Cosmic whack-a-mole

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, Mb/R ​≪ ​MDM/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 (Σdyn) 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 md [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 (md ​∝ ​Vf), 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 md (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 md ​∝ ​Vf (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

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 Ωm → 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

Mb = AVf4 (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 Mb(R), and one for the dark matter MDM(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(Mb/R + MDM/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.

ObservationDDSH
Evolutionary rate++
Size distribution++
Clustering+X
Tully-Fisher relationX?
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 Mb/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

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

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

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 (Ωm ​≈ ​0.3, e.g., Faber and Gallagher, 1979; Davis et al., 1980, 1992) clearly exceeds the mass density in baryons as constrained by BBN (Ωb ​≈ ​0.05, e.g., Walker et al., 1991). There is something gravitating that is not ordinary matter: Ωm ​> ​Ωb.

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.

Galaxy models in compressed halos

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?

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 (gbar) and that of the dark matter (gobs-gbar). 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 gobs < gbar. Moreover, since gobs = gbar+gDM, 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 gDM. 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 gDM to the gobs 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 gbar, 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 gbar. In contrast, the acceleration gDM of an NFW halo continues to increase all the way to zero radius. Consequently, the predicted gobs continues to increase even after gbar 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.

What JWST will see

What JWST will see

Big galaxies at high redshift!

That’s my prediction, anyway. A little context first.

New Year, New Telescope

First, JWST finally launched. This has been a long-delayed NASA mission; the launch had been put off so many times it felt like a living example of Zeno’s paradox: ever closer but never quite there. A successful launch is always a relief – rockets do sometimes blow up on lift off – but there is still sweating to be done: it has one of the most complex deployments of any space mission. This is still a work in progress, but to start the new year, I thought it would be nice to look forward to what we hope to see.

JWST is a major space telescope optimized for observing in the near and mid-infrared. This enables observation of redshifted light from the earliest galaxies. This should enable us to see them as they would appear to our eyes had we been around at the time. And that time is long, long ago, in galaxies very far away: in principle, we should be able to see the first galaxies in their infancy, 13+ billion years ago. So what should we expect to see?

Early galaxies in LCDM

A theory is only as good as its prior. In LCDM, structure forms hierarchically: small objects emerge first, then merge into larger ones. It takes time to build up large galaxies like the Milky Way; the common estimate early on was that it would take at least a billion years to assemble an L* galaxy, and it could easily take longer. Ach, terminology: an L* galaxy is the characteristic luminosity of the Schechter function we commonly use to describe the number density of galaxies of various sizes. L* galaxies like the Milky Way are common, but the number of brighter galaxies falls precipitously. Bigger galaxies exist, but they are rare above this characteristic brightness, so L* is shorthand for a galaxy of typical brightness.

We expect galaxies to start small and slowly build up in size. This is a very basic prediction of LCDM. The hierarchical growth of dark matter halos is fundamental, and relatively easy to calculate. How this translates to the visible parts of galaxies is more fraught, depending on the details of baryonic infall, star formation, and the many kinds of feedback. [While I am a frequent critic of model feedback schemes implemented in hydrodynamic simulations on galactic scales, there is no doubt that feedback happens on the much smaller scales of individual stars and their nurseries. These are two very different things for which we confusingly use the same word since the former is the aspirational result of the latter.] That said, one only expects to assemble mass so fast, so the natural expectation is to see small galaxies first, with larger galaxies emerging slowly as their host dark matter halos merge together.

Here is an example of a model formation history that results in the brightest galaxy in a cluster (from De Lucia & Blaizot 2007). Little things merge to form bigger things (hence “hierarchical”). This happens a lot, and it isn’t really clear when you would say the main galaxy had formed. The final product (at lookback time zero, at redshift z=0) is a big galaxy composed of old stars – fairly typically for a giant elliptical. But the most massive progenitor is still rather small 8 billion years ago, over 4 billion years after the Big Bang. The final product doesn’t really emerge until the last major merger around 4 billion years ago. This is just one example in one model, and there are many different models, so your mileage will vary. But you get the idea: it takes a long time and a lot of mergers to assemble a big galaxy.

Brightest cluster galaxy merger tree. Time progresses upwards from early in the universe at bottom to the present day at top. Every line is a small galaxy that merges to ultimately form the larger galaxy. Symbols are color-coded by B−V color (red meaning old stars, blue young) and their area scales with the stellar mass (bigger circles being bigger galaxies. From De Lucia & Blaizot 2007).

It is important to note that in a hierarchical model, the age of a galaxy is not the same as the age of the stars that make up the galaxy. According to De Lucia & Blaizot, the stars of the brightest cluster galaxies

“are formed very early (50 per cent at z~5, 80 per cent at z~3)”

but do so

“in many small galaxies”

– i.e., the little progenitor circles in the plot above. The brightest cluster galaxies in their model build up rather slowly, such that

“half their final mass is typically locked-up in a single galaxy after z~0.5.”

De Lucia & Blaizot (2007)

So all the star formation happens early in the little things, but the final big thing emerges later – a lot later, only reaching half its current size when the universe is about 8 Gyr old. (That’s roughly when the solar system formed: we are late-comers to this party.) Given this prediction, one can imagine that JWST should see lots of small galaxies at high redshift, their early star formation popping off like firecrackers, but it shouldn’t see any big galaxies early on – not really at z > 3 and certainly not at z > 5.

Big galaxies in the data at early times?

