloquent. They didn’t want to believe it, and plenty of people got busy questioning the data.

Stage 1: It is not true.

I spent the next five years expanding and improving the data. More rotation curves became available thanks to the work of many, particularly Erwin de Blok, Marc Verheijen, and Rob Swaters. That was great, but the more serious limitation was how well we could measure the stellar mass distribution needed to predict the baryonic acceleration.

The mass models we could build at the time were based on optical images. A mass model takes the observed light distribution, assigns a mass-to-light ratio, and makes a numerical solution of the Poisson equation to obtain the the gravitational force corresponding to the observed stellar mass distribution. This is how we obtain the stellar contribution to the predicted baryonic force; the same procedure is applied to the observed gas distribution. The blue part of the spectrum is the best place in which to observe low contrast, low surface brightness galaxies as the night sky is darkest there, at least during new moon. That’s great for measuring the light distribution, but what we want is the stellar mass distribution. The mass-to-light ratio is expected to have a lot of scatter in the blue band simply from the happenstance of recent star formation, which makes bright blue stars that are short-lived. If there is a stochastic uptick in the star formation rate, then the mass-to-light ratio goes down because there are lots of bright stars. Wait a few hundred million years and these die off, so the mass-to-light ratio gets bigger (in the absence of further new star formation). The time-integrated stellar mass may not change much, but the amount of blue light it produces does. Consequently, we expect to see well-observed galaxies trace distinct lines in the radial acceleration plane, even if there is a single universal relation underlying the phenomenon. This happens simply because we expect to get M*/L wrong from one galaxy to the next: in 1998, I had simply assumed all galaxies had the same M*/L for lack of any better prescription. Clearly, a better prescription was warranted.

In those days, I traveled through Tucson to observe at Kitt Peak with some frequency. On one occasion, I found myself with a few hours to kill between coming down from the mountain and heading to the airport. I wandered over to the Steward Observatory at the University of Arizona to see who I might see. A chance meeting in the wild west: I encountered Eric Bell and Roelof de Jong, who were postdocs there at the time. I knew Eric from his work on the stellar populations of low surface brightness galaxies, an interest closely aligned with my own, and Roelof from my visits to Groningen.

As we got to talking, Eric described to me work they were doing on stellar populations, and how they thought it would be possible to break the age-metallicity degeneracy using near-IR colors in addition to optical colors. They were mostly focused on improving the age constraints on stars in LSB galaxies, but as I listened, I realized they had constructed a more general, more powerful tool. At my encouragement (read their acknowledgements), they took on this more general task, ultimately publishing the classic Bell & de Jong (2001). In it, they built a table that enabled one to look up the expected mass-to-light ratio of a complex stellar population – one actively forming stars – as a function of color. This was a big step forward over my educated guess of a constant mass-to-light ratio: there was now a way to use a readily observed property, color, to improve the estimated M*/L of each galaxy in a well-calibrated way.

Combining the new stellar population models with all the rotation curves then available, I obtained an improved mass discrepancy-acceleration relation:

The Radial Acceleration Relation from the data in McGaugh (2004); version using Bell’s stellar population synthesis models to estimate M*/L (see Fig. 5 for other versions). Plot credit: Federico Lelli.

Again, the relation is clear, but with scatter. Even with the improved models of Bell & de Jong, some individual galaxies have M*/L that are wrong – that’s inevitable in this game. What you cannot know is which ones! Note, however, that there are now 74 galaxies in this plot, and almost all of them fall on top of each other where the point density is large. There are some obvious outliers; those are presumably just that: the trees that fall outside the forest because of the expected scatter in M*/L estimates.

I tried a variety of prescriptions for M*/L in addition to that of Bell & de Jong. Though they differed in texture, they all told a consistent story. A relation was clearly present; only its detailed form varied with the adopted prescription.

The prescription that minimized the scatter in the relation was the M*/L obtained in MOND fits. That’s a tautology: by construction, a MOND fit finds the M*/L that puts a galaxy on this relation. However, we can generalize the result. Maybe MOND is just a weird, unexpected way of picking a number that has this property; it doesn’t have to be the true mass-to-light ratio in nature. But one can then define a ratio Q

that relates the “true” mass-to-light ratio to the number that gives a MOND fit. They don’t have to be identical, but MOND does return M*/L that are reasonable in terms of stellar populations, so Q ~ 1. Individual values could vary, and the mean could be a bit more or less than unity, but not radically different. One thing that impressed me at the time about the MOND fits (most of which were made by Bob Sanders) was how well they agreed with the stellar population models, recovering the correct amplitude, the correct dependence on color in different bandpasses, and also giving the expected amount of scatter (more in the blue than in the near-IR).

