One of the strongest correlations in extragalactic astronomy is the Tully–Fisher relation . Originally identified as an empirical relation between a galaxy’s luminosity and its HI line-width, it has been widely employed as a distance indicator. Though extensively studied for decades, the physical basis of the relation remains unclear.
Luminosity and line-width are readily accessible observational quantities. The optical luminosity of a galaxy is a proxy for its stellar mass, and the HI line-width is a proxy for its rotation velocity. The quality of the correlation improves as more accurate indicators of these quantities are employed. For example, resolved rotation curves, where the flat portion of the rotation curve or the maximum peak velocity can be measured, give relations that are tighter than those utilizing only line-width information . Similarly, the scatter declines as we shift from optical luminosities to those in the near-infrared  as the latter are expected to give a more reliable mapping of starlight to stellar mass .
It was then realized [322, 157, 283] that a more fundamental relation was that between the total observed baryonic mass and the rotation velocity. In most bright galaxies, the stars harbor the majority of the detected baryonic mass, so luminosity suffices as a proxy for mass. The next-most–important known reservoir of baryons is the neutral atomic hydrogen (HI) of the interstellar medium. As studies have probed down the mass spectrum to lower mass, more slowly rotating systems, a higher preponderance of gas rich galaxies is found. The luminous Tully–Fisher relation breaks down [283, 272], but a tight relation persists if instead of luminosity, the detected baryonic mass is used [283, 475, 42, 272, 353, 31, 445, 462, 276]. This is the Baryonic Tully–Fisher Relation (BTFR), plotted on Figure 3.
The luminous Tully–Fisher relation extends over about two decades in luminosity. Recent work extending the relation to low mass, typically LSB and gas rich galaxies [31, 445, 462] extends the dynamic range of the BTFR to five decades in baryonic mass. Over this range, the BTFR has remarkably little intrinsic scatter (consistent with zero given the observational errors) and is well described as a power law, or equivalently, as a straight line in log-log space:[272, 445, 276]. This slope is consistent with a constant acceleration scale such that10 the normalization constant .
The acceleration scale (Eq. 1) is thus present in the data. Figure 4 shows the distribution of this acceleration , around the best fit line in Figure 3, strongly peaked around in natural units. As we shall see, this acceleration scale arises empirically in a variety of distinct situations involving the mass discrepancy problem.
A BTFR of the observed form does not arise naturally in CDM. The naive expectation is and 11 where is the Hubble constant and is a factor of order unity (currently estimated to be ) that relates the observed to the circular velocity of the potential at the virial radius12. This modest fudge factor is necessary because CDM does not explicitly predict either axis of the observed BTFR. Rather, there is a relationship between total (baryonic plus dark) mass and rotation velocity at very large radii. This simple scaling fails (dashed line in Figure 3), obliging us to introduce an additional fudge factor [273, 284] that relates the detected baryonic mass to the total mass of baryons available in a halo. This mismatch drives the variation in the detected baryon fraction seen in Figure 2. A constant is excluded by the difference between the observed and predicted slopes; must vary with , or , or the gravitational potential .
This brings us to the first fine-tuning problem posed by the data. There is essentially zero intrinsic scatter in the BTFR , while the detected baryon fraction could, in principle, obtain any value between zero and unity. Somehow galaxies must “know” what the circular velocity of the halo they reside in is so that they can make observable the correct fraction of baryons.