While JWST is eagerly awaited, people have not been idle about looking into this. There have been many deep surveys made with the Hubble Space Telescope, augmented by the infrared capable (and now sadly defunct) Spitzer Space Telescope. These have already spied a number of big galaxies at surprisingly high redshift. So surprising that Steinhardt et al. (2016) dubbed it “The Impossibly Early Galaxy Problem.” This is their key plot:

The observed (points) and predicted (lines) luminosity functions of galaxies at various redshifts (colors). If all were well, the points would follow the lines of the same color. Instead, galaxies appear to be brighter than expected, already big at the highest redshifts probed. From Steinhardt et al. (2016).

There are lots of caveats to this kind of work. Constructing the galaxy luminosity function is a challenging task at any redshift; getting it right at high redshift especially so. While what counts as “high” varies, I’d say everything on the above plot counts. Steinhardt et al. (2016) worry about these details at considerable length but don’t find any plausible way out.

Around the same time, one of our graduate students, Jay Franck, was looking into similar issues. One of the things he found was that not only were there big galaxies in place early on, but they were also in clusters (or at least protoclusters) early and often. That is to say, not only are the galaxies too big too soon, so are the clusters in which they reside.

Dr. Franck made his own comparison of data to models, using the Millennium simulation to devise an apples-to-apples comparison:

The apparent magnitude m* at 4.5 microns of L* galaxies in clusters as a function of redshift. Circles are data; squares represent the Millennium simulation. These diverge at z > 2: galaxies are brighter (smaller m*) than predicted (Fig. 5.5 from Franck 2017).

The result is that the data look more like big galaxies formed early already as big galaxies. The solid lines are “passive evolution” models in which all the stars form in a short period starting at z=10. This starting point is an arbitrary choice, but there is little cosmic time between z = 10 and 20 – just a few hundred million years, barely one spin around the Milky Way. This is a short time in stellar evolution, so is practically the same as starting right at the beginning of time. As Jay put it,

“High redshift cluster galaxies appear to be consistent with an old stellar population… they do not appear to be rapidly assembling stellar mass at these epochs.”

Franck 2017

We see old stars, but we don’t see the predicted assembly of galaxies via mergers, at least not at the expected time. Rather, it looks like some galaxies were already big very early on.

As someone who has worked mostly on well resolved, relatively nearby galaxies, all this makes me queasy. Jay, and many others, have worked desperately hard to squeeze knowledge from the faint smudges detected by first generation space telescopes. JWST should bring these into much better focus.

Early galaxies in MOND

To go back to the first line of this post, big galaxies at high redshift did not come as a surprise to me. It is what we expect in MOND.

Structure formation is generally considered a great success of LCDM. It is straightforward and robust to calculate on large scales in linear perturbation theory. Individual galaxies, on the other hand, are highly non-linear objects, making them hard to beasts to tame in a model. In MOND, it is the other way around – predicting the behavior of individual galaxies is straightforward – only the observed distribution of mass matters, not all the details of how it came to be that way – but what happens as structure forms in the early universe is highly non-linear.

The non-linearity of MOND makes it hard to work with computationally. It is also crucial to how structure forms. I provide here an outline of how I expect structure formation to proceed in MOND. This page is now old, even ancient in internet time, as the golden age for this work was 15 – 20 years ago, when all the essential predictions were made and I was naive enough to think cosmologists were amenable to reason. Since the horizon of scientific memory is shorter than that, I felt it necessary to review in 2015. That is now itself over the horizon, so with the launch of JWST, it seems appropriate to remind the community yet again that these predictions exist.

This 1998 paper by Bob Sanders is a foundational paper in this field (see also Sanders 2001 and the other references given on the structure formation page). He says, right in the abstract,

“Objects of galaxy mass are the first virialized objects to form (by z = 10), and larger structure develops rapidly.”

Sanders (1998)

This was a remarkable prediction to make in 1998. Galaxies, much less larger structures, were supposed to take much longer to form. It takes time to go from the small initial perturbations that we see in the CMB at z=1000 to large objects like galaxies. Indeed, the it takes at least a few hundred million years simply in free fall time to assemble a galaxy’s worth of mass, a hard limit. Here Sanders was saying that an L* galaxy might assemble as early as half a billion years after the Big Bang.

So how can this happen? Without dark matter to lend a helping hand, structure formation in the very early universe is inhibited by the radiation field. This inhibition is removed around z ~ 200; exactly when being very sensitive to the baryon density. At this point, the baryon perturbations suddenly find themselves deep in the MOND regime, and behave as if there is a huge amount of dark matter. Structure proceeds hierarchically, as it must, but on a highly compressed timescale. To distinguish it from LCDM hierarchical galaxy formation, let’s call it prompt structure formation. In prompt structure formation, we expect

  • Early reionization (z ~ 20)
  • Some L* galaxies by z ~ 10
  • Early emergence of the cosmic web
  • Massive clusters already at z > 2
  • Large, empty voids
  • Large peculiar velocities
  • A very large homogeneity scale, maybe fractal over 100s of Mpc

There are already indications of all of these things, nearly all of which were predicted in advance of the relevant observations. I could elaborate, but that is beyond the scope of this post. People should read the references* if they’re keen.

*Reading the science papers is mandatory for the pros, who often seem fond of making straw man arguments about what they imagine MOND might do without bothering to check. I once referred some self-styled experts in structure formation to Sanders’s work. They promptly replied “That would mean structures of 1018 M!” when what he said was

“The largest objects being virialized now would be clusters of galaxies with masses in excess of 1014 M. Superclusters would only now be reaching maximum expansion.”

