6.5 MOND in rotationally-supported stellar systems

6.5.1 Rotation curves of disk galaxies

The root and heart of MOND, as modified inertia or modified gravity, is Milgrom’s formula (Eq. 7View Equation). Up to some small corrections outside of symmetrical situations, this formula yields (once a 0 and the form of the transition function μ 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 ΛCDM 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 [168Jump To The Next Citation Point, 402Jump To The Next Citation Point]. 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 μ-functions.

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Figure 20: Comparison of theoretical rotation curves for the inner parts (before the rotation curve flattens) of an HSB exponential disk [145Jump To The Next Citation Point], computed with three different formulations of MOND. Green: Milgrom’s formula; Blue: Bekenstein–Milgrom MOND (AQUAL); Red: TeVeS-like multi-field theory. Image reproduced by permission from [145Jump To The Next Citation Point], copyright by APS.
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Figure 21: Examples of detailed MOND rotation curve fits of the HSB and LSB galaxies of Figure 13View Image (NGC 6946 on the left and NGC 1560 on the right). The black line represents the Newtonian contribution of stars and gas as determined by numerical solution of the Newtonian Poisson equation for the observed light distribution, as per Figure 13View Image. The blue line is the MOND fit with the γ = δ = 1 function of Eq. 52View Equation and Eq. 53View Equation, the only free parameter being the stellar mass-to-light ratio. In the K-band, the best fit value is 0.37M ⊙∕L ⊙ for NGC 6946 and 0.18M ⊙∕L ⊙ for NGC 1560. In practice, the best fit mass-to-light ratio can co-vary with the distance to the galaxy and a 0; here a 0 is held fixed (1.2 × 10−10 m s−2) and the distance has been held fixed to the best observed value (5.9 Mpc for NGC 6946 [220] and 3.45 Mpc for NGC 1560 [219]). Milgrom’s formula provides an effective mapping between the rotation curve predicted by the observed baryons and the observed rotation, including the bumps and wiggles.
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Figure 22: The rotation curve [124] and MOND fit [384Jump To The Next Citation Point] of the Local Group spiral M33 assuming a constant stellar mass-to-light ratio (top panel). While the overall shape is a good match, there is a slight mismatch at ∼ 3 kpc and above 7 kpc. The observed color gradient implies a slight variation in the mass-to-light ratio, in the sense that the stars at small radii are slightly redder and heavier than those at large radii. Applying stellar population models [42Jump To The Next Citation Point] to the observed color gradient produces a slight adjustment of the Newtonian mass model. The dotted line in the lower panel reiterates the constant M ∕L model from the top panel, while the solid line has been corrected for the observed color gradient. This slight adjustment to the baryonic mass distribution considerably improves the fit.
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Figure 23: Residuals of MOND fits to the rotation curves of 78 nearby galaxies (all data to which authors have access) including about two thousand individual resolved measurements. Data for 21 galaxies are either new or improved in terms of spatial resolution and velocity accuracy over those in [401Jump To The Next Citation Point]. More accurate points are illustrated with larger symbols. The histogram of residuals is plotted on the right panel, and is well fitted by a Gaussian of width Δv ∕v ∼ 0.04. The bulk of the more accurate data are in good accord with MOND. There are a few deviant points, mostly at small radii where non-circular motions are ubiquitous and observational resolution (beam smearing) can be a challenge. These are but a few trees outlying from a very clear forest.

In order to illustrate this, we plot in Figure 20View Image the theoretical rotation curve of an HSB exponential disk (see [145Jump To The Next Citation Point] for exact parameters) computed with three different formulations of MOND38: Milgrom’s formula (Eq. 7View Equation), representative of circular orbits in modified inertia, AQUAL (Eq. 17View Equation), and a multi-field theory (Eq. 40View Equation) representative of a whole class of relativistic theories (see Sections 7.1 to 7.4), all with the α = n = 1 “simple” μ-function of Eq. 46View Equation and Eq. 49View Equation. 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 [76]. 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 [18].