Fig. 7 *of McGaugh (2004).* Stellar mass-to-light ratios of galaxies in the blue B-band (*top*) and near-IR K-band (*bottom*) as a function of B–V color for the prescription of maximum disk (*left*) and MOND (*right*). Each point represents one galaxy for which the requisite data were available at the time. The line represents the mean expectation of stellar population synthesis models from Bell et al. (2003). These lines are completely independent of the data: neither the normalization nor the slope has been fit to the dynamical data. The red points are due to Sanders & Verheijen (1998); note the weak dependence of M*/L on color in the near-IR.

The obvious interpretation is that we should take seriously a theory that obtains good fits with a single free parameter that checks out admirably well with independent astrophysical constraints, in this case the M*/L expected for stellar populations. But I knew many people would not want to do that, so I defined Q to generalize to any M*/L in any (dark matter) context one might want to consider.

Indeed, Q allows us to write a general expression for the rotation curve of the dark matter halo (essentially the HAR alluded to above) in terms of that of the stars and gas:

The stars and the gas are observed, and μ is the MOND interpolation function assumed in the fit that leads to Q. Except now the interpolation function isn’t part of some funny new theory; it is just the shape of the radial acceleration relation – a relation that is there empirically. The only fit factor between these data and any given model is Q – a single number of order unity. This does leave some wiggle room, but not much.

I went off to a conference to describe this result. At the 2006 meeting Galaxies in the Cosmic Web in New Mexico, I went out of my way at the beginning of the talk to show that even if we ignore MOND, this relation is present in the data, and it provides a strong constraint on the required distribution of dark matter. We may not know why this relation happens, but we can use it, modulo only the modest uncertainty in Q.

Having bent over backwards to distinguish the data from the theory, I was disappointed when, immediately at the end of my talk, prominent galaxy formation theorist Anatoly Klypin loudly shouted

“We don’t have to explain MOND!”

But you do have to explain the data. The problem was and is that the data look like MOND. It is easy to conflate one with the other; I have noticed that a lot of people have trouble keeping the two separate. Just because you don’t like the theory doesn’t mean that the data are wrong. What Anatoly was saying was that

2. It is contrary to orthodoxy.

Despite phrasing the result in a way that would be useful to galaxy formation theorists, they did not, by and large, claim to explain it at the time – it was contrary to orthodoxy so didn’t need to be explained. Looking at the list of papers that cite this result, the early adopters were not the target audience of galaxy formation theorists, but rather others citing it to say variations of “no way dark matter explains this.”

At this point, it was clear to me that further progress required a better way to measure the stellar mass distribution. Looking at the stellar population models, the best hope was to build mass models from near-infrared rather than optical data. The near-IR is dominated by old stars, especially red giants. Galaxies that have been forming stars actively for a Hubble time tend towards a quasi-equilibrium in which red giants are replenished by stellar evolution at about the same rate they move on to the next phase. One therefore expects the mass-to-light ratio to be more nearly constant in the near-IR. Not perfectly so, of course, but a 2 or 3 micron image is as close to a map of the stellar mass of a galaxy as we’re likely to get.

Around this time, the University of Maryland had begun a collaboration with Kitt Peak to build a big infrared camera, NEWFIRM, for the 4m telescope. Rob Swaters was hired to help write software to cope with the massive data flow it would produce. The instrument was divided into quadrants, each of which had a field of view sufficient to hold a typical galaxy. When it went on the telescope, we developed an efficient observing method that I called “four-shooter”, shuffling the target galaxy from quadrant to quadrant so that in processing we could remove the numerous instrumental artifacts intrinsic to its InSb detectors. This eventually became one of the standard observing modes in which the instrument was operated.

NEWFIRM in the lab in Tucson. Most of the volume is for cryogenics: the IR detectors are *helium*–*cooled to 30 K. Partial student for scale.*

I was optimistic that we could make rapid progress, and at first we did. But despite all the work, despite all the active cooling involved, we were still on the ground. The night sky was painfully bright in the IR. Indeed, the thermal component dominated, so we could observe during full moon. To an observer of low surface brightness galaxies attuned to any hint of scattered light from so much as a crescent moon, I cannot describe how discombobulating it was to walk outside the dome and see the full fricking moon. So bright. So wrong. And that wasn’t even the limiting factor: the thermal background was.

We had hit a surface brightness wall, again. We could do the bright galaxies this way, but the LSBs that sample the low acceleration end of the radial acceleration relation were rather less accessible. Not inaccessible, but there was a better way.

