Luminous elliptical galaxies are dense bodies of old stars with very little gas and typically large internal accelerations. The age of the stellar populations suggest they formed early and all the gas has been used to form stars. To form early, one might expect the presence of a massive dark-matter halo, but the study of, e.g.,  showed that actually, there is very little evidence for dark matter within the effective radius, and even several effective radii, in ellipticals. On the other hand, these are very-HSB objects and would thus not be expected to show a large mass discrepancy within the bright optical object in MOND. And indeed, the results of  were shown to be in perfect agreement with MOND predictions, assuming very reasonable anisotropy profiles . On the theoretical side, it was also importantly shown that triaxial elliptical galaxies can be reproduced using the Schwarzschild orbit superposition technique , and that these models are stable 41.
Interestingly, some observational studies circumvented the mass-anisotropy degeneracy by constructing non-parametric models of observed elliptical galaxies, from which equivalent circular velocity curves, radial profiles of mass-to-light ratio, and anisotropy profiles, as well as high-order moments, could be computed . Thanks to these studies, it was, e.g., shown  that, although not much dark matter is needed, the equivalent circular velocity curves (see also  where the rotation curve could be measured directly) tend to become flat at much larger accelerations than in thin exponential disk galaxies. This would seem to contradict the MOND prescription, for which flat circular velocities typically occur well below the acceleration threshold , but not at accelerations on the order of a few times as in ellipticals. However, as shown in , if one assumes the simple interpolating function ( in Eq. 46 and Eq. 49), known to yield excellent fits to spiral galaxy rotation curves (see Section 6.5.1), one finds that MONDian galaxies exhibit a flattening of their circular velocity curve at high accelerations if they can be described by a Jaffe profile  in the region where the circular velocity is constant. Since this flattening at high accelerations is not possible for exponential profiles, it is remarkable that such flattenings of circular velocity curves at high accelerations are only observed in elliptical galaxies. What is more, , as well as , derived from their models scaling relations for the configuration space and phase-space densities of dark matter in ellipticals, and these DM scaling relations have been shown  to be in very good agreement with the MOND predictions on “phantom DM” (Eq. 33) scaling relations. This is displayed on Figure 36. Of course, some of these galaxies are residing in clusters, and the external field effect (see Section 6.3) could modify the predictions, but this was shown to be negligible for most of the analyzed sample, because the galaxies are far away from the cluster center . Note that when closer to the center of galaxy clusters, interesting behaviors such as lopsidedness caused by the external field effect could allow new tests of MOND in the near future . However, this would require modelling both the orbit of the galaxy in the cluster to take into account time-variations of the external field, as well as a precise estimate of the external field from the cluster itself, which can be tricky as the whole cluster should be modelled at once due to the non-linearity of MOND [113, 259].
At a more detailed level, precise full line-of-sight velocity dispersion profiles of individual ellipticals, typically measured with tracers such as PNe or globular-cluster populations, have been reproduced by solving Jeans equation in spherical symmetry:) shows an example. In general, it was found that field galaxies all fit very naturally with MOND [461, 410] (see also ). On the other hand, the MOND modification has been found to slightly underpredict the velocity dispersions in large elliptical galaxies at the very center of galaxy clusters , which is just the small-scale equivalent of the problem of MOND in clusters, pointing towards missing baryons (see Section 6.6.4).
On the other hand,  used satellite galaxies of ellipticals to test MOND at distances of several 100 kpcs. They used the stacked SDSS satellites to generate a pair of mock galaxy groups with reasonably precise line-of-sight velocity dispersions as a function of radius across the group. When these systems were first analysed by  they claimed that MOND was excluded by 10, but this was only for models that had constant velocity anisotropy. It was then found  that with varying anisotropy profiles similar to those found in simulations of the formation of ellipticals by dissipationless collapse in MOND , excellent fits to the line-of-sight velocity dispersions of both mock galaxies could be found. This can be taken as strong evidence that MOND describes the dynamics in the surroundings of relatively isolated ellipticals very well.