Quantitatively, in the CDM picture, the baryonic mass plotted in the BTFR (Figure 3) is while the total baryonic mass available in a halo is . The difference between these quantities implies a reservoir of dark baryons in some undetected form, . It is commonly speculated that the undetected baryons could be in a hard-to-detect hot, diffuse, ionized phase mixed in with the dark matter halo (and extending to comparable radius), or that the missing baryons have been entirely blown away by winds from supernovae. For the purposes of this argument, it does not matter which form the dark baryons take. All that matters is that a substantial mass of them are required so that 
Another remarkable fact about the BTFR is that it shows no residuals with variations in the distribution of baryons [517, 443, 109, 271]. Figure 5 shows deviations from the BTFR as a function of the characteristic baryonic surface density of the galaxies, as defined in , i.e., where is the radius at which the rotation curve of baryons peaks. Over several decades in surface density, the BTFR is completely insensitive to variations in the mass distribution of the baryons. This is odd because, a priori, , and thus . Yet the BTFR is with no dependence on . This brings us to a second fine-tuning problem. For some time, it was thought  that spiral galaxies all had very nearly the same surface brightness (a condition formerly known as “Freeman’s Law”). If this is indeed the case, the observed BTFR naturally follows from the constancy of . However, there do exist many LSB galaxies  that violate the constancy of surface brightness implied in Freeman’s Law. Thus, one would expect them to deviate systematically from the Tully–Fisher relation, with lower surface brightness galaxies having lower rotation velocities at a given mass. Yet they do not. Thus, one must fine-tune the mass surface density of the dark matter to precisely make up for that of the baryons . As the surface density of baryons declines, that of the dark matter must increase just so as to fill in the difference (Figure 6 ). The relevant quantity is the dynamical surface density enclosed within the radius, where the velocity is measured. The latter matters little along the flat portion of the rotation curve, but the former is the sum of dark and baryonic matter.
One might be able to avoid fine-tuning if all galaxies are dark-matter dominated . In the limit , the dynamics are entirely dark-matter dominated and the distribution of the baryons is irrelevant. There is some systematic uncertainty in the mass-to-light ratios of stellar populations , making such an approach a priori tenable. In effect, we return to the interpretation of constant originally made by  in the context of Freeman’s Law, but now we invoke a constant surface density of CDM rather than of baryons. But as we will see, such an interpretation, i.e., that in all disk galaxies, is flatly contradicted by other observations (e.g., Figure 9 and Figure 13).
The Tully–Fisher relation is remarkably persistent. Originally posited for bright spirals, it applies to galaxies that one would naively expected to deviate from it. This includes low-luminosity, gas-dominated irregular galaxies [445, 462, 276], LSB galaxies of all luminosities [517, 443], and even tidal dwarfs formed in the collision of larger galaxies . Such tidal dwarfs may be especially important in this context (see also Section 6.5.4). Galactic collisions should be very effective at segregating dark and baryonic matter. The rotating gas disks of galaxies that provide the fodder for tidal tails and the tidal dwarfs that form within them initially have nearly circular, coplanar orbits. In contrast, the dark-matter particles are on predominantly radial orbits in a quasi-spherical distribution. This difference in phase space leads to tidal tails that themselves contain very little dark matter . When tidal dwarfs form from tidal debris, they should be largely devoid13 of dark matter. Nevertheless, tidal dwarfs do appear to contain dark matter  and obey the BTFR .
The critical acceleration scale of Eq. 1 also appears in non-rotating galaxies. Elliptical galaxies are three-dimensional stellar systems supported more by random motions than organized rotation. First of all, in such systems of measured velocity dispersion , the typical acceleration is also on the order of within a factor of a few, where is the effective radius of the system . Moreover, they obey an analogous relation to the Tully–Fisher one, known as the Faber–Jackson relation (Figure 7). In bulk, the data for these star-dominated galaxies follow the relation (dotted line in Figure 7). This is not strictly analogous to the flat part of the rotation curves of spiral galaxies, the dispersion typically being measured at smaller radii, where the equivalent circular velocity curve is often falling [367, 323], or in a temporary plateau before falling again (see also Section 6.6.1). Indeed, unlike the case in spiral galaxies, where the distribution of stars is irrelevant, it clearly does matter in elliptical galaxies (the Faber–Jackson relation is just one projection of the “fundamental plane” of elliptical galaxies ). This is comforting: at small radii in dense stellar systems where the baryonic mass of stars is clearly important, the data behave as Newton predicts.
The acceleration scale is clearly imprinted on the data for local galaxies. This is an empirical statement that might not hold at all times, perhaps evolving over cosmic time or evaporating altogether. Substantial efforts have been made to investigate the Tully–Fisher relation to high redshift. To date, there is no persuasive evidence of evolution in the zero point of the BTFR out to [356, 357] and perhaps even to . One must exercise caution in interpreting such results given the difficulty inherent in peering many Gyr back in cosmic time. Nonetheless, it appears that the scale remains present in the data and has not obviously changed over the more recent half of the age of the Universe.
