6.6 MOND in pressure-supported stellar systems

We have already outlined (Section 5.2) how Milgrom’s formula accounts for general scaling relations of pressure-supported systems such as the Faber–Jackson relation (Figure 7View Image and see [395Jump To The Next Citation Point]), and that isothermal systems have a finite mass in MOND with the density at large radii falling approximately as −4 r [296]. Note also that, in order to match the observed fundamental plane, MOND models must actually deviate somewhat from being strictly isothermal and isotropic: a radial orbit anisotropy in the outer regions is needed [388Jump To The Next Citation Point, 86]. Here we concentrate on slightly more detailed predictions and scaling relations. In general, these detailed predictions are less obvious to make than in rotationally-supported systems, precisely because of the new degree of freedom introduced by the anisotropy of the velocity distribution, very difficult to constrain observationally (as higher-order moments than the velocity dispersions would be needed to constrain it). As we shall see, the successes of MOND are in general a bit less impressive in pressure-supported systems than in rotationally-supported ones, and even in some cases really problematic (e.g., in the case of galaxy clusters, see Section 6.6.4). Whether this is due to the fact that predictions are less obvious to make, or whether this truly reflects a breakdown of Milgrom’s formula for these objects (or the fact that certain theoretical versions of MOND would explicitly deviate from Milgrom’s formula in pressure-supported systems, see Section 6.1.1) remains unclear.

6.6.1 Elliptical galaxies

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., [367Jump To The Next Citation Point] 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 [367] were shown to be in perfect agreement with MOND predictions, assuming very reasonable anisotropy profiles [323Jump To The Next Citation Point]. On the theoretical side, it was also importantly shown that triaxial elliptical galaxies can be reproduced using the Schwarzschild orbit superposition technique [482], and that these models are stable [493]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 [171Jump To The Next Citation Point]. Thanks to these studies, it was, e.g., shown  [171Jump To The Next Citation Point] that, although not much dark matter is needed, the equivalent circular velocity curves (see also [484Jump To The Next Citation Point] 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 a0, but not at accelerations on the order of a few times a0 as in ellipticals. However, as shown in [363Jump To The Next Citation Point], if one assumes the simple interpolating function (α = n = 1 in Eq. 46View Equation and Eq. 49View Equation), 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 [208] 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, [171Jump To The Next Citation Point], as well as [454Jump To The Next Citation Point], 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 [363Jump To The Next Citation Point] to be in very good agreement with the MOND predictions on “phantom DM” (Eq. 33View Equation) scaling relations. This is displayed on Figure 36View Image. 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 [363Jump To The Next Citation Point]. 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 [491]. 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 [113Jump To The Next Citation Point, 259Jump To The Next Citation Point].

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Figure 36: MOND phantom dark matter scaling relations in ellipticals. The circles display central density ρ0, and central phase space density f of the phantom dark halos predicted by MOND for different masses of baryonic Hernquist profiles (with scale-radius r H related to the effective radius by Reff = 1.815rH). The dotted lines are the scaling relations of [171], and the dashed lines those of [454], which exhibit a very large observational scatter in good agreement with the MOND prediction [363Jump To The Next Citation Point]. Image reproduced by permission from [363], copyright by ESO.

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:

2 d-σ- + σ2(2β-+--α) = − g(r), (65 ) dr r
where σ is the radial velocity dispersion, α = d ln ρ∕d ln r is the slope of the tracer density ρ, and 2 2 2 β = 1 − (σ 𝜃 + σ ϕ)∕2σ is the velocity anisotropy. Note that on the left-hand side, one uses the density and the velocity dispersion of the tracers only, which can be different from the density producing the gravity on the right-hand side, if a specific population of tracers such as globular clusters is used. When the global kinematics of a galaxy is analyzed, we do expect in MOND that the gravity on the right-hand side of Eq. 65View Equation is generated by the observed mass distribution, so both should be fit simultaneously: Figure 37View Image (provided by [399Jump To The Next Citation Point]) shows an example. In general, it was found that field galaxies all fit very naturally with MOND [461, 410] (see also [484]). 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 [364Jump To The Next Citation Point], which is just the small-scale equivalent of the problem of MOND in clusters, pointing towards missing baryons (see Section 6.6.4).
View Image

Figure 37: The surface brightness (a) and velocity dispersion (b) profiles of the elliptical galaxy NGC 7507 [375Jump To The Next Citation Point] fitted by MOND (lines [399Jump To The Next Citation Point]). Elliptical galaxies can be approximated in MOND as high-order polytropes with some radial orbit anisotropy [388]. This particular case has a polytropic index of 14 with anisotropy of the Osipkov–Merritt form with an anisotropy radius of 5 kpc and maximum anisotropy β = 0.75 at large radii [399]. The stellar mass-to-light ratio is ϒB = 3.03M ⊙∕L ⊙ ∗. This simple model captures the gross properties of both the surface brightness and velocity dispersion profiles. The galaxy is well-fitted by MOND, contrary to the claim of [375].