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Figure 24: Examples of MOND fits (blue lines, using Eq. 53View Equation with δ = 1) to two massive galaxies [402Jump To The Next Citation Point]. With baryonic masses in excess of 1011M ⊙, these are among the most massive, rapidly rotating disk galaxies known. Stars dominate the mass, and Newtonian dynamics suffices to explain the innermost regions because of the high acceleration, but the mass discrepancy becomes apparent as the Keplerian decline (black lines) falls well below the data at the enormous radii spanned by these giant disks (the diameter of UGC 2487 spans half a million lightyears).
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Figure 25: Examples of MOND fits (blue lines) to two dwarf galaxies [324Jump To The Next Citation Point]. The data for DDO 210 come from [29], and those for UGC 11583 (also known as KK98 250) are from [30] augmented with high resolution data from [281, 242Jump To The Next Citation Point]. The high gas content of these galaxies make them strong tests of MOND, as the one fit-parameter – the mass-to-light ratio of the stars – has only a minor impact on the fit. What is more, as they are deep in the MOND regime, the exact form of the interpolating function (Section 6.2) also has little impact on the fits, making them the cleanest tests of MOND, with essentially no wiggle room. Note that, with a mass of only a few million solar masses (comparable in mass to the largest globular clusters), the Local Group dwarf DDO 210 is the smallest galaxy known to show clear rotation (Vf ∼ 15 km ∕s). It is the lowest point in Figure 3View Image.
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Figure 26: MOND rotation curve fits for representative galaxies from the THINGS survey [121, 166Jump To The Next Citation Point, 481Jump To The Next Citation Point]. Galaxies are chosen to illustrate a broad range of mass, from Mb ∼ 3 × 108M ⊙ to ∼ 3 × 1011M ⊙. All galaxies have high resolution interferometric 21 cm data for the gas and 3.6μ photometry for mapping the stars. The Newtonian baryonic mass model is shown as a black line and the MOND fit as a blue line (as in Figure 21View Image). The fits use the interpolating function of Eq. 53View Equation with δ = 1.
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Figure 27: MOND rotation curve fits for LSB galaxies [120Jump To The Next Citation Point] updated with high resolution Hα data [242Jump To The Next Citation Point, 241Jump To The Next Citation Point] and using Eq. 53View Equation with δ = 1. LSB galaxies are important tests of MOND because their low surface densities (Σ ≪ a0∕G) place them well into the MOND regime everywhere, and the exact form of the interpolating function is rather unimportant. Their baryonic mass models fall well short of explaining the observed rotation at any but the smallest radii in Newtonian dynamics, and MOND nevertheless provides the necessary additional force everywhere (lines as per Figure 21View Image).
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Figure 28: A comparison of the mass-to-light ratios obtained from MOND rotation curve fits (points) with the independent expectations of stellar population synthesis models (lines) [42Jump To The Next Citation Point]. The mass-to-light ratio in the optical (blue B-band, left) and near-infrared (2.2 μm K-band, right) are shown as a function of B − V color (the ratio of blue to green light). The one free parameter of MOND rotation curve fits reproduces the normalization, slope, and scatter expected from what we know about stars. Not all galaxies illustrated here have both B and K-band data. Some have neither, instead having photometry in some other bandpass (e.g., V or R or I).

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. 7View Equation or Eq. 10View Equation. First of all, an interpolating function must be chosen, then one can determine the value of a0 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 (α = n = 1, see Eqs. 46View Equation and 49View Equation) yields a value of −10 − 2 a0 = 1.2 × 10 m s, and excellent fits to galaxy rotation curves [166Jump To The Next Citation Point]. 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 γ = δ = 1 interpolating function of Eqs. 52View Equation and 53View Equation (solid blue line on Figure 19View Image), very similar to the simple interpolating function in the intermediate to weak gravity regime.

Figure 21View Image shows two examples of detailed MOND fits to rotation curves of Figure 13View Image. 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 23View Image. 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 24View Image fits to rotation curves of extremely massive HSB early-type disk galaxies [402] with V f up to 400 km/s, and in Figure 25View Image fits to very low mass LSB galaxies [324Jump To The Next Citation Point] with Vf 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 26View Image eight fits for representative galaxies from the latest high-resolution THINGS survey [166Jump To The Next Citation Point, 481], and in Figure 27View Image 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 [120Jump To The Next Citation Point], updated with high resolution Hα data [242, 241Jump To The Next Citation Point]. The overall results for the whole 78 nearby galaxies (Figure 23View Image) 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 [384Jump To The Next Citation Point] that one begins to notice small disparities. These are sometimes attributable to external disturbances that invalidate the assumption of equilibrium [403Jump To The Next Citation Point], non-circular motions or bad observational resolution. For targets that are intrinsically difficult to observe, minor problems become more common [120Jump To The Next Citation Point, 448Jump To The Next Citation Point]. 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 [68Jump To The Next Citation Point, 166Jump To The Next Citation Point], but this could simply be due to a problem with the potentially too high Cepheids-based distance (reddening problem mentioned in [254]). Indeed, the adopted distance plays an important role in the MOND fitting procedure, as the value of the centripetal acceleration V2c ∕R depends on the distance through the conversion of the observed angular radius in arcsec into the physical radius R 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 [166Jump To The Next Citation Point] and Figure 26View Image).