The Freeman limit  is the maximum central surface brightness in the distribution of galaxy surface brightnesses. Originally thought to be a universal surface brightness, it has since become clear that instead galaxies exist over a wide range in surface brightness . In the absence of a perverse and fine-tuned anti-correlation between surface brightness and stellar mass-to-light ratio , this implies a comparable range in baryonic surface density (Figure 8).
An upper limit to the surface brightness distribution is interesting in the context of disk stability. Recall that dynamically cold, purely Newtonian disks are subject to potentially–self-destructive instabilities, one cure being to embed them in the potential wells of spherical dark-matter halos . While the proper criterion for stability is much debated [131, 415], it is clear that the dark matter halo moderates the growth of instabilities and that the ratio of halo to disk self gravity is a relevant quantity. The more self-gravitating a disk is, the more likely it is to suffer undamped growth of instabilities. But, in principle, galaxies with a baryonic disk and a dark matter halo are totally scalable: if a galaxy model has a certain dynamics, and one multiplies all densities by any (positive) constant (and also scales the velocities appropriately) one gets another galaxy with exactly the same dynamics (with scaled time scales). So if one is stable, so is the other. In turn, the mere fact that there might be an upper limit to is a priori surprising, and even more so that there might be a coincidence of this upper limit with the acceleration scale identified dynamically.
The scale is clearly present in the data (Figure 8). Selection effects make high–surface-brightness (HSB) galaxies easy to detect and hence discover, but their intrinsic numbers appear to decline exponentially when the central surface density of the stellar disk . It seems natural to associate the dynamical scale with the disk stability scale since they are numerically indistinguishable and both arise in the context of the mass discrepancy. However, there is no reason to expect this in CDM, which predicts denser dark matter halos than observed [280, 169, 167, 241, 243, 478, 118]. Such dense dark matter halos could stabilize much higher density disks than are observed to exist. Lacking a clear mechanism to specify this scale, it is introduced into models by hand .
Poisson’s equation provides a direct relation between the force per unit mass (centripetal acceleration in the case of circular orbits in disk galaxies), the gradient of the potential, and the surface density of gravitating mass. If there is no dark matter, the observed surface density of baryons must correlate perfectly with the dynamical acceleration. If, on the other hand, dark matter dominates the dynamics of a system, as we might infer from Figure 5 [279, 109], then there is no reason to expect a correlation between acceleration and the dynamically-insignificant baryons. Figure 9 shows the dynamical acceleration as a function of baryonic surface density in disk galaxies. The acceleration is measured at the radius , where the rotation curve of baryons peaks. Given the systematic variation of rotation curve shape [376, 495], the specific choice of radii is unimportant. Nevertheless, this radius is advocated by  since this maximizes the possibility of perceiving the baryonic contribution in the plot of Figure 5. That this contribution is not present leads to the inference that in all disk galaxies . This is directly contradicted by Figure 9, which shows a clear correlation between and .
The higher the surface density of baryons, the higher the observed acceleration. The slope of the relation is not unity, , as we would expect in the absence of a mass discrepancy, but rather . To simultaneously explain Figure 5 and Figure 9, there must be a strong fine-tuning between dark and baryonic surface densities (i.e., Figure 6), a sort of repulsion between them, a repulsion which is however contradicted by the correlations between baryonic and dark matter bumps and wiggles in rotation curves (see Section 4.3.4).
So far we have discussed total quantities. For the BTFR, we use the total observed mass of a galaxy and its characteristic rotation velocity. Similarly, the dynamical acceleration–baryonic surface density relation uses a single characteristic value for each galaxy. These are not the only ways in which the “magical” acceleration constant appears in the data. In general, the mass discrepancy only appears at very low accelerations and not (much) above . Equivalently, the need for dark matter only becomes clear at very low baryonic surface densities . Indeed, the amplitude of the mass discrepancy in galaxies anti-correlates with acceleration .