Finally, let us note an intriguing possibility in a MONDian universe (see also Section 9.2). While massive ellipticals would form at  from monolithic dissipationless collapse , dwarf ellipticals could be more difficult to form. A possibility to form those would then be that tidal dwarf galaxies would be formed and survive more easily (see Section 6.5.4) in major mergers, and could then evolve to lead to the population of dwarf ellipticals seen today, thereby providing a natural explanation for the observed density-morphology relation  (more dwarf ellipticals in denser environments).
Dwarf spheroidal (dSph) satellites of the Milky Way [427, 477] exhibit some of the largest mass discrepancies observed in the universe. In this sense, they are extremely interesting objects in which to test MOND. Observationally, let us note that there are essentially two classes of objects in the galactic stellar halo: globular clusters (see Section 6.6.3) and dSph galaxies. These overlap in baryonic mass, but not in surface brightness, nor in age or uniformity of the stellar populations. The globular clusters are generally composed of old stellar populations, they are HSB objects and mostly exhibit no mass discrepancy problem, as expected for HSB objects in MOND. The dSphs, on the contrary, generally contain slightly younger stellar populations covering a range of ages, they are extreme LSB objects and exhibit, as said before, an extreme mass discrepancy, as generically expected from MOND. So, contrary to the case of CDM where different formation scenarios have to be invoked (see Section 6.6.3), the different mass discrepancies in these objects find a natural explanation in MOND.
At a more detailed level, MOND should also be able to fit the whole velocity dispersion profiles, and not only give the right ballpark prediction. This analysis has recently been possible for the eight “classical” dSph around the Milky Way . Solving Jeans equation (Eq. 65), it was found  that the four most massive and distant dwarf galaxies (Fornax, Sculptor, Leo I and Leo II) have typical stellar mass-to-light ratios, exactly within the expected range. Assuming equilibrium, two of the other four (smallest and most nearby) dSphs have mass-to-light ratios that are a bit higher than expected (Carina and Ursa Minor), and two have very high ones (Sextans and Draco). For all these dSphs, there is a remarkable correlation between the stellar inferred from MOND and the ages of their stellar populations . Concerning the high inferred stellar , note that it has been shown  that a dSph will begin to suffer tidal disruption at distances from the Milky Way that are 4 – 7 times larger in MOND than in CDM, Sextans and Draco could thus actually be partly tidally disrupted in MOND. And indeed, after subjecting the five dSphs with published data to an interloper removal algorithm , it was found that Sextans was probably littered with unbound stars, which inflated the computed , while Draco’s projected distance-l.o.s. velocity diagram actually looks as out-of-equilibrium as Sextans’ one. Ursa Minor, on the other hand, is the typical example of an out-of-equilibrium system, elongated and showing evidence of tidal tails. In the end, only Carina has a suspiciously high (; see ).
What is more, there is a possibility that, in a MONDian Universe, dSphs are not primordial objects but have been tidally formed in a major merger (see Section 9.2 as a solution to the phase-space correlation challenge of Section 4.2). In addition to the MOND effect, it would be possible that these objects never really reach a stable equilibrium , and exhibit an artificially high ratio. This is even more true for the recently discovered “ultra-faint” dwarf spheroidals, that are also, due to to their extremely low-density, very much prone to tidal heating in MOND. Indeed, at face value, if these ultrafaints are equilibrium objects, their velocity dispersions are much too high compared to what MOND predicts, and rule out MOND straightforwardly. However, unless this is due to systematic errors linked with the smallness of the velocity dispersion to measure (one must distinguish between and ), and/or to high intrinsic stellar ratios related to stochastic effects linked with the small number of stars , it was also found  that these objects are all close to filling their MONDian tidal radii, and that their stars can complete only a few orbits for every orbit of the satellite itself around the Milky Way (see Figure 38). As Brada & Milgrom  have shown, it then comes as no surprise that they are displaying out-of-equilibrium dynamics in MOND (and even more so in the case of a tidal formation scenario ).