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 [42Jump To The Next Citation Point]. There is absolutely nothing built into MOND that would require that redder galaxies should have higher stellar mass-to-light ratios in the B-band, but this is what the rotation curve fits require. This is shown on Figure 28View Image, where the best-fit mass-to-light ratio in the B-band is plotted against B − V color index (left panel), and the same for the K-band (right panel).

6.5.2 The Milky Way

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, 274Jump To The Next Citation Point], measured with the tangent point method, compared to the baryonic content of the inner Galaxy [53Jump To The Next Citation Point, 155Jump To The Next Citation Point], have shown full agreement between the rotation curve and MOND, assuming, as usual, the simple interpolating function (α = n = 1 in Eqs. 46View Equation and 49View Equation) or the γ = δ = 1 interpolating function (Eqs. 52View Equation and 53View Equation). The inverse problem was also tackled, i.e., deriving the surface density of the inner Milky Way disk from its rotation curve (see Figure 29View Image): this exercise [274Jump To The Next Citation Point] 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 [494Jump To The Next Citation Point].

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Figure 29: The mass distribution of the Milky Way disk (left) inferred from fitting in MOND the observed bumps and wiggles in the rotation curve of the galaxy (right) [274]. The Newtonian contributions of the stellar and gas disk are shown as dashed and dotted lines as per Figure 13View Image. The resulting model is consistent with independent star count data [155Jump To The Next Citation Point] and compares favorably to constraints on the rotation curve at radii beyond those included in the fit [494]. The prominent feature at R ≈ 6 kpc corresponds to the Centaurus spiral arm.

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 18View Image) 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. 17View Equation).

Based on the baryonic distribution from, e.g., the Besançon model of the Milky Way [366Jump To The Next Citation Point], one can compute the MOND gravitational field of the Galaxy by solving the BM-equation (Eq. 17View Equation). This has been done in [490Jump To The Next Citation Point]. 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 [50Jump To The Next Citation Point, 140]. In this context, as already shown (Figure 18View Image), 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 [378Jump To The Next Citation Point] 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 [358]. 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 [474] 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. 61View Equation or Eq. 63View Equation) instead of a logarithmic (Eq. 20View Equation) 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 [144] that with an external field of 0.01a0, the local escape speed at the Sun’s radius was about 550 km/s, exactly as observed (within the observational error range [433]). This was later confirmed by rigorous modeling in the context of Bekenstein–Milgrom MOND and with the Besançon baryonic model of the Milky Way [492]. This value of the external field, 10 −2 × a 0, 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.

Table 1: Values predicted from the Besançon model of the Milky Way in MOND as seen by a Newtonist (i.e., in terms of phantom dark matter contributions) compared to current observational constraints in the Milky Way, for the local dynamical surface density and the tilt of the stellar velocity ellipsoid [50Jump To The Next Citation Point]. Predictions for a round dark halo without a dark disk are also compatible with the current constraints, though [194Jump To The Next Citation Point, 422Jump To The Next Citation Point]. The tilt at z = 2 kpc should be more discriminating.
MOND predictions Observations
Surface density Σ ⊙(z = 1.1 kpc) 2 78M ⊙∕pc 2 74 ± 6M ⊙∕pc [194]
Velocity ellipsoid tilt at z = 1 kpc 6 degrees 7.3 ± 1.8 degrees [422]

6.5.3 Disk stability and interacting galaxies

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. 17View Equation) can be solved numerically using various methods [50Jump To The Next Citation Point, 77Jump To The Next Citation Point, 96, 147Jump To The Next Citation Point, 250Jump To The Next Citation Point, 457Jump To The Next Citation Point]. 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 [77Jump To The Next Citation Point] investigated the important problem of stability of disk galaxies. They demonstrated that MOND, as anticipated [299Jump To The Next Citation Point], 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 30View Image). 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.