Sanders (1998)

The exact numbers are very sensitive to cosmological parameters, as he discussed, but I have no idea where they got 1018, other than just making stuff up. More importantly, Sanders’s statement clearly presaged the observation of very massive clusters at surprisingly high redshift and the discovery of the Laniakea Supercluster.

These are just the early predictions of prompt structure formation, made in the same spirit that enabled me to predict the second peak of the microwave background and the absorption signal observed by EDGES at cosmic dawn. Since that time, at least two additional schools of thought as to how MOND might impact cosmology have emerged. One of them is the sterile neutrino MOND cosmology suggested by Angus and being actively pursued by the Bonn-Prague research group. Very recently, there is of course the new relativistic theory of Skordis & Złośnik which fits the cosmologists’ holy grail of the power spectrum in both the CMB at z = 1090 and galaxies at z = 0. There should be an active exchange and debate between these approaches, with perhaps new ones emerging.

Instead, we lack critical mass. Most of the community remains entirely obsessed with pursuing the vain chimera of invisible mass. I fear that this will eventually prove to be one of the greatest wastes of brainpower (some of it my own) in the history of science. I can only hope I’m wrong, as many brilliant people seem likely to waste their career running garbage in-garbage out computer simulations or at the bottom of a mine shaft failing to detect what isn’t there.

A beautiful mess

JWST can’t answer all of these questions, but it will help enormously with galaxy formation, which is bound to be messy. It’s not like L* galaxies are going to spring fully formed from the void like Athena from the forehead of Zeus. The early universe must be a chaotic place, with clumps of gas condensing to form the first stars that irradiate the surrounding intergalactic gas with UV photons before detonating as the first supernovae, and the clumps of stars merging to form giant elliptical galaxies while elsewhere gas manages to pool and settle into the large disks of spiral galaxies. When all this happens, how it happens, and how big galaxies get how fast are all to be determined – but now accessible to direct observation thanks to JWST.

It’s going to be a confusing, beautiful mess, in the best possible way – one that promises to test and challenge our predictions and preconceptions about structure formation in the early universe.

Galaxy Stellar and Halo Masses: tension between abundance matching and kinematics

Galaxy Stellar and Halo Masses: tension between abundance matching and kinematics

Mass is a basic quantity. How much stuff does an astronomical object contain? For a galaxy, mass can mean many different things: that of its stars, stellar remnants (e.g., white dwarfs, neutron stars), atomic gas, molecular clouds, plasma (ionized gas), dust, Bok globules, black holes, habitable planets, biomass, intelligent life, very small rocks… these are all very different numbers for the same galaxy, because galaxies contain lots of different things. Two things that many scientists have settled on as Very Important are a galaxy’s stellar mass and its dark matter halo mass.

The mass of a galaxy’s dark matter halo is not well known. Most measurement provide only lower limits, as tracers fade out before any clear end is reached. Consequently, the “total” mass is a rather notional quantity. So we’ve adopted as a convention the mass M200 contained within an over-density of 200 times the critical density of the universe. This is a choice motivated by an ex-theory that would take an entire post to explain unsatisfactorily, so do not question the convention: all choices are bad, so we stick with it.

One of the long-standing problems the cold dark matter paradigm has is that the galaxy luminosity function should be steep but is observed to be shallow. This sketch shows the basic issue. The number density of dark matter halos as a function of mass is expected to be a power law – one that is well specified once the cosmology is known and a convention for the mass is adopted. The obvious expectation is that the galaxy luminosity function should just be a downshifted version of the halo mass function: one galaxy per halo, with the stellar mass proportional to the halo mass. This was such an obvious assumption [being provision (i) of canonical galaxy formation in LCDM] that it was not seriously questioned for over a decade. (Minor point: a turn down at the high mass end could be attributed to gas cooling times: the universe didn’t have time to cool and assemble a galaxy above some threshold mass, but smaller things had plenty of time for gas to cool and form stars.)

The number density of galaxies (blue) and dark matter halos (red) as a function of their mass. Our original expectation is on the left: the galaxy mass function should be a down-shifted version of the halo mass function, up to a gas cooling limit. Dashed grey lines illustrate the correspondence of galaxies with dark matter halos of proportional mass: M* = md M200. On the right is the current picture of abundance matching with the grey lines connecting galaxies with dark matter halos of equal cosmic density in which they are supposed to reside. In effect, we make the proportionality factor md a rolling, mass-dependent fudge factor.

The galaxy luminosity function does not look like a shifted version of the halo mass function. It has the wrong slope at the faint end. At no point is the size of the shift equal to what one would expect from the mass of available baryons. The proportionality factor md is too small; this is sometimes called the over-cooling problem, in that a lot more baryons should have cooled to form stars than apparently did so. So, aside from the shape and the normalization, it’s a great match.

We obsessed about this problem all through the ’90s. At one point, I thought I had solved it. Low surface brightness galaxies were under-represented in galaxy surveys. They weren’t missed entirely, but their masses could be systematically underestimated. This might matter a lot because the associated volume corrections are huge. A small systematic in mass would get magnified into a big one in density. Sadly, after a brief period of optimism, it became clear that this could not work to solve the entire problem, which persists.