In , one examined the role of various possible scales, as well as the effects of different stellar mass-to-light ratio estimators, on the mass discrepancy problem. The amplitude of the mass discrepancy, as measured by , the ratio of observed velocity to that predicted by the observed baryons, depends on the choice of estimator for stellar . However, for any plausible (non-zero) , the amplitude of the mass discrepancy correlates with acceleration (Figure 10) and baryonic surface density, as originally noted in [382, 266, 406]. It does not correlate with radius and only weakly with orbital frequency14.
There is no reason in the dark matter picture why the mass discrepancy should correlate with any physical scale. Some systems might happen to contain lots of dark matter; others very little. In order to make a prediction with a dark matter model, it is necessary to model the formation of the dark matter halo, the condensation of gas within it, the formation of stars therefrom, and any feedback processes whereby the formation of some stars either enables or suppresses the formation of further stars. This complicated sequence of events is challenging to model. Baryonic “gastrophysics” is particularly difficult, and has thus far precluded the emergence of a clear prediction for galaxy dynamics from CDM.
CDM does make a prediction for the distribution of mass in baryonless dark matter halos: the NFW halo [332, 333]. These are remarkable for being scale free. Small halos have a profile similar to large halos. No feature stands out that marks a unique physical scale as observed. Galaxies do not resemble pure NFW halos , even when dark matter dominates the dynamics as in LSB galaxies [241, 243, 118]. The inference in CDM is that gastrophysics, especially the energetic feedback from stellar winds and supernova explosions, plays a critical role in sculpting observed galaxies. This role is not restricted to the minority baryonic constituents; it must also affect the majority dark matter . Simulations incorporating these effects in a quasi-realistic way are extremely expensive computationally, so a comprehensive survey of the plausible parameter space occupied by such models has yet to be made. We have no reason to expect that a particular physical scale will generically emerge as the result of baryonic gastrophysics. Indeed, feedback from star formation is inherently a random process. While it is certainly possible for simple laws to emerge from complicated physics (e.g., the fact that SNIa are standard candles despite the complicated physics involved), the more common situation is for chaos to beget chaos. Therefore, it seems unnatural to imagine feedback processes leading to the orderly behavior that is observed (Figure 10); nor is it obvious how they would implicate any particular physical scale. Indeed, the dark matter halos formed in CDM simulations [332, 333] provide an initial condition with greater scatter than the final observed one [280, 478], so we must imagine that the chaotic processes of feedback not only impart order, but do so in a way that cancels out some of the scatter in the initial conditions.
In any case, and whatever the reason for it, a physical scale is clearly observationally present in the data: (Eq. 1). At high accelerations , there is no indication of the need for dark matter. Below this acceleration, the mass discrepancy appears. It cannot be emphasized enough that the role played by in the BTFR and this role as a transition acceleration have strictly no intrinsic link with each other, they are fully independent of each other. There is nothing in CDM that stipulates that these two relations (the existence of a transition acceleration and the BTFR) should exist at all, and even less that these should harbour an identical acceleration scale.
Thus, it is important to realize not only that the relevant dynamical scale is one of acceleration, not size, but also that the mass discrepancy appears only at extremely low accelerations. Just as galaxies are much bigger than the Solar system, so too are the centripetal accelerations experienced by stars orbiting within a galaxy much smaller than those experienced by planets in the Solar system. Many of the precise tests of gravity that have been made in the Solar system do not explore the relevant regime of physical parameter space. This is emphasized in Figure 11, which extends the mass discrepancy–acceleration relation to Solar system scales. Many decades in acceleration separate the Solar system from galaxies. Aside from the possible exception of the Pioneer anomaly, there is no hint of a discrepancy in the Solar system: . Even the Pioneer anomaly15 is well removed from the regime where the mass discrepancy manifests in galaxies, and is itself much too subtle to be perceptible in Figure 11. Indeed, to within a factor of , no system exhibits a mass discrepancy at accelerations .