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Figure 30: The scaled growth-rate of the m = 2 instability in Newtonian disks with a dark halo (dotted line) and MONDian disks (solid line) as a function of disk mass. In the MOND case, as the disk mass decreases, the surface density decreases and the disk sinks deeper into the MOND regime. However, at very low masses the growth-rate saturates. In the equivalent Newtonian case, the rotation curve is maintained at the MOND level by supplementing the force with a round stabilizing dark halo, which causes the growth-rate to crash [77Jump To The Next Citation Point, 401Jump To The Next Citation Point]. An ad-hoc dark disk could help maintain the growth rate in the dark matter context. Image reproduced by permission from [401Jump To The Next Citation Point].

This was confirmed in recent simulations [104Jump To The Next Citation Point, 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 [458Jump To The Next Citation Point]. 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 31View Image and Figure 32View Image). All these results have been shown to be independent of the exact choice of interpolating μ-function [458Jump To The Next Citation Point].

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 [25Jump To The Next Citation Point, 241Jump To The Next Citation Point], instead of 2 km/s for Newtonian disks with −2 Σ = 1M ⊙ pc (depending on the thickness of the disk). However, the observed value is usually attributed to non-gravitational phenomena. Note that [279Jump To The Next Citation Point] 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 [159Jump To The Next Citation Point, 371Jump To The Next Citation Point]. 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 [497], 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 [79Jump To The Next Citation Point]. 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 [79], 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 [104Jump To The Next Citation Point]. 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 [95Jump To The Next Citation Point] 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, 459Jump To The Next Citation Point]. Concerning bars, Nipoti [335Jump To The Next Citation Point] found that they were indeed slowed down more in MOND, as predicted analytically [95Jump To The Next Citation Point], 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 [95Jump To The Next Citation Point], 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 ΛCDM, 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 [239Jump To The Next Citation Point]. 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 [104Jump To The Next Citation Point].

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 [459Jump To The Next Citation Point]. This is illustrated in Figure 33View Image. 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 [130Jump To The Next Citation Point]. 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).

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Figure 31: (a) The galaxy ESO 509-98. (b) The galaxy NGC 1543. These are two examples of galaxies that exhibit clear ring and pseudo-ring structures. Image courtesy of Tiret, reproduced by permission from [458Jump To The Next Citation Point], copyright by ESO.
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Figure 32: Simulations of ESO 509-98 and NGC 1543 in MOND, to be compared with Figure 31View Image. Rings and pseudo-ring structures are well reproduced with modified gravity. Image courtesy of Tiret, reproduced by permission from [458Jump To The Next Citation Point], copyright by ESO.
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Figure 33: Simulation of the Antennae with MOND (right, [459Jump To The Next Citation Point]) compared to the observations (left, [190]). In the observations, the gas is represented in blue and the stars in green. In the simulation the gas is in blue and the stars are in yellow/red. Image courtesy of Tiret, reproduced by permission from [459Jump To The Next Citation Point], copyright by ASP.

6.5.4 Tidal dwarf galaxies

As seen in, e.g., Figure 33View Image, 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 ΛCDM model, these objects are difficult to form, and require very extended dark matter distribution [71]. In MOND simulations [459Jump To The Next Citation Point, 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.

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Figure 34: The NGC 5291 system [72Jump To The Next Citation Point]. VLA atomic hydrogen 21-cm map (blue) superimposed on an optical image (white). The UV emission observed by GALEX (red) traces dense star-forming concentrations. The most massive of these objects are rotating with the projected spin axis as indicated by dashed arrows. The three most massive ones are denoted as NGC5291N, NGC5291S, and NGC5291W. Image courtesy of Bournaud, reproduced by permission from [72Jump To The Next Citation Point].
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Figure 35: Rotation curves of the three TDGs in the NGC 5291 system. In red: ΛCDM prediction (with no additional cold molecular gas), with the associated uncertainties. In black: MOND prediction with the associated uncertainties (prediction with zero free parameter, “simple” μ-function assumed). Image reproduced by permission from [165Jump To The Next Citation Point], copyright by ESO.

What is more, in the ΛCDM 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 ΛCDM 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 34View Image) has revealed the presence of dark matter in these three objects [72Jump To The Next Citation Point]. 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 (∘ i = 45, see [72]), and apply Milgrom’s formula to the visible matter distribution with zero free parameters [165Jump To The Next Citation Point, 309Jump To The Next Citation Point], one gets very reasonable curves (Figure 35View Image). Playing around a little bit with the inclinations allows perfect fits to these rotation curves [165Jump To The Next Citation Point], while the influence of the external field effect has been shown not to significantly change the result. Therefore, we can conclude that ΛCDM 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 [239Jump To The Next Citation Point], 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 ΛCDM (see Section 4.2), it could actually make sense in a MONDian Universe (see Section 9.2).

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