The root and heart of MOND, as modified inertia or modified gravity, is Milgrom’s formula (Eq. 7). Up to some small corrections outside of symmetrical situations, this formula yields (once and the form of the transition function are chosen) a unique prediction for the total effective gravity as a function of the gravity produced by the visible baryons. It is absolutely remarkable that this formula, devised 30 years ago, has been able to successfully predict an impressive number of galactic scaling relations (the “Kepler-like” laws of Section 5.2, backed by the modern data of Section 4.3) that were very unprecise and/or unobserved at the time, and which still are a puzzle to understand in the CDM framework. What is more, this formula not only predicts global scaling relations successfully, we show in this section that it also predicts the shape and amplitude of galactic rotation curves at all radii with uncanny precision, and this for all disk galaxy Hubble types [168, 402]. Of course, the absolute exact prediction of MOND depends on the exact formulation of MOND (as modified inertia or some form or other of modified gravity), but the differences are small compared to observational error bars, and even compared with the differences between various -functions.
In order to illustrate this, we plot in Figure 20 the theoretical rotation curve of an HSB exponential disk (see  for exact parameters) computed with three different formulations of MOND38: Milgrom’s formula (Eq. 7), representative of circular orbits in modified inertia, AQUAL (Eq. 17), and a multi-field theory (Eq. 40) representative of a whole class of relativistic theories (see Sections 7.1 to 7.4), all with the “simple” -function of Eq. 46 and Eq. 49. One can see velocity differences of only a few percents in this case, while, in general, it has been shown that the maximum difference between formulations is on the order of 10% for any type of disk . This justifies using Milgrom’s formula as a proxy for MOND predictions on rotation curves, keeping in mind that, in order to constrain MOND within the modified gravity framework, one should actually calculate predictions of the various modified Poisson formulations of Section 6.1 for each galaxy model, and for each choice of galaxy parameters .
The procedure is then the following (see Section 4.3.4 for more detail). One usually assumes that light traces stellar mass (constant mass-to-light ratio, but see the counter-example M33), and one adds to this baryonic density the contribution of observed neutral hydrogen, scaled up to account for the contribution of primordial helium. The Newtonian gravitational force of baryons is then calculated via the Newtonian Poisson equation, and the MOND force is simply obtained via Eq. 7 or Eq. 10. First of all, an interpolating function must be chosen, then one can determine the value of by fitting, all at once, a sample of high-quality rotation curves with small distance uncertainties and no obvious non-circular motions. Then, all individual rotation curve fits can be performed with the mass-to-light ratio of the disk as the single free parameter of the fit39. It turns out that using the simple interpolating function (, see Eqs. 46 and 49) yields a value of , and excellent fits to galaxy rotation curves . However, as already pointed out in Sections 6.3 and 6.4, this interpolating function yields too strong a modification in the solar system, so hereafter we use the interpolating function of Eqs. 52 and 53 (solid blue line on Figure 19), very similar to the simple interpolating function in the intermediate to weak gravity regime.