The Spitzer Space Telescope was active at this time. Jim Schombert and I started winning time to observe LSB galaxies with it. We discovered that space is dark. There was no atmosphere to contend with. No scattered light from the clouds or the moon or the OH lines that afflict that part of the sky spectrum. No ground-level warmth. The data were fantastic. In some sense, they were too good: the biggest headache we faced was blotting out all the background galaxies that shown right through the optically thin LSB galaxies.

Still, it took a long time to collect and analyze the data. We were starting to get results by the early-teens, but it seemed like it would take forever to get through everything I hoped to accomplish. Fortunately, when I moved to Case Western, I was able to hire Federico Lelli as a postdoc. Federico’s involvement made all the difference. After many months of hard, diligent, and exacting work, he constructed what is now the SPARC database. Finally all the elements were in place to construct an empirical radial acceleration relation with absolutely minimal assumptions about the stellar mass-to-light ratio.

In parallel with the observational work, Jim Schombert had been working hard to build realistic stellar population models that extended to the 3.6 micron band of Spitzer. Spitzer had been built to look redwards of this, further into the IR. 3.6 microns was its shortest wavelength passband. But most models at the time stopped at the K-band, the 2.2 micron band that is the reddest passband that is practically accessible from the ground. They contain pretty much the same information, but we still need to calculate the band-specific value of M*/L.

Being a thorough and careful person, Jim considered not just the star formation history of a model stellar population as a variable, and not just its average metallicity, but also the metallicity distribution of its stars, making sure that these were self-consistent with the star formation history. Realistic metallicity distributions are skewed; it turn out that this subtle effect tends to counterbalance the color dependence of the age effect on M*/L in the near-IR part of the spectrum. The net results is that we expect M*/L to be very nearly constant for all late type galaxies.

This is the best possible result. To a good approximation, we expected all of the galaxies in the SPARC sample to have the same mass-to-light ratio. What you see is what you get. No variable M*/L, no equivocation, just data in, result out.

We did still expect some scatter, as that is an irreducible fact of life in this business. But even that we expected to be small, between 0.1 and 0.15 dex (roughly 25 – 40%). Still, we expected the occasional outlier, galaxies that sit well off the main relation just because our nominal M*/L didn’t happen to apply in that case.

One day as I walked past Federico’s office, he called for me to come look at something. He had plotted all the data together assuming a single M*/L. There… were no outliers. The assumption of a constant M*/L in the near-IR didn’t just work, it worked far better than we had dared to hope. The relation leapt straight out of the data:

*The Radial Acceleration Relation from the data in McGaugh et al. (2016)*. Plot credit: Federico Lelli.

Over 150 galaxies, with nearly 2700 resolved measurements within each galaxy, each with their own distinctive mass distribution, all pile on top of each other without effort. There was plenty of effort in building the database, but once it was there, the result appeared, no muss, no fuss. No fitting or fiddling. Just the measurements and our best estimate of the mean M*/L, applied uniformly to every individual galaxy in the sample. The scatter was only 0.12 dex, within the range expected from the population models.

No MOND was involved in the construction of this relation. It may look like MOND, but we neither use MOND nor need it in any way to see the relation. It is in the data. Perhaps this is the sort of result for which we would have to invent MOND if it did not already exist. But the dark matter paradigm is very flexible, and many papers have since appeared that claim to explain the radial acceleration relation. We have reached

3. We knew it all along.

On the one hand, this is good: the community is finally engaging with a startling fact that has been pointedly ignored for decades. On the other hand, many of the claims to explain the radial acceleration relation are transparently incorrect on their face, being nothing more than elaborations of models I considered and discarded as obviously unworkable long ago. They do not provide a satisfactory explanation of the predictive power of MOND, and inevitably fail to address important aspects of the problem, like disk stability. Rather than grapple with the deep issues the new and startling fact poses, it has become fashionable to simply assert that one’s favorite model explains the radial acceleration relation, and does so naturally.

There is nothing natural about the radial acceleration relation in the context of dark matter. Indeed, it is difficult to imagine a less natural result – hence stages one and two. So on the one hand, I welcome the belated engagement, and am willing to consider serious models. On the other hand, if someone asserts that this is natural and that we expected it all along, then the engagement isn’t genuine: they’re just fooling themselves.

Early Days. This was one of Vera Rubin’s favorite expressions. I always had a hard time with it, as many things are very well established. Yet it seems that we have yet to wrap our heads around the problem. Vera’s daughter, Judy Young, once likened the situation to the parable of the blind men and the elephant. Much is known, yes, but the problem is so vast that each of us can perceive only a part of the whole, and the whole may be quite different from the part that is right before us.

So I guess Vera is right as always: these remain Early Days.

Triton Station

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