Circa 2000, a local version of the problem became known as the missing satellites problem. This is a down-shifted version of the mismatch between the galaxy luminosity function and the halo mass function that pervades the entire universe: few small galaxies are observed where many are predicted. To give visual life to the numbers we’re talking about, here is an image of the dark matter in a simulation of a Milky Way size galaxy:

Dark Matter in the Via Lactea simulation (Diemand et al. 2008). The central region is the main dark matter halo which would contain a large galaxy like the Milky Way. All the lesser blobs are subhalos. A typical galaxy-sized dark matter halo should contain many, many subhalos. Naively, we expect each subhalo to contain a dwarf satellite galaxy. Structure is scale-free in CDM, so major galaxies should look like miniature clusters of galaxies.

In contrast, real galaxies have rather fewer satellites that meet the eye:

NGC 6946 and environs. The points are foreground stars, ignore them. The neighborhood of NGC 6946 appears to be pretty empty – there is no swarm of satellite galaxies as in the simulation above. I know of two dwarf satellite galaxies in this image, both of low surface brightness. The brighter one (KK98-250) the sharp-eyed may find between the bright stars at top right. The fainter one (KK98-251) is nearby KK98-250, a bit down and to the left of it; good luck seeing it on this image from the Digital Sky Survey. That’s it. There are no other satellite galaxies visible here. There can of course be more that are too low in surface brightness to detect. The obvious assumption of a one-to-one relation between stellar and halo mass cannot be sustained; there must instead be a highly non-linear relation between mass and light so that subhalos contain only contain dwarfs of extraordinarily low surface brightness.

By 2010, we’d thrown in the towel, and decided to just accept that this aspect of the universe was too complicated to predict. The story now is that feedback changes the shape of the luminosity function at both the faint and the bright ends. Exactly how depends on who you ask, but the predicted halo mass function is sacrosanct so there must be physical processes that make it so. (This is an example of the Frenk Principle in action.)

Lacking a predictive theory, theorists instead came up with a clever trick to relate galaxies to their dark matter halos. This has come to be known as abundance matching. We measure the number density of galaxies as a function of stellar mass. We know, from theory, what the number density of dark matter halos should be as a function of halo mass. Then we match them up: galaxies of a given density live in halos of the corresponding density, as illustrated by the horizontal gray lines in the right panel of the figure above.

There have now been a number of efforts to quantify this. Four examples are given in the figure below (see this paper for references), together with kinematic mass estimates.

The ratio of stellar to halo mass as a function of dark matter halo mass. Lines represent the abundance matching relations derived by assigning galaxies to dark matter halos based on their cosmic abundance. Points are independent halo mass estimates based on kinematics (McGaugh et al. 2010). The horizontal dashed line represents the maximum stellar mass that would result if all available baryons were turned into stars. (Mathematically, this happens when md equals the cosmic baryon fraction, about 15%.)

The abundance matching relations have a peak around a halo mass of 1012 M and fall off to either side. This corresponds to the knee in the galaxy luminosity function. For whatever reason, halos of this mass seem to be most efficient at converting their available baryons into stars. The shape of these relations mean that there is a non-linear relation between stellar mass and halo mass. At the low mass end, a big range in stellar mass is compressed into a small range in halo mass. The opposite happens at high mass, where the most massive galaxies are generally presumed to be the “central” galaxy of a cluster of galaxies. We assign the most massive halos to big galaxies understanding that they may be surrounded by many subhalos, each containing a cluster galaxy.

Around the same time, I made a similar plot, but using kinematic measurements to estimate halo masses. Both methods are fraught with potential systematics, but they seem to agree reasonably well – at least over the range illustrated above. It gets dodgy above and below that. The agreement is particularly good for lower mass galaxies. There seems to be a departure for the most massive individual galaxies, but why worry about that when the glass is 3/4 full?

Skip ahead a decade, and some people think we’ve solved the missing satellite problem. One key ingredient of that solution is that the Milky Way resides in a halo that is on the lower end of the mass range that has traditionally been estimated for it (1 to 2 x 1012 M). This helps because the number of subhalos scales with mass: clusters are big halos with lots of galaxy-size halos; the Milky Way is a galaxy-sized halo with lots of smaller subhalos. Reality does not look like that, but having a lower mass means fewer subhalos, so that helps. It does not suffice. We must invoke feedback effects to make the relation between light and mass nonlinear. Then the lowest mass satellites may be too dim to detect: selection effects have to do a lot of work. It also helps to assume the distribution of satellites is isotropic, which looks to be true in the simulation, but not so much in reality where known dwarf satellites occupy a planar distribution. We also need to somehow fudge the too-big-to-fail problem, in which the more massive subhalos appear not to be occupied by luminous galaxies at all. Given all that, we can kinda sorta get in the right ballpark. Kinda, sorta, provided that we live in a galaxy whose halo mass is closer to 1012 M than to 2 x 1012 M.

At an IAU meeting in Shanghai (in July 2019, before travel restrictions), the subject of the mass of the Milky Way was discussed at length. It being our home galaxy, there are many ways in which to constrain the mass, some of which take advantage of tracers that go out to greater distances than we can obtain elsewhere. Speaker after speaker used different methods to come to a similar conclusion, with the consensus hedging on the low side (roughly 1 – 1.5 x 1012 M). A nice consequence would be that the missing satellite problem may no longer be a problem.

Galaxies in general and the Milky Way in particular are different and largely distinct subfields. Different data studied by different people with distinctive cultures. In the discussion at the end of the session, Pieter van Dokkum pointed out that from the perspective of other galaxies, the halo mass ought to follow from abundance matching, which for a galaxy like the Milky Way ought to be more like 3 x 1012 M, considerably more than anyone had suggested, but hard to exclude because most of that mass could be at distances beyond the reach of the available tracers.

This was not well received.