Figure 21 shows two examples of detailed MOND fits to rotation curves of Figure 13. The black line represents the Newtonian contribution of stars and gas and the blue line is the MOND fit, the only free parameter being the stellar mass-to-light ratio40. Not only does MOND predict the general trend for LSB and HSB galaxies, it also predicts the observed rotation curves in great detail. This procedure has been carried out for 78 nearby galaxies (all galaxy rotation curves to which the authors have access), and the residuals between the observed and predicted velocities, at every point in all these galaxies (thus about two thousand individual measurements), are plotted in Figure 23. As an illustration of the variety and richness of rotation curves fitted by MOND, as well as of the range of magnitude of the discrepancies covered, we display in Figure 24 fits to rotation curves of extremely massive HSB early-type disk galaxies  with up to 400 km/s, and in Figure 25 fits to very low mass LSB galaxies  with down to 15 km/s. In the latter, gas-rich, small galaxies, the detailed fits are insensitive to the exact form of the interpolating function (Section 6.2) and to the stellar mass-to-light ratio [168, 324]. We then display in Figure 26 eight fits for representative galaxies from the latest high-resolution THINGS survey [166, 481], and in Figure 27 six fits of yet other LSB galaxies (as these provide strong tests of MOND and depend less on the exact form of the interpolating function than HSB ones) from , updated with high resolution H data [242, 241]. The overall results for the whole 78 nearby galaxies (Figure 23) are globally very impressive, although there are a few outliers among the 2000 measurements. These are but a few trees outlying from a very clear forest. It is actually only as the quality of the data decline  that one begins to notice small disparities. These are sometimes attributable to external disturbances that invalidate the assumption of equilibrium , non-circular motions or bad observational resolution. For targets that are intrinsically difficult to observe, minor problems become more common [120, 448]. These typically have to do with the challenges inherent in combining disparate astronomical data sets (e.g., rotation curves measured independently at optical and radio wavelengths) and constraining the inclinations. A single individual galaxy that can be considered as a bit problematic is NGC 3198 [68, 166], but this could simply be due to a problem with the potentially too high Cepheids-based distance (reddening problem mentioned in ). Indeed, the adopted distance plays an important role in the MOND fitting procedure, as the value of the centripetal acceleration depends on the distance through the conversion of the observed angular radius in arcsec into the physical radius in kpc. Note that other galaxies such as NGC 2841 had historically-posed problems to MOND but that these have largely gone away with modern data (see  and Figure 26).
We finally note that what makes all these rotation curve fits really impressive is that either (i) stellar mass-to-light ratios are unimportant (in the case of gas-rich galaxies) yielding excellent fits with essentially zero free parameters (apart from some wiggle room on the distance), or (ii) stellar mass-to-light ratios are important, and their best-fit value, obtained on purely dynamical grounds assuming MOND, vary with galaxy color as one would expect on purely astrophysical grounds from stellar population synthesis models . There is absolutely nothing built into MOND that would require that redder galaxies should have higher stellar mass-to-light ratios in the -band, but this is what the rotation curve fits require. This is shown on Figure 28, where the best-fit mass-to-light ratio in the -band is plotted against color index (left panel), and the same for the -band (right panel).
Our own Milky Way galaxy (an HSB galaxy) is a unique laboratory within which present and future surveys will allow us to perform many precision tests of MOND (at a level of precision that might even discriminate between the various versions of MOND described in Section 6.1) that are not feasible with external galaxies. However, concerning the rotation curve, the test is, at present, not the most conclusive, as the outer rotation curve of the Milky Way is paradoxically much less precisely known than that of external galaxies (the forthcoming Gaia mission should allow improvement to this situation, although the rotation curve will not be measured directly). Nevertheless, past studies of the inner rotation curve of the Milky Way [141, 142, 274], measured with the tangent point method, compared to the baryonic content of the inner Galaxy [53, 155], have shown full agreement between the rotation curve and MOND, assuming, as usual, the simple interpolating function ( in Eqs. 46 and 49) or the interpolating function (Eqs. 52 and 53). The inverse problem was also tackled, i.e., deriving the surface density of the inner Milky Way disk from its rotation curve (see Figure 29): this exercise  led to a derived surface density fully consistent with star count data, and also even reproducing the details of bumps and wiggles in the surface brightness (Renzo’s rule, Section 4.3.4), while being fully consistent with the (somewhat imprecise) constraints on the outer rotation curve of the galaxy .
However, especially with the advent of present and future astrometric and spectroscopic surveys, the Milky Way offers a unique opportunity to test many other predictions of MOND. These include the effect of the “phantom dark disk” (see Figure 18) on vertical velocity dispersions and on the tilt of the stellar velocity ellipsoid, the precise shape of tidal streams around the galaxy, or the effects of the external gravitational field in which the Milky Way is embedded on fundamental parameters such as the local escape speed. However, all these predictions can vary slightly depending on the exact formulation of MOND (mainly Bekenstein–Milgrom MOND, QUMOND, or multi-field theories, the predictions being anyway difficult to make in modified inertia versions of MOND when non-circular orbits are considered). Most of the predictions made until today and reviewed hereafter have been using the Bekenstein–Milgrom version of MOND (Eq. 17).