Star clusters come in two types: open clusters and globular clusters. Most observed open clusters are in the inner parts of the Milky Way disk, and for that reason, the prediction of MOND is that their internal dynamics is Newtonian  with, perhaps, a slightly renormalized gravitational constant and slightly squashed isopotentials, due to the external field effect (Section 6.3). Therefore, the possibility of distinguishing Newtonian dynamics from MOND in these objects would require extreme precision. On the other hand, globular clusters are mostly HSB halo objects (see Section 6.6.2), and are consequently predicted to be Newtonian, and most of those that are fluffy enough to display MONDian behavior are close enough to the Galactic disk to be affected by the external field effect (Section 6.3), and so are Newtonian, too. Interestingly, MOND thus provides a natural explanation for the dichotomy between dwarf spheroidals and globular clusters. In CDM, this dichotomy is rather explained by the formation history [235, 397]: globular clusters are supposedly formed in primordial disk-bound supermassive molecular clouds with high baryon-to-dark matter ratio, and later become more spheroidal due to subsequent mergers. In MOND, it is, of course, not implied that the two classes of objects have necessarily the same formation history, but the different dynamics are qualitatively explained by MOND itself, not by the different formation scenarios.
However, there exist a few globular clusters (roughly, less than 10 compared to the total number of 150) both fluffy enough to display typical internal accelerations well below , and far away enough from the galactic plane to be more or less immune from the external field effect [27, 182, 181, 436]. Thus, these should, in principle, display a MONDian mass discrepancy. They include, e.g., Pal 14 and Pal 3, or the large fluffy globular cluster NGC 2419. Pal 3 is interesting, because it indeed tends to display a larger-than-Newtonian global velocity dispersion, broadly in agreement with the MOND prediction (Baumgardt & Kroupa, private communication). However, it is difficult to draw too strong a conclusion from this (e.g., on excluding Newtonian dynamics), since there are not many stars observed, and one or two outliers would be sufficient to make the dispersion grow artificially, while a slightly-higher-than-usual mass-to-light ratio could reconcile Newtonian dynamics with the data. Other clusters such as NGC 1851 and NGC 1904 apparently display the same MONDian behavior  (see also ). On the other hand, Pal 14 displays exactly the opposite behavior: the measured velocity dispersion is Newtonian , but again the number of observed stars is too small to draw a statistically significant conclusion , and it is still possible to reconcile the data with MOND assuming a slightly low stellar mass-to-light ratio . Note that if the cluster is on a highly eccentric orbit, the external gravitational field could vary very rapidly both in amplitude and direction, and it is possible that the cluster could take some time to accomodate this by still displaying a Newtonian signature in its kinematics after a sudden decrease of the external field.
NGC 2419 is an interesting case, because it allows not only for a measure of the global velocity dispersion, but also of the detailed velocity dispersion profile . And, again, like in the case of Pal 14 (but contrary to Pal 3), it displays Newtonian behavior. More precisely, it was found, solving Jeans equations (Eq. 65), that the best MOND fit, although not extremely bad in itself, was 350 times less likely than the best Newtonian fit without DM [199, 200]. However, the stability  of this best MOND fit has not been checked in detail. These results are heavily debated as they rely on the small quoted measurement errors on the surface density, and even a slight rotation of only the outer parts of this system near the plane of the sky (which would not show up in th velocity data) would make a considerable difference in the right direction for MOND . However, these observations, together with the results on Pal 14, although not ruling out any theory, are not a resounding success for MOND. However, it could perhaps indicate that globular clusters are generically on highly eccentric orbits, and out of equilibrium due to this (however, the effect would have to be opposite to that prevailing in ultra-faint dwarfs, where the departure from equilibrium would boost the velocity dispersion instead of decreasing it). A stronger view on these results could indicate that MOND as formulated today is an incomplete paradigm (see, e.g., Eq. 27), or that MOND is an effect due to the fundamental nature of the DM fluid in galaxies (see Sections 7.6 and 7.9), which is absent from globular clusters. Concerning NGC 2419, it is perhaps useful to remind oneself that it is very plausibly not a globular cluster. It is part of the Virgo stream and is thus most probably the remaining nucleus of a disrupting satellite galaxy in the halo of the Milky Way, on a generically-highly-eccentric orbit. Detailed N-body simulations of such an event, and of the internal dynamics of the remaining nucleus, would thus be the key to confront MOND with observations in this object. All in all, the situation regarding MOND and the internal dynamics of globular clusters remains unclear.