On the other hand, [225Jump To The Next Citation Point] 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 [225] they claimed that MOND was excluded by 10σ, but this was only for models that had constant velocity anisotropy. It was then found [14] that with varying anisotropy profiles similar to those found in simulations of the formation of ellipticals by dissipationless collapse in MOND [337Jump To The Next Citation Point], 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 z ≈ 10 [393Jump To The Next Citation Point] from monolithic dissipationless collapse [337], 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 [239Jump To The Next Citation Point] (more dwarf ellipticals in denser environments).

6.6.2 Dwarf spheroidal galaxies

Dwarf spheroidal (dSph) satellites of the Milky Way [427Jump To The Next Citation Point, 477Jump To The Next Citation Point] 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 ΛCDM 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 [477Jump To The Next Citation Point]. Solving Jeans equation (Eq. 65View Equation), it was found [8Jump To The Next Citation Point] 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 M ∕L inferred from MOND and the ages of their stellar populations [189]. Concerning the high inferred stellar M ∕L, note that it has been shown [78Jump To The Next Citation Point] 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 [418Jump To The Next Citation Point], it was found that Sextans was probably littered with unbound stars, which inflated the computed M ∕L, 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 M ∕L (> 4; see [418]).

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 [237Jump To The Next Citation Point], and exhibit an artificially high M ∕L 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 −1 σ ≈ 2 km s and − 1 σ ≈ 5 km s), and/or to high intrinsic stellar M ∕L ratios related to stochastic effects linked with the small number of stars [186], it was also found [285Jump To The Next Citation Point] 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 38View Image). As Brada & Milgrom [78Jump To The Next Citation Point] 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 [237]).

View Image

Figure 38: The characteristic acceleration, in units of a 0, in the smallest galaxies known: the dwarf satellites of the Milky Way (orange squares) and M31 (pink squares) [285Jump To The Next Citation Point]. The classical dwarfs, with thousands of velocity measurements of individual stars [477Jump To The Next Citation Point], are largely consistent with MOND. The more recently discovered “ultrafaint” dwarfs, tiny systems with only a handful of stars [427Jump To The Next Citation Point], typically are not, in the sense that their measured velocity dispersions and accelerations are too high. This could be due to systematic uncertainties in the data [230], as we must distinguish between −1 σ ≈ 2 km s and − 1 σ ≈ 5 km s. Nevertheless, there may be a good physical reason for the non-compliance of the ultrafaint galaxies in the context of MOND. The deviation of these objects only occurs in systems where the stars are close to filling their MONDian tidal radii: the left panel shows the half light radius relative to the tidal radius. Such systems may not be in equilibrium. Brada & Milgrom [78Jump To The Next Citation Point] note that systems will no longer respond adiabatically to the influence of their host galaxy when a star in a satellite galaxy can complete only a few orbits for every orbit the satellite makes about its host. The deviant dwarfs are in this regime (right panel).

6.6.3 Star clusters

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 [293] 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 ΛCDM, this dichotomy is rather explained by the formation history [235, 397Jump To The Next Citation Point]: 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 a0, 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 [408] (see also [187]). On the other hand, Pal 14 displays exactly the opposite behavior: the measured velocity dispersion is Newtonian [212], but again the number of observed stars is too small to draw a statistically significant conclusion [164], and it is still possible to reconcile the data with MOND assuming a slightly low stellar mass-to-light ratio [437]. 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 [199Jump To The Next Citation Point]. 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. 65View Equation), that the best MOND fit, although not extremely bad in itself, was 350 times less likely than the best Newtonian fit without DM [199Jump To The Next Citation Point, 200]. However, the stability [336] 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 [398]. 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. 27View Equation), 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 [397Jump To The Next Citation Point]. 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, 377Jump To The Next Citation Point] 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 [95], which could cause the globular clusters to spiral in and merge within at most 2 Gyrs [377]. However, this strongly depends on the orbits of the globular clusters, and, in particular, on their initial radius [10], which can allow for a Hubble time survival of the orbits in MOND.

6.6.4 Galaxy groups and clusters

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 σ2∕R is always observed to be on the order of a0, thereby naturally explaining the linear relation between size and temperature for galaxy clusters [327, 392Jump To The Next Citation Point]. However, one of the main predictions of Milgrom’s formula is the baryonic Tully–Fisher relation (circular velocity vs. baryonic mass, Figure 3View Image), and its equivalent for isotropic pressure-supported systems, the Faber–Jackson relation (stellar velocity dispersion vs. baryonic mass, Figure 7View Image), 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, Mb ∝ T 2, plotted on Figure 39View Image, as the line log(Mb ∕M ⊙ ) = 2log(T∕keV ) + 12.9 (note that this differs slightly from [389Jump To The Next Citation Point] 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 3∕2 M ∝ T in ΛCDM, without the need to invoke preheating (a need that may arise as an artifact of the mismatch in slopes).