Based on the baryonic distribution from, e.g., the Besançon model of the Milky Way , one can compute the MOND gravitational field of the Galaxy by solving the BM-equation (Eq. 17). This has been done in . Then one can apply the Newtonian Poisson equation to it, in order to find back the density distribution that would have yielded this potential within Newtonian dynamics [50, 140]. In this context, as already shown (Figure 18), MOND predicts a disk of “phantom dark matter” allowing one to clearly differentiate it from a Newtonian model with a dark halo:
Such tests of MOND could be applied with the first release of future Gaia data. To fix the ideas on the current local constraints, the predictions of the Besançon MOND model are compared with the relevant observations in Table 1. However, let us note that these predictions are extremely dependent on the baryonic content of the model [53, 155, 366], so that testing MOND at the precision available in the Milky Way heavily relies on star counts, stellar population synthesis, census of the gaseous content (including molecular gas), and inhomogeneities in the baryonic distribution (clusters, gas clouds).
Another test of the predictions of MOND for the gravitational potential of the Milky Way is the thickness of the HI layer as a function of position in the disk (see Section 6.5.3): it has been found  that Bekenstein–Milgrom MOND and it phantom disk successfully accounts for the most recent and accurate flaring of the HI layer beyond 17 kpc from the center, but that it slightly underpredicts the scale-height in the region between 10 and 15 kpc. This could indicate that the local stellar surface density in this region should be slightly smaller than usually assumed, in order for MOND to predict a less massive phantom disk and hence a thicker HI layer. Another explanation for this discrepancy would rely on non-gravitational phenomena, namely ordered and small-scale magnetic fields and cosmic rays contributing to support the disk.
Yet another test would be the comparison of the observed Sagittarius stream [198, 248] with the predictions made for a disrupting galaxy satellite in the MOND potential of the Milky Way. Basic comparisons of the stream with the orbit of a point mass have shown accordance at the zeroth order . In reality, such an analysis is not straightforward because streams do not delineate orbits, and because of the non-linearity of MOND. However, combining a MOND N-body code with a Bayesian technique  in order to efficiently explore the parameter space, it should be possible to rigorously test MOND with such data in the near future, including for external galaxies, which will lead to an exciting battery of new observational tests of MOND.
Finally, a last test of MOND in the Milky Way involves the external field effect of Section 6.3. As explained there, the return to a Newtonian (Eq. 61 or Eq. 63) instead of a logarithmic (Eq. 20) potential at large radii is defining the escape speed in MOND. By observationally estimating the escape speed from a system (e.g., the Milky Way escape speed from our local neighborhood), one can estimate the amplitude of the external field in which the system is embedded. With simple analytical arguments, it was found  that with an external field of , the local escape speed at the Sun’s radius was about 550 km/s, exactly as observed (within the observational error range ). This was later confirmed by rigorous modeling in the context of Bekenstein–Milgrom MOND and with the Besançon baryonic model of the Milky Way . This value of the external field, , corresponds to the order of magnitude of the gravitational field exerted by Large Scale Structure, estimated from the acceleration endured by the Local Group during a Hubble time in order to attain a peculiar velocity of 600 km/s.
A lot of questions in galaxy dynamics require using N-body codes. This is notably necessary for studying stability of galaxy disks, the formation of bars and spirals, or highly time-varying configurations such as galaxy mergers. As we have seen in Section 6.1.2, the BM modified Poisson equation (Eq. 17) can be solved numerically using various methods [50, 77, 96, 147, 250, 457]. Such a Poisson solver can then be used in particle-mesh N-body codes. More general codes based on QUMOND (Section 6.1.3) are currently under development.