On the other hand, it has been noted that MOND seems to overpredict the Roche lobe volume of globular clusters [499, 500, 512]. Again, the fact that globular clusters could generically be on highly eccentric orbits could come to the rescue here. What is more, it was shown that, in MOND, globular clusters can have a cutoff radius, which is unrelated to the tidal radius when non-isothermal . In general, the cutoff radii of dwarf spheroidals, which have comparable baryonic masses, are larger than those of the globular clusters, meaning that those may well extend to their tidal radii because of a possibly different formation history than globular clusters.
Finally, a last issue for MOND related to globular clusters [335, 377] is the existence of five such objects surrounding the Fornax dwarf spheroidal galaxy. Indeed, under similar environmental conditions, dynamical friction occurs on significantly shorter timescales in MOND than standard dynamics , which could cause the globular clusters to spiral in and merge within at most 2 Gyrs . However, this strongly depends on the orbits of the globular clusters, and, in particular, on their initial radius , which can allow for a Hubble time survival of the orbits in MOND.
As pointed out earlier (3rd Kepler-like law of Section 5.2), it is a natural consequence of Milgrom’s law that, at the effective baryonic radius of the system, the typical acceleration is always observed to be on the order of , thereby naturally explaining the linear relation between size and temperature for galaxy clusters [327, 392]. However, one of the main predictions of Milgrom’s formula is the baryonic Tully–Fisher relation (circular velocity vs. baryonic mass, Figure 3), and its equivalent for isotropic pressure-supported systems, the Faber–Jackson relation (stellar velocity dispersion vs. baryonic mass, Figure 7), both for their slope and normalization. For systems such as galaxy clusters, where the hot intra-cluster gas is the major baryonic component, this relation can also be translated into a “gas temperature vs. baryonic mass” relation, , plotted on Figure 39, as the line (note that this differs slightly from  where solar metallicity gas is assumed). Note on this figure that observations are closer to the MOND predicted slope than to the conventional prediction of in CDM, without the need to invoke preheating (a need that may arise as an artifact of the mismatch in slopes).
So, interestingly, the data are still reasonably consistent with the slope predicted by MOND , but not with the normalization. There is roughly a factor of two of residual missing mass in these objects [170, 354, 387, 389, 392, 453]. This conclusion, reached from applying the hydrostatic equilibrium equation to the temperature profile of the X-ray emitting gas of these objects, has also been reached for low mass X-ray emitting groups . This is essentially because, contrary to the case of galaxies, there is observationally a need for “Newtonian” missing mass in the central parts42 of clusters, where the observed acceleration is usually slightly larger than , meaning that the MOND prescription is not enough to explain the observed discrepancy between visible and dynamical mass there. For this reason, the residual missing mass in MOND is essentially concentrated in the central parts of clusters, where the ratio of MOND dynamical mass to observed baryonic mass reaches a value of 10, to then only decrease to a value of roughly 2 in the very outer parts, where almost no residual mass is present. Thus, the profile of this residual mass would thus consist of a large constant density core of about 100 – 200 kpc in size (depending on the size of the group/cluster in question), followed by a sharp cutoff.
The need for this residual missing mass in MOND might be taken in one of the five following ways:
If (i) is correct, one still needs to explain the success of MOND on galaxy scales with CDM. Such an explanation has yet to be offered. Thus, tempting as case (i) is, it is worth giving a closer inspection to the four other possibilities.