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Figure 39: The baryonic mass–X-ray temperature relation for rich clusters (gray triangles [359, 389Jump To The Next Citation Point]) and groups of galaxies (green triangles [12Jump To The Next Citation Point]). The solid line indicates the prediction of MOND: the data are reasonably consistent with the slope (M ∝ T 2), but not with the normalization. This is the residual missing baryon problem in MOND: there should be roughly twice as much mass (on average) as observed. Also shown is the scaling relation expected in ΛCDM (dashed line [137]). This is in better (if not perfect) agreement with the normalization of the data for rich clusters, but not the slope. The difference is sometimes attributed to preheating of the gas [496], which might also occur in MOND.
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Figure 40: The baryon budget in the low redshift universe adopted from [421Jump To The Next Citation Point]. The census of baryons includes the detected Warm-Hot Intergalactic Medium (WHIM), the Lymanα forest, stars in galaxies, detected cold gas in galaxies (atomic HI and molecular H2), other gas associated with galaxies (the Circumgalactic Medium, CGM), and the Intracluster Medium (ICM) of groups and clusters of galaxies. The sum of known baryons falls short of the density of baryons expected from BBN: ∼ 30% are missing. These missing baryons presumably exist in some as yet undetected (i.e., dark) form. If a fraction of these dark baryons reside in clusters (an amount roughly comparable to that in the ICM) it would suffice to explain the residual mass discrepancy problem MOND suffers in galaxy clusters.

So, interestingly, the data are still reasonably consistent with the slope predicted by MOND [383Jump To The Next Citation Point], but not with the normalization. There is roughly a factor of two of residual missing mass in these objects [170, 354, 387, 389Jump To The Next Citation Point, 392Jump To The Next Citation Point, 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 [12Jump To The Next Citation Point]. 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 a0, 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 ΛCDM. 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 [24], the ratio of dark to baryonic mass is only ∼ 8 conventionally [175Jump To The Next Citation Point, 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 40View Image), 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 40View Image). 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 [310Jump To The Next Citation Point] 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 [310Jump To The Next Citation Point] rather relies on the work of Pfenniger & Combes [352], 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 10− 4, 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, [310Jump To The Next Citation Point] 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 [330]. Milgrom [310Jump To The Next Citation Point] 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 √ ----- m ν ≈ Δm2 [434Jump To The Next Citation Point], 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 ΛCDM 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 β-decay of Tritium [234Jump To The Next Citation Point] are m ν < 2.2 eV. 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 [392Jump To The Next Citation Point] would be

Ων = 0.062m ν, (66 )
where m ν is the mass of a single neutrino type in eV. If one assumes that clusters of galaxies respect the baryon-neutrino cosmological ratio, and that the MOND missing mass is mostly made of neutrinos as suggested by [389Jump To The Next Citation Point, 392Jump To The Next Citation Point], 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 [430Jump To The Next Citation Point]), 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 [463Jump To The Next Citation Point], 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:
i=∑6 4 f = 1- m-νi. (67 ) max 2 h3 i=1
Converting this into configuration space, the maximum density for a cluster of a given temperature, T, is defined for a given mass of one neutrino type as [463]:
ρmax ( T )1.5 ( m ν)4 ------−ν5-------−3-= ------ ---- . (68 ) 7 × 10 M ⊙ pc 1keV 2eV
Assuming the temperature of the neutrino fluid as being equal (due to violent relaxation) to the mean emission weighted temperature of the gas, Sanders [389Jump To The Next Citation Point] 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 T > 4 keV (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 T3∕2, as observed in [392Jump To The Next Citation Point]. However, looking at the central region of low-temperature X-ray emitting galaxy groups, it was found [12] 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 m ∼ 10 eV to reach the required densities. One exotic possibility is then the idea of right-handed eV-scale sterile neutrinos [13Jump To The Next Citation Point]: 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 ∼ 10 eV-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 a0 in galaxy clusters, while lowering it to its preferred value in galaxies. There are several ways to implement such an idea. For instance, Bekenstein [36Jump To The Next Citation Point] 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. 27View Equation). This idea should be investigated more in the future, but it is not clear that such a simple rescaling of a 0 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 [82Jump To The Next Citation Point, 143Jump To The Next Citation Point] 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 [112Jump To The Next Citation Point] 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.

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