The main results obtained via these simulations are the following (the comparison with observations will be discussed below):
Concerning the first point (i), Brada & Milgrom  investigated the important problem of stability of disk galaxies. They demonstrated that MOND, as anticipated , has an effect similar to a dark halo in stabilizing a rotationally-supported disk, thereby explaining the upper limit in surface density seen in the data (Section 4.3.2), and also showing how it damps the growth-rate of bar-forming modes in the weak gravitational field regime. In a comparison of MOND disks with the equivalent Newtonian+halo counterpart (with identical rotation curves), they found that, as the surface density of the disk decreases, the growth-rate of the bar-forming mode decreases similarly in both cases. However, in the limit of very low surface densities, typical of LSB galaxies, the MOND growth rate stops decreasing, contrary to the Newton+dark halo case (Figure 30). This could provide a solution to the stability challenge of Section 4.2, as observed LSBs do exhibit bars and spirals, which would require an ad hoc dark component within the self-gravitating disk of the Newtonian system. One can also see on this figure that if the surface density is typical of intermediate HSB galaxies, the bar systematically forms quicker in MOND.
This was confirmed in recent simulations [104, 457], where it was additionally found that (ii) the bar is sustained longer, and is not slowed down by dynamical friction against the dark halo, which leads to fast bars, consistent with the observed fast bars in disk galaxies (measured through the position of resonances). However, when gas inflow and external gas accretion are included, a larger range of situations are met regarding pattern speeds in MOND, all compatible with observations . Since the bar pattern speed has a tendency to stay constant, the resonances remain at the same positions, and particles are trapped on these orbits more easily than in the Newtonian case, which leads to the formation of rings and pseudo-rings as observed (see Figure 31 and Figure 32). All these results have been shown to be independent of the exact choice of interpolating -function .
What is more, (iii) LSB disks can be both very thin and extended in MOND thanks to the stabilizing effect of the “phantom disk”, and vertical velocity dispersions level off at 8 km/s, as typically observed [25, 241], instead of 2 km/s for Newtonian disks with (depending on the thickness of the disk). However, the observed value is usually attributed to non-gravitational phenomena. Note that  utilized this fact to predict that conventional analyses of LSB disks would infer abnormally high mass-to-light ratios for their stellar populations – a prediction that was subsequently confirmed [159, 371]. But let us also note that this stabilizing effect of the phantom disk, leading to very thin stellar and gaseous layers, could even be too strong in the region between 10 and 15 kpc from the galactic center in the Milky Way (see Section 6.5.2), and in external galaxies , even though, as said, non-gravitational effects such as ordered and small-scale magnetic fields and cosmic rays could significantly contribute to the prediction in these regions.
Via these simulations, it has also been shown (iv) that the external field effect of MOND (Section 6.3) offers a mechanism other than the relatively weak effect of tides in inducing and maintaining warps . It was demonstrated that a satellite at the position and with the mass of the Magellanic clouds can produce a warp in the plane of the galaxy with the right amplitude and form , and even more importantly, that isolated galaxies could be affected by the external field of large scale structure, inducing a differential precession over the disk, in turn causing a warp . This could provide a new explanation for the puzzle of isolated warped galaxies.
Interactions and mergers of galaxies are (v) very important in the cosmological context of galaxy formation (see also Section 9.2). It has been found  from analytical arguments that dynamical friction should be much more efficient in MOND, for instance for bar slowing down or mergers occurring more quickly. But simulations display exactly the opposite effect, in the sense of bars not slowing down and merger time-scales being much larger in MOND [338, 459]. Concerning bars, Nipoti  found that they were indeed slowed down more in MOND, as predicted analytically , but this is because their bars were unrealistically small compared to observed ones. In reality, the bar takes up a significant fraction of the baryonic mass, and the reservoir of particles to interact with, assumed infinite in the case of the analytic treatment , is in reality insufficient to affect the bar pattern speed in MOND. Concerning long merging time-scales, an important constraint from this would be that, in a MONDian cosmology, there should perhaps be fewer mergers, but longer ones than in CDM, in order to keep the total observed amount of interacting galaxies unchanged. This is indeed what is expected (see Section 9.2). What is more, the long merging time-scales would imply that compact galaxy groups do not evolve statistically over more than a crossing time. In contrast, in the Newtonian+dark halo case, the merging time scale would be about one crossing time because of dynamical friction, such that compact galaxy groups ought to undergo significant merging over a crossing time, contrary to what is observed . Let us also note that, in MOND, many passages in binary galaxies will happen before the final merging, with a starburst triggered at each passage, meaning that the number of observed starbursts as a function of redshift cannot be used as an estimate of the number of mergers .