The systematic increase in the amplitude of the mass discrepancy with decreasing acceleration and baryonic surface density has a remarkable implication. Even though the observed velocity is not correctly predicted by the observed baryons, it is predictable from them. Independent of any theory, we can simply fit a function to describe the variation of the discrepancy with baryonic surface density . We can then apply it to any new system we encounter to predict . In effect, boosts the velocity already predicted by the observed baryons. While this is a purely empirical exercise with no underlying theory, it is quite remarkable that the distribution of dark matter required in a galaxy is entirely predictable from the distribution of its luminous mass (see also ). In the conventional picture, dark matter outweighs baryonic matter by a factor of five, and more in individual galaxies given the halo-by-halo missing baryon problem (Figure 2), but apparently the baryonic tail wags the dark matter dog. And it does so again through the acceleration scale . Indeed, at very low accelerations, the mass discrepancy is precisely defined by the inverse of the square-root of the gravitational acceleration generated by the baryons in units of . This actually asymptotically leads to the BTFR.
So, up to now, we have seen five roles of in galaxy dynamics. (i) It defines the zero point of the Tully–Fisher relation, (ii) it appears as the characteristic acceleration at the effective radius of spheroidal systems, (iii) it defines the Freeman limit for the maximum surface density of pure disks, (iv) it appears as a transition-acceleration above which no dark matter is needed, and below which it appears, and (v) it defines the amplitude of the mass-discrepancy in the weak-field regime (this last point is not a fully independent role as it leads to the Tully–Fisher relation). Let us eventually note that there is yet a final role played by , which is that it defines the central surface density of all dark matter halos as being on the order of [129, 167, 313].
The relation between dynamical and baryonic surface densities appears as a global scaling relation in disk galaxies (Figure 9) and as a local correspondence within each galaxy (Figure 10). When all galaxies are plotted together as in Figure 10, this connection appears as a single smooth function . This does not suffice to illustrate that individual galaxies have features in their baryon distribution that are reflected in their dynamics. While the above correlations could be interpreted as a sort of repulsion between dark and baryonic matter, the following rather indicates closer-than-natural attraction.
Figure 12 shows the spiral galaxy NGC 6946. Two multi-color images of the stellar component are given. The optical bands provide a (nearly) true color picture of the galaxy, which is perceptibly redder near the center and becomes progressively more blue further out. This is typical of spiral galaxies and reflects real differences in stellar content: the stars towards the center tend to be older and more dominated by the light of red giants, while those further out are younger on average so the light has a greater fractional contribution from bright-but-short-lived main sequence stars. The near-infrared bands  give a more faithful map of stellar mass, and are less affected by dust obscuration. Radio synthesis imaging of the 21 cm emission from the hydrogen spin-flip transition maps the atomic gas in the interstellar medium, which typically extends to rather larger radii than the stars.
Surface density profiles of galaxies are constructed by fitting ellipses to images like those illustrated in Figure 12. The ellipses provide an axisymmetric representation of the variation of surface brightness with radius. This is shown in the top panels of Figure 13 for NGC 6946 (Figure 12) and the nearby, gas rich, LSB galaxy NGC 1560. The -band light distribution is thought to give the most reliable mapping of observed light to stellar mass , and has been used to trace the run of stellar surface density in Figure 13. The sharp feature at the center is a small bulge component visible as the red central region in Figure 12. The bulge contains only 4% of the -band light. The remainder is the stellar disk; a straight line fit to the data outside the central bulge region gives the parameters of the exponential disk approximation, and . Similarly, the surface density of atomic gas is traced by the 21 cm emission, with a correction for the cosmic abundance of helium – the detected hydrogen represents 75% of the gas mass believed to be present, with most of the rest being helium, in accordance with BBN.
Mass models (bottom panels of Figure 13) are constructed from the surface density profiles by numerical solution of the Poisson equation [52, 472]. No approximations (like sphericity or an exponential disk) are made at this step. The disks are assumed to be thin, with radial scale length exceeding their vertical scale by 8:1, as is typical of edge-on disks . Consequently, the computed rotation curves (various broken lines in Figure 13) are not smooth, but reflect the observed variations in the observed surface density profiles of the various components. The sum (in quadrature) leads to the total baryonic rotation curve (the solid lines in Figure 13): this is what would be observed if no dark matter were implicated. Instead, the observed rotation (data points in Figure 13) exceeds that predicted by : this is the mass discrepancy.
