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 = gN, where gN 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 → √(a0gN) 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 Mp in the deep MOND regime (eq. (5)) will experience a centripetal acceleration
Vc2/R = √(a0GMp/R2). (6)
Note that the term for the radius R cancels out, so eq. (6) reduces to
Vc4 = a0GMp (7)
which the reader will recognize as the Baryonic Tully-Fisher relation
Mb = A Vf4 (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: Vf 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 ~ Vx 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, Mdyn/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.
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 = Vobs/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).
|Slope = 4||+|
|No size or surface brightness residuals||+|
|Mdyn/L depends on surface brightness||+|
|Central density relation||+|
|Rotation curve fits||+|
|Stellar population mass-to-light ratios||+|
|Mb alone specifies Vf||+|
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.