The second case (ii) would be most in line with the elegant absence of need for any non-baryonic mass in MOND (however, see the “dark fields” invoked in Section 7). It has happened before that most of the baryonic mass was in an unobserved component. From the 1930s when Zwicky first discovered the missing mass problem in clusters till the 1980s, it was widely presumed that the stars in the observed galaxies represented the bulk of baryonic mass in clusters. Only after the introduction of MOND (in 1983) did it become widely appreciated that the diffuse X-ray emitting intracluster gas (the ICM) greatly outweighed the stars. That is to say, some of the missing mass problem in clusters was due to optically dark baryons — instead of the enormous mass discrepancies implied by cluster dynamical mass to optical light ratios in excess of 100 , the ratio of dark to baryonic mass is only 8 conventionally [175, 278]. So we should not be too hasty in presuming we now have a complete census of baryons in clusters. Indeed, in the global baryon inventory of the universe, 30% of the baryons produced during BBN are missing (Figure 40), and presumably reside in some, as yet undetected, (dark) form. It is estimated [160, 421] that the observed baryons in clusters only account for about 4% of those produced during BBN (Figure 40). This is much less than the 30% of baryons that are still missing. Consequently, only a modest fraction of the dark baryons need to reside in clusters to solve the problem of missing mass in the central regions of clusters in MOND. It should be highlighted that this missing mass only appears in MOND for systems with a high abundance of ionised gas and X-ray emission. Indeed, for even smaller galaxy groups, devoid of gas, the MOND predictions for the velocity dispersions of individual galaxies are again perfectly in line with the observations [303, 307]. It is then no stretch of the imagination to surmise that these gas rich systems, where the residual–missing-baryons problem have equal quantities of molecular hydrogen or other molecules. Milgrom  has, e.g., proposed that the missing mass in MOND could entirely be in the form of cold, dense gas clouds. There is an extensive literature discussing searches for cold gas in the cores of galaxy clusters, but what is usually meant there is quite different from what is meant here, since those searches consisted in trying to find the signature of diffuse cold molecular gas at a temperature of 30 K. The proposition of Milgrom  rather relies on the work of Pfenniger & Combes , where dense gas clouds with a temperature of only a few Kelvin ( 3 K), solar-system size, and of a Jupiter mass, were considered to be possible candidates for both galactic and extragalactic dark matter. These clouds would behave in a collisionless way, just like stars. However, since the dark mass considered in the context of MOND cannot be present in galaxies, it is not subject to the galactic constraints on such gas clouds. Note that the total sky covering factor of such clouds in the core of the clusters would be on the order of only , so that they would only occult a minor fraction of the X-rays emitted by the hot gas (and it would be a rather constant fraction). For the same reason, the chances of a given quasar having light absorbed by them is very small. Still,  notes that these clouds could be probed through X-ray flashes coming out of individual collisions between them. Of course, this speculative idea also raises a number of questions, the most serious one being how these clumps form and stabilize, and why they form only in clusters, X-ray emitting groups and some ellipticals at the center of these groups and clusters, but not in individual spiral galaxies. As noted above, the fact that missing mass in MOND is necessarily associated with an abundance of ionised gas could be a hint at a formation and stabilization process somehow linked with the presence of hot gas and X-ray emission themselves. Then, there is the issue of knowing whether the cloud formation would be prior to or posterior to the cluster formation. We note that a rather late formation mechanism could help increase the metal abundance, solving the problem of small-scale variations of metallicity in clusters when the clouds are destroyed . Milgrom  also noted that these clouds could alleviate the cooling flow conundrum, because whatever destroys them (e.g., cloud-cloud collisions and dynamical friction between the clouds and the hot gas) is conducive to heating the core gas, and thus preventing it from cooling too quickly. Such a heating source would not be transient and would be quite isotropic, contrary to AGN heating.