It is often merely stated that flat rotation curves require dark matter. But there is considerably more information in rotation curve data than asymptotic flatness. For example, it is common that the rotation curve in the inner parts of HSB galaxies like NGC 6946 is well described by the baryons alone. The data are often consistent with a very low density of dark matter at small radii with baryons providing the bulk of the gravitating mass. This condition is referred to as maximum disk , and also runs contrary to our inferences of dark matter dominance from Figure 5 . More generally, features in the baryonic rotation curve often correspond to features in the total rotation .
Perhaps the most succinct empirical statement of the detailed connection between baryons and dynamics has been given by Renzo Sancisi, and known as Renzo’s rule : “For any feature in the luminosity profile there is a corresponding feature in the rotation curve.” Both galaxies shown in Figure 13 illustrate this statement. In the inner region of NGC 6946, the small but compact bulge component causes a sharp feature in that declines rapidly before the rotation curve rises again, as mass from the disk begins to contribute. The up-down-up morphology predicted by the observed distribution of the baryons is observed in high resolution observations [54, 114]. A dark matter halo with a monotonically-varying density profile cannot produce such a morphology; the stellar bulge must be the dominant mass component at small radii in this galaxy.
A surprising aspect of Renzo’s rule is that it applies to LSB galaxies as well as those of high surface brightness. That the baryons should have some dynamical impact where their surface density is highest is natural, though there is no reason to demand that they become competitive with dark matter. What is distinctly unnatural is for the baryons to have a perceptible impact where dark matter must clearly dominate. NGC 1560 provides an example where they appear to do just that. The gas distribution in this galaxy shows a substantial kink in its surface density profile  (recently confirmed by ) that has a distinct impact on . This occurs at a radius where , so dark matter should be dominant. A spherical–dark-matter halo with particles on randomly oriented, highly radial orbits cannot support the same sort of structure as seen in the gas disk, and the spherical geometry, unlike a disk geometry, would smear the effect on the local acceleration. And yet the wiggle in the baryonic rotation curve is reflected in the total, as per Renzo’s rule.16
One inference that might be made from these observations is that the dark matter is baryonic. This is unacceptable from a cosmological perspective, but it is possible to have a multiplicity of dark matter components. That is, we could have baryonic dark matter in the disks of galaxies in addition to a halo of non-baryonic cold dark matter. It is often possible to scale up the atomic gas component to fit the total rotation . That implies a component of mass that is traced by the atomic gas – presumably some other dynamically cold gas component – that outweighs the observed hydrogen by a factor of six to ten . One hypothesis for such a component is very cold molecular gas . It is difficult to exclude such a possibility, though it also appears to be hard to sustain in LSB galaxies. Dynamically, one might expect the extra mass to destabilize the LSB disk. One also returns to a fine-tuning between baryonic surface density and mass-to-light ratio. In order to maintain the balance observed in Figure 5, relatively more dark molecular gas will be required in LSB galaxies so as to maintain a constant surface density of gravitating mass, but given the interactions at hand, this might be at least a bit more promising than explaining it with CDM halos.
As a matter of fact, LSB galaxies play a critical role in testing many of the existing models for dark matter. This happens in part because they were appreciated as an important population of galaxies only after many relevant hypotheses were established, and thus provide good tests of their a priori expectations. Observationally, we infer that LSB disks exhibit large mass discrepancies down to small radii . Conventionally, this means that dark matter completely dominates their dynamics: the surface density of baryons in these systems is never high enough to be relevant. Nevertheless, the observed distribution of baryons suffices to predict the total rotation [279, 120]. Once again, the baryonic tail wags the dark matter dog, with the observations of the minority baryonic component sufficing to predict the distribution of the dominant dark matter. Note that, conversely, nothing is “observable” about the dark matter, in present-day simulations, that predicts the distribution of baryons.
Thus, we see that there are many observations, mostly on galaxy scales, that are unpredicted, and perhaps unpredictable, in the standard dark matter context. They mostly involve a unique relationship between the distribution of baryons and the gravitational field, as well as an acceleration constant on the order of the square-root of the cosmological constant, and they represent the most significant challenges to the current CDM model.
Living Rev. Relativity 15, (2012), 10
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