The session was followed by a coffee break, and I happened to find myself standing in line next to Pieter. I was still processing his comment, and decided he was right – from a certain point of view. So we got to talking about it, and wound up making the plot below, which appears in a short research note. (For those who know the field, it might be assumed that Pieter and I hate each other. This is not true, but we do frequently disagree, so the fact that we do agree about this is itself worthy of note.)

The Local Group and its two most massive galaxies, the Milky Way and Andromeda (M31), in the stellar mass-halo mass plane. Lines are the abundance matching relations from above. See McGaugh & van Dokkum for further details. The remaining galaxies of the Local Group all fall off the edge of this plot, and do not add up to anything close to either the Milky Way or Andromeda alone.

The Milky Way and Andromeda are the 1012 M gorillas of the Local Group. There are many dozens of dwarf galaxies, but none of them are comparable in mass, even with the boost provided by the non-linear relation between mass and luminosity. To astronomical accuracy, in terms of mass, the Milky Way plus Andromeda are the Local Group. There are many distinct constraints, on each galaxy as an individual, and on the Local Group as a whole. Any way we slice it, all three entities lie well off the relation expected from abundance matching.

There are several ways one could take it from here. One might suppose that abundance matching is correct, and we have underestimated the mass with other measurements. This happens all the time with rotation curves, which typically do not extend far enough out into the halo to give a good constraint on the total mass. This is hard to maintain for the Local Group, where we have lots of tracers in the form of dwarf satellites, and there are constraints on the motions of galaxies on still larger scales. Moreover, a high mass would be tragic for the missing satellite problem.

One might instead imagine that there is some scatter in the abundance matching relation, and we just happen to live in a galaxy that has a somewhat low mass for its luminosity. This is almost reasonable for the Milky Way, as there is some overlap between kinematic mass estimates and the expectations of abundance matching. But the missing satellite problem bites again unless we are pretty far off the central value of the abundance matching relation. Other Milky Way-like galaxies ought to fall on the other end of the spectrum, with more mass and more satellites. A lot of work is going on to look for satellites around other spirals, which is hard work (see NGC 6946 above). There is certainly scatter in the number of satellites from system to system, but whether this is theoretically sensible or enough to explain our Milky Way is not yet apparent.

There is a tendency in the literature to invoke scatter when and where needed. Here, it is important to bear in mind that there is little scatter in the Tully-Fisher relation. This is a relation between stellar mass and rotation velocity, with the latter supposedly set by the halo mass. We can’t have it both ways. Lots of scatter in the stellar mass-halo mass relation ought to cause a corresponding amount of scatter in Tully-Fisher. This is not observed. It is a much stronger than most people seem to appreciate, as even subtle effects are readily perceptible. Consequently, I think it unlikely that we can nuance the relation between halo mass and observed rotation speed to satisfy both relations without a lot of fine-tuning, which is usually a sign that something is wrong.

There are a lot of moving parts in modern galaxy formation simulations that need to be fine-tuned: the effects of halo mass, merging, dissipation, [non]adiabatic compression, angular momentum transport, gas cooling, on-going accretion of gas from the intergalactic medium, expulsion of gas in galactic winds, re-accretion of expelled gas via galactic fountains, star formation and the ensuing feedback from radiation pressure, stellar winds, supernovae, X-rays from stellar remnants, active galactic nuclei, and undoubtedly other effects I don’t recall off the top of my head. Visualization from the Dr. Seuss suite of simulations.

A lot of effort has been put into beating down the missing satellite problem around the Milky Way. Matters are worse for Andromeda. Kinematic halo mass estimates are typically in the same ballpark as the Milky Way. Some are a bit bigger, some are lower. Lower is a surprise, because the stellar mass of M31 is clearly bigger than that of the Milky Way, placing it is above the turnover where the efficiency of star formation is maximized. In this regime, a little stellar mass goes a long way in terms of halo mass. Abundance matching predicts that a galaxy of Andromeda’s stellar mass should reside in a dark matter halo of at least 1013 M. That’s quite a bit more than 1 or 2 x 1012 M, even by astronomical standards. Put another way, according to abundance matching, the Local Group should have the Milky Way as its most massive occupant. Just the Milky Way. Not the Milky Way plus Andromeda. Despite this, the Local Group is not anomalous among similar groups.

Words matter. A lot boils down to what we consider to be “close enough” to call similar. I do not consider the Milky Way and Andromeda to be all that similar. They are both giant spirals, yes, but galaxies are all individuals. Being composed of hundreds of billions of stars, give or take, leaves a lot of room for differences. In this case, the Milky Way and Andromeda are easily distinguished in the Tully-Fisher plane. Andromeda is about twice the baryonic mass of the Milky Way. It also rotates faster. The error bars on these quantities do not come close to overlapping – that would be one criterion for considering them to be similar – a criterion they do not meet. Even then, there could be other features that might be readily distinguished, but let’s say a rough equality in the Tully-Fisher plane would indicate stellar and halo masses that are “close enough” for our present discussion. They aren’t: to me, the Milky Way and M31 are clearly different galaxies.

I spent a fair amount of time reading the recent literature on satellites searches, and I was struck by the ubiquity with which people make the opposite assumption, treating the Milky Way and Andromeda as interchangeable galaxies of similar mass. Why would they do this? If one looks at the kinematic halo mass as the defining characteristic of a galaxy, they’re both close to 1012 M, with overlapping error bars on M200. By that standard, it seems fair. Is it?