Finally, (vi) at a more detailed level, the Antennae system, the prototype of a major merger, has been shown to be nicely reproducible in MOND . This is illustrated in Figure 33. On the contrary, while it is well established that CDM models can result in nice tidal tails, it turns out to be difficult to simultaneously match the narrow morphology of many observed tidal tails with rotation curves of the systems from which they come . In MOND, reproducing the Antennae requires relatively fine-tuned initial conditions, but the resulting tidal tails are narrow and the galaxy is more extended and thus closer to observations than with CDM, thanks to the absence of angular momentum transfer to the dark halo (solution to the angular momentum challenge of Section 4.2).
As seen in, e.g., Figure 33, left panel, major mergers between spiral galaxies are frequently observed with dwarf galaxies at the extremity of their tidal tails, called Tidal Dwarf Galaxies (TDG). These young objects are formed through gravitational instabilities within the tidal tails, leading to local collapse of gas and star formation. These objects are very common in interacting systems: in some cases dozens of such condensations are seen in the tidal tails, with a few ones having a mass typical of other dwarf galaxies in the Universe. However, in the CDM model, these objects are difficult to form, and require very extended dark matter distribution . In MOND simulations [459, 104], the exchange of angular momentum occurs within the disks, whose sizes are inflated. For this reason, it is much easier with MOND to form TDGs in extended tidal tails.
What is more, in the CDM context, these objects are not expected to drag CDM around them, the reason being that these objects are formed out of the material in the tidal tails, itself made of the dynamically cold, rotating, material in the progenitor disk galaxies. In these disks, the local ratio of dark matter to baryons is close to zero. For this reason, the CDM prediction is that these objects should not exhibit a mass discrepancy problem. However, the first ever measurement of the rotation curve of three TDGs in the NGC 5291 ring system (Figure 34) has revealed the presence of dark matter in these three objects . A solution to explain this in the standard picture could then be to resort to dark baryons in the form of cold molecular gas in the disks of the progenitor galaxies. However, it is very surprising that a very different kind of dark matter, in this case baryonic dark matter, would conspire to assemble itself precisely in the right way such as to put the three TDGs (see Section 4.3.1) on the baryonic Tully–Fisher relation (when this baryonic dark matter is not taken into account in the baryonic budget of the BTF). Another possibility, not resorting to baryonic dark matter, would be that, by chance, the three TDGs have been observed precisely edge-on. However, if we simply consider the most natural inclination coming from the geometry of the ring (, see ), and apply Milgrom’s formula to the visible matter distribution with zero free parameters [165, 309], one gets very reasonable curves (Figure 35). Playing around a little bit with the inclinations allows perfect fits to these rotation curves , while the influence of the external field effect has been shown not to significantly change the result. Therefore, we can conclude that CDM has severe problems with these objects, while MOND does exceedingly well in explaining their observed rotation curves.
However, the observations of only three TDGs are, of course, not enough, from a statistical point of view, in order for this result to be as robust as needed. Many other TDGs should be observed to randomize the uncertainties, and consolidate (or invalidate) this potentially extremely important result, that could allow one to really discriminate between Milgrom’s law being either a consequence of some fundamental aspect of gravity (or of the nature of dark matter), or simply a mere recipe for how CDM organizes itself inside spiral galaxies. As a summary, since the internal dynamics of tidal dwarfs should not be affected by CDM, they cannot obey Milgrom’s law for a statistically-significant sample of TDGs if Milgrom’s law is only linked to the way CDM assembles itself in galaxies. Thus, observations of the internal dynamics of TDGs should be one of the observational priorities of the coming years in order to settle this debate.
Finally, let us note that it has been suggested , as a possible solution to the satellites phase-space correlation problem of Section 4.2, that most dwarf satellites of the Milky Way could have been formed tidally, thereby being old tidal dwarf galaxies. They would then naturally appear in closely related planes, explaining the observed disk-of-satellites. While this scenario would lead to a missing satellites catastrophe in CDM (see Section 4.2), it could actually make sense in a MONDian Universe (see Section 9.2).
Living Rev. Relativity 15, (2012), 10
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