Another possibility (iii) would be that this residual missing mass in clusters is in the form of non-baryonic matter. There is one obviously existing form of such matter: neutrinos. If , then the neutrino mass is too small to be of interest in this context. But there is nothing that prevents it from being larger (note that the “cosmological” constraints from structure formation in the CDM context obviously do not apply in MOND). Actual model-independent experimental limits on the electron neutrino mass from the Mainz/Troitsk experiments, counting the highest energy electrons in the -decay of Tritium  are . Interestingly, the KATRIN experiment (the KArlsruhe TRItium Neutrino experiment, under construction) will be able to falsify these 2 eV electron neutrinos at 95% confidence. If the neutrino mass is substantially larger than the mass differences, then all types have about the same mass, and the cosmological density of three left-handed neutrinos and their antiparticles  would be[389, 392], then the mass of neutrinos must indeed be around 2 eV. Combined with the effect of additional degrees of freedom in relativistic MOND theories (Section 7), it has been shown that the CMB anisotropies could also be reproduced (see Section 9.2 and ), while this hot dark matter would obviously free-stream out of spiral galaxies and would thus not perturb the MOND fits of Section 6.5.1. The main limit on the neutrino ability to condense in clusters comes from the Tremaine–Gunn limit , stating that the phase space density must be preserved during collapse. This is a density level half the quantum mechanical degeneracy level in phase-space: :  showed that such 2 eV neutrinos at the limit of experimental detection could indeed account for the bulk of the dynamical mass in his sample of galaxy clusters of (see also Section 8.3 for gravitational lensing constraints). This has the great advantage of naturally reproducing the proportionality of the electron density in the cores of clusters to , as observed in . However, looking at the central region of low-temperature X-ray emitting galaxy groups, it was found  that the needed central density of missing mass far exceeded this limit by a factor of several hundred. One would need one neutrino species with to reach the required densities. One exotic possibility is then the idea of right-handed eV-scale sterile neutrinos : as strange as this sounds, this mass for sterile neutrinos could also provide a good fit to the CMB acoustic peaks (see Section 9.2). This could indeed sound like the strangest and most complicated universe possible, combining true non-baryonic (hot) dark matter with a modification of gravity, but if this is what it takes to simultaneously explain the Kepler-like laws of galactic dynamics and the extragalactic evidence for dark matter, it is useful to remember that there are both good reasons for there being more particles than those of the standard model of particle physics and that there is no reason that general relativity should be valid over a wide range of scales where it has never been tested. In any case, experiments that can address the existence of such a -scale sterile neutrino would thus be very interesting, as this kind of particle could provide the dark matter candidate only in a modified gravity framework, since such a hot dark matter particle would be unable to form small structures and to provide the dark matter that would be needed in galaxies.
Yet another possibility (iv) would be that MOND is incomplete, and that a new scale should be introduced, in order to effectively enhance the value of in galaxy clusters, while lowering it to its preferred value in galaxies. There are several ways to implement such an idea. For instance, Bekenstein  proposed adding a second scale in order to allow for effective variations of the acceleration constant as a function of the deepness of the potential (Eq. 27). This idea should be investigated more in the future, but it is not clear that such a simple rescaling of would account for the exact spatial distribution of the residual missing mass in MOND clusters, especially in cases where it is displaced from the baryonic distribution (see Section 8.3). However, as even Gauss’ theorem would not be valid anymore in spherical symmetry, the high non-linearity might provide non-intuitive results, and it would thus clearly be worth investigating this suggestion in more detail, as well as developing similar ideas with other additional scales in the future (such as, for instance, the baryonic matter density; see [82, 143] and Section 7.6).
Finally, as we shall see in Section 7, parent relativistic theories of MOND often require additional degrees of freedom in the form of “dark fields”, which can nevertheless be globally subdominant to the baryon density, and thus do not necessarily act precisely as true “dark matter”. Thus, the last possibility (v) is that these fields, which are obviously not included in Milgrom’s formula, are responsible for the cluster missing mass in MOND. An example of such fields are the vector fields of TeVeS (Section 7.4) and Generalized Einstein-Aether theories (Section 7.7). It has been shown (see Section 9.2) that the growth of the spatial part of the vector perturbation in the course of cosmological evolution can successfully seed the growth of baryonic structures, just as dark matter does. If these seeds persist, it was shown  that they could behave in very much the same way as a dark matter halo in relatively unrelaxed galaxy clusters. However, it remains to be seen whether the spatially-concentrated distribution of missing mass in MOND would be naturally reproduced in all clusters. In other relativistic versions of MOND (see, e.g., Sections 7.6 and 7.9), the “dark fields” are truly massive and can be thought of as true dark matter (although more complex than simple collisionless dark matter), whose energy density outweighs the baryonic one, and could provide the missing mass in clusters. However, again, it is not obvious that the centrally-concentrated distribution of residual missing mass in clusters would be naturally reproduced. All in all, there is no obviously satisfactory explanation for the problem of residual missing mass in the center of galaxy clusters, which remains one of the most serious problems facing MOND.
Living Rev. Relativity 15, (2012), 10
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