Luminosity is observable. Rotation speed is observable. There are arguments to be had about how to convert luminosity into stellar mass, and what rotation speed measure is “best.” These are sometimes big arguments, but they are tiny in scale compared to estimating notional quantities like the halo mass. The mass M200 is not an observable quantity. As such, we have no business using it as a defining characteristic of a galaxy. You know a galaxy when you see it. The same cannot be said of a dark matter halo. Literally.

If, for some theoretically motivated reason, we want to use halo mass as a standard then we need to at least use a consistent method to assess its value from directly observable quantities. The methods we use for the Milky Way and M31 are not applicable beyond the Local Group. Nowhere else in the universe do we have such an intimate picture of the kinematic mass from a wide array of independent methods with tracers extending to such large radii. There are other standards we could apply, like the Tully-Fisher relation. That we can do outside the Local Group, but by that standard we would not infer that M31 and the Milky Way are the same. Other observables we can fairly apply to other galaxies are their luminosities (stellar masses) and cosmic number densities (abundance matching). From that perspective, what we know from all the other galaxies in the universe is that the factor of ~2 difference in stellar mass between Andromeda and the Milky Way should be huge in terms of halo mass. If it were anywhere else in the universe, we wouldn’t treat these two galaxies as interchangeably equal. This is the essence of Pieter’s insight: abundance matching is all about the abundance of dark matter halos, so that would seem to be the appropriate metric by which to predict the expected number of satellites, not the kinematic halo mass that we can’t measure in the same way anywhere else in the universe.

That isn’t to say we don’t have some handle on kinematic halo masses, it’s just that most of that information comes from rotation curves that don’t typically extend as far as the tracers that we have in the Local Group. Some rotation curves are more extended than others, so one has to account for that variation. Typically, we can only put a lower limit on the halo mass, but if we assume a profile like NFW – the standard thing to do in LCDM, then we can sometimes exclude halos that are too massive.

Abundance matching has become important enough to LCDM that we included it as a prior in fitting dark matter halo models to rotation curves. For example:

The stellar mass-halo mass relation from rotation curve fits (Li et al 2020). Each point is one galaxy; the expected abundance matching relation (line) is not recovered (left) unless it is imposed as a prior (right). The data are generally OK with this because the amount of mass at radii beyond the end of the rotation curve is not strongly constrained. Still, there are some limits on how crazy this can get.

NFW halos are self-similar: low mass halos look very much like high mass halos over the range that is constrained by data. Consequently, if you have some idea what the total mass of the halo should be, as abundance matching provides, and you impose that as a prior, the fits for most galaxies say “OK.” The data covering the visible galaxy have little power to constrain what is going on with the dark matter halo at much larger radii, so the fits literally fall into line when told to do so, as seen in Pengfei‘s work.

That we can impose abundance matching as a prior does not necessarily mean the result is reasonable. The highest halo masses that abundance matching wants in the plot above are crazy talk from a kinematic perspective. I didn’t put too much stock in this, as the NFW halo itself, the go-to standard of LCDM, provides the worst description of the data among all the dozen or so halo models that we considered. Still, we did notice that even with abundance matching imposed as a prior, there are a lot more points above the line than below it at the high mass end (above the bend in the figure above). The rotation curves are sometimes pushing back against the imposed prior; they often don’t want such a high halo mass. This was explored in some detail by Posti et al., who found a similar effect.

I decided to turn the question around. Can we use abundance matching to predict the halo and hence rotation curve of a massive galaxy? The largest spiral in the local universe, UGC 2885, has one of the most extended rotation curves known, meaning that it does provide some constraint on the halo mass. This galaxy has been known as an important case since Vera Rubin’s work in the ’70s. With a modern distance scale, its rotation curve extends out 80 kpc. That’s over a quarter million light-years – a damn long way, even by the standards of galaxies. It also rotates remarkably fast, just shy of 300 km/s. It is big and massive.

(As an aside, Vera once offered a prize for anyone who found a disk that rotated faster than 300 km/s. Throughout her years of looking at hundreds of galaxies, UGC 2885 remained the record holder, with 300 seeming to be a threshold that spirals did not exceed. She told me that she did pay out, but on a technicality: someone showed her a gas disk around a supermassive black hole in Keplerian rotation that went up to 500 km/s at its peak. She lamented that she had been imprecise in her language, as that was nothing like what she meant, which was the flat rotation speed of a spiral galaxy.)

That aside aside, if we take abundance matching at face value, then the stellar mass of a galaxy predicts the mass of its dark matter halo. Using the most conservative (in that it returns the lowest halo mass) of the various abundance matching relations indicates that with a stellar mass of about 2 x 1011 M, UGC 2885 should have a halo mass of 3 x 1013 M. Combining this with a well-known relation between halo concentration and mass for NFW halos, we then know what the rotation curve should be. Doing this for UGC 2885 yields a tragic result:

The extended rotation curve of UGC 2885 (points). The declining dotted line is the rotation curve predicted by the observed stars and gas. The rising dashed line is the halo predicted by abundance matching. Combining this halo with the observed stars and gas should result in the solid line. This greatly exceeds the data. UGC 2885 does not reside in an NFW halo that is anywhere near as massive as predicted by abundance matching.

The data do not allow for the predicted amount of dark matter. If we fit the rotation curve, we obtain a “mere” M200 = 5 x 1012 M. Note that this means that UGC 2885 is basically the Milky Way and Andromeda added together in terms of both stellar mass and halo mass – if added to the M*-M200 plot above, it would land very close to the open circle representing the more massive halo estimate for the combination of MW+M31, and be just as discrepant from the abundance matching relations. We get the same result regardless of which direction we look at it from.

Objectively, 5 x 1012 M is a huge dark matter halo for a single galaxy. It’s just not the yet-more massive halo that is predicted by abundance matching. In this context, UGC 2885 apparently has a serious missing satellites problem, as it does not appear to be swimming in a sea of satellite galaxies the way we’d expect for the central galaxy of such high mass halo.

UGC 2885 appears to be pretty lonely in this image from the DSS. I see a few candidate satellite galaxies amidst the numerous foreground stars, but nothing like what you’d expect for dark matter subhalos from a simulation like the via Lactea. This impression does not change when imaged in more detail with HST.

It is tempting to write this off as a curious anecdote. Another outlier. Sure, that’s always possible, but this is more than a bit ridiculous. Anyone who wants to go this route I refer to Snoop Dog.

I spent much of my early career obsessed with selection effects. These preclude us from seeing low surface brightness galaxies as readily as brighter ones. However, it isn’t binary – a galaxy has to be extraordinarily low surface brightness before it becomes effectively invisible. The selection effect is a bias – and a very strong one – but not an absolute screen that prevents us from finding low surface brightness galaxies. That makes it very hard to sustain the popular notion that there are lots of subhalos that simply contain ultradiffuse galaxies that cannot currently be seen. I’ve been down this road many times as an optimist in favor of this interpretation. It hasn’t worked out. Selection effects are huge, but still nowhere near big enough to overcome the required deficit.

Having the satellite galaxies that inhabit subhalos be low in surface brightness is a necessary but not sufficient criterion. It is also necessary to have a highly non-linear stellar mass-halo mass relation at low mass. In effect, luminosity and halo mass become decoupled: satellite galaxies spanning a vast range in luminosity must live in dark matter halos that cover only a tiny range. This means that it should not be possible to predict stellar motions in these galaxies from their luminosity. The relation between mass and light has just become too weak and messy.

And yet, we can do exactly that. Over and over again. This simply should not be possible in LCDM.

The Star Forming Main Sequence – Dwarf Style

The Star Forming Main Sequence – Dwarf Style

A subject of long-standing interest in extragalactic astronomy is how stars form in galaxies. Some galaxies are “red and dead” – most of their stars formed long ago, and have evolved as stars will: the massive stars live bright but short lives, leaving the less massive ones to linger longer, producing relatively little light until they swell up to become red giants as they too near the end of their lives. Other galaxies, including our own Milky Way, made some stars in the ancient past and are still actively forming stars today. So what’s the difference?

The difference between star forming galaxies and those that are red and dead turns out to be both simple and complicated. For one, star forming galaxies have a supply of cold gas in their interstellar media, the fuel from which stars form. Dead galaxies have very little in the way of cold gas. So that’s simple: star forming galaxies have the fuel to make stars, dead galaxies don’t. But why that difference? That’s a more complicated question I’m not going to begin to touch in this post.

One can see current star formation in galaxies in a variety of ways. These usually relate to the ultraviolet (UV) photons produced by short-lived stars. Only O stars are hot enough to produce the ionizing radiation that powers the emission of HII (pronounced `H-two’) regions – regions of ionized gas that are like cosmic neon lights. O stars power HII regions but live less than 10 million years. That’s a blink of the eye on the cosmic timescale, so if you see HII regions, you know stars have formed recently enough that the short-lived O stars are still around.

f549_1_small
The dwarf LSB galaxy F549-1 and companion. The pink knots are HII regions detected in the light of H-alpha, the first emission line in the Balmer sequence of hydrogen. HII regions are ionized by short-lived O-stars, serving as cosmic shingles that shout “Hey! We’re forming stars here!”

Measuring the intensity of the Hα Balmer line emission provides a proxy for the number of UV photons that ionize the gas, which in turn basically counts the number of O stars that produce the ionizing radiation. This number, divided by the short life-spans of O stars, measures the current star formation rate (SFR).

There are many uncertainties in the calibration of this SFR: how many UV photons do O stars emit? Over what time span? How many of these ionizing photons are converted into Hα, and how many are absorbed by dust or manage to escape into intergalactic space? For every O star that comes and goes, how many smaller stars are born along with it? This latter question is especially pernicious, as most stellar mass resides in small stars. The O stars are only the tip of the iceberg; we are using the tip to extrapolate the size of the entire iceberg.

Astronomers have obsessed over these and related questions for a long time. See, for example, the review by Kennicutt & Evans. Suffice it to say we have a surprisingly decent handle on it, and yet the systematic uncertainties remain substantial. Different methods give the same answer to within an order of magnitude, but often differ by a factor of a few. The difference is often in the mass spectrum of stars that is assumed, but even rationalizing that to the same scale, the same data can be interpreted to give different answers, based on how much UV we estimate to be absorbed by dust.

In addition to the current SFR, one can also measure the stellar mass. This follows from the total luminosity measured from starlight. Many of the same concerns apply, but are somewhat less severe because more of the iceberg is being measured. For a long time we weren’t sure we could do better than a factor of two, but this work has advanced to the point where the integrated stellar masses of galaxies can be estimated to ~20% accuracy.

A diagram that has become popular in the last decade or so is the so-called star forming main sequence. This name is made in analogy with the main sequence of stars, the physics of which is well understood. Whether this is an appropriate analogy is debatable, but the terminology seems to have stuck. In the case of galaxies, the main sequence of star forming galaxies is a plot of star formation rate against stellar mass.

The star forming main sequence is shown in the graph below. It is constructed from data from the SINGS survey (red points) and our own work on dwarf low surface brightness (LSB) galaxies (blue points). Each point represents one galaxy. Its stellar mass is determined by adding up the light emitted by all the stars, while the SFR is estimated from the Hα emission that traces the ionizing UV radiation of the O stars.

SFMSannotated.001
The star formation rate measured as a function of stellar mass for star forming galaxies, the “star forming main sequence” (from McGaugh, Schombert, & Lelli 2017). Each point represents one galaxy. Star formation is rapid in the most luminous spirals, which contain tens of thousands of O stars. In contrast, some dwarf galaxies contain only a single HII region that is so faint that it may be ionized by a single O star.

The data show a nice correlation, albeit with plenty of intrinsic scatter. This is hardly surprising, as the two axes are not physically independent. They are measuring different quantities that trace the same underlying property: star formation over different time scales. The y-axis is a measure of the quasi-instantaneous star formation rate; the x-axis is the SFR integrated over the age of the galaxy.

Since the stellar mass is the time integral of the SFR, one expects the slope of the star forming main sequence (SFMS) to be one. This is illustrated by the diagonal line marked “Hubble time.” A galaxy forming stars at a constant rate for the age of the universe will fall on this line.

The data for LSB galaxies scatter about a line with slope unity. The best-fit line has a normalization a bit less than that of a constant SFR for a Hubble time. This might mean that the galaxies are somewhat younger than the universe (a little must be true, but need not be much), have a slowly declining SFR (an exponential decline with an e-folding time of a Hubble time works well), or it could just be an error in the calibration of one or both axes. The systematic errors discussed above are easily large enough to account for the difference.

To first order, the SFR in LSB galaxies is constant when averaged over billions of years. On the millions of years timescale appropriate to O stars, the instantaneous SFR bounces up and down. Looks pretty stochastic: galaxies form stars at a steady average rate that varies up and down on short timescales.

Short-term fluctuations in the SFR explain the data with current SFR higher than the past average. These are the points that stray into the gray region of the plot, which becomes increasingly forbidden towards the top left. This is because galaxies that form stars so fast for too long will build up their entire stellar mass in the blink of a cosmic eye. This is illustrated by the lines marked as 0.1 and 0.01 of a Hubble time. A galaxy above these lines would make all their stars in < 2 Gyr; it would have had to be born yesterday. No galaxies reside in this part of the diagram. Those that approach it are called “starbursts:” they’re forming stars at a high specific rate (relative to their mass) but this is presumably a brief-lived phenomenon.

Note that the most massive of the SINGS galaxies all fall below the extrapolation of the line fit to the LSB galaxies (dotted line). The are forming a lot of stars in an absolute sense, simply because they are giant galaxies. But the current SFR is lower than the past average, as if they were winding down. This “quenching” seems to be a mass-dependent phenomenon: more massive galaxies evolve faster, burning through their gas supply before dwarfs do. Red and dead galaxies have already completed this process; the massive spirals of today are weary giants that may join the red and dead galaxy population in the future.

One consequence of mass-dependent quenching is that it skews attempts to fit relations to the SFMS. There are very many such attempts in the literature; these usually have a slope less than one. The dashed line in the plot above gives one specific example. There are many others.

If one looks only at the most massive SINGS galaxies, the slope is indeed shallower than one. Selection effects bias galaxy catalogs strongly in favor of the biggest and brightest, so most work has been done on massive galaxies with M* > 1010 M. That only covers the top one tenth of the area of this graph. If that’s what you’ve got to work with, you get a shallow slope like the dashed line.

The dashed line does a lousy job of extrapolating to low mass. This is obvious from the dwarf galaxy data. It is also obvious from the simple mathematical considerations outlined above. Low mass galaxies could only fall on the dashed line if they were born yesterday. Otherwise, their high specific star formation rates would over-produce their observed stellar mass.

Despite this simple physical limit, fits to the SFMS that stray into the forbidden zone are ubiquitous in the literature. In addition to selection effects, I suspect the calibrations of both SFR and stellar mass are in part to blame. Galaxies will stray into the forbidden zone if the stellar mass is underestimated or the SFR is overestimated, or some combination of the two. Probably both are going on at some level. I suspect the larger problem is in the SFR. In particular, it appears that many measurements of the SFR have been over-corrected for the effects of dust. Such a correction certainly has to be made, but since extinction corrections are exponential, it is easy to over-do. Indeed, I suspect this is why the dashed line overshoots even the bright galaxies from SINGS.

This brings us back to the terminology of the main sequence. Among stars, the main sequence is defined by low mass stars that evolve slowly. There is a turn-off point, and an associated mass, where stars transition from the main sequence to the sub giant branch. They then ascend the red giant branch as they evolve.

If we project this terminology onto galaxies, the main sequence should be defined by the low mass dwarfs. These are nowhere near to exhausting their gas supplies, so can continue to form stars far into the future. They establish a star forming main sequence of slope unity because that’s what the math says they must do.

Most of the literature on this subject refers to massive star forming galaxies. These are not the main sequence. They are the turn-off population. Massive spirals are near to exhausting their gas supply. Star formation is winding down as the fuel runs out.

Red and dead galaxies are the next stage, once star formation has stopped entirely. I suppose these are the red giants in this strained analogy to individual stars. That is appropriate insofar as most of the light from red and dead galaxies is produced by red giant stars. But is this really they right way to think about it? Or are we letting our terminology get the best of us?