10 Summary and Discussion

In this review, after briefly presenting the currently favored ΛCDM model of cosmology (which clearly works overwhelmingly well on large scales despite its slightly unelegant mixture of currently unknown elements, Sections 2 and 3), we reviewed the few most outstanding challenges that this model is still facing (Section 4), which will have to be addressed one way or the other in the coming years. These include coincidences at z = 0 between the scale of the energy density in dark energy, dark matter, and baryonic matter, as well as a common natural scale for the behavior of the dark matter and dark energy sectors. What is more, as far as galaxy formation is concerned, many predictions made by the model (keeping in mind that baryon physics could modify these predictions) were ruled out by observations: these include many observations indicating that structure formation should take place earlier than predicted, the low number of observed satellites around the Milky Way (especially the missing satellites at the low and high mass ends of the mass function), the phase-space correlation of satellite galaxies of the Milky Way as opposed to their predicted isotropic distribution, the apparent presence of constant DM density cores in the central parts of galaxies instead of the predicted cuspy dark halos, the over-abundance of large bulgeless thin disk galaxies that are extremely difficult to produce in simulations, or the presence of spiral arms in disks that should be immune to such instabilities. But even more challenging is the appearance (Figure 48View Image) of an acceleration constant − 10 −2 a0 ≃ 10 m s (i.e., the common scale of the dark matter and dark energy sectors as 1∕2 a0 ∼ Λ in natural units) in many unrelated scaling relations for DM and baryons in galaxies. These scaling relations involve a possibly devastating amount of fine-tuning for all collisionless dark matter models (Section 4.3), and can all be summarized by Milgrom’s empirical formula (Section 5), meaning that the observed gravitational field in galaxies is mimicking a universal force law generated by the baryons alone.

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Figure 48: The acceleration parameter ∼ V4f ∕ (GMb ) of extragalactic systems, spanning ten decades in baryonic mass M b. X-ray emitting galaxy groups and clusters are visibly offset from smaller systems, but by a remarkably modest amount over such a long baseline. The characteristic acceleration scale √ -- a0 ∼ Λ is in the data, irrespective of the interpretation. And it actually plays various other independent roles in observed galaxy phenomenology. This is natural in MOND (see Section 5.2), but not in ΛCDM (see Section 4.3).

With inert, collisionless and dissipationless DM, making Milgrom’s law emerge requires a huge, and perhaps even unreasonable, amount of fine-tuning in the expected feedback from the baryons. Indeed, the relation between the distribution of baryons and DM should depend on the various different histories of formation, intrinsic evolution, and interaction with the environment of the various different galaxies, whereas Milgrom’s law provides a sucessful unique and history-independent relation. Given this puzzle, the central idea of Modified Newtonian Dynamics (MOND) is rather to explore the possibility that the force law is indeed effectively modified (Section 6). The main motivation for studying MOND is thus a fully empiricist one, as it is driven by the observed phenomenology on galaxy scales, and not by an aesthetic wish of getting rid of DM. The corollary is that it is not a problem for a theory designed to reproduce the uncanny successes of the MOND phenomenology to replace CDM by “dark fields” (see Section 7) or more exotic forms of DM, different from simple collisionless DM particles, contrary to the common belief that this would be against the spirit of the MOND paradigm (although it is true that it would be more elegant to avoid too many additional degrees of freedom). It is perhaps more important that, if MOND is correct in the sense of the acceleration a0 being a truly fundamental quantity, the strong equivalence principle cannot hold anymore, and local Lorentz invariance could perhaps be spontaneously violated too.

At this juncture, it is worthwhile to summarize the general predictions of MOND, as a paradigm, and their observational tests (Table 2). As a mathematical description of the effective force law, MOND works remarkably well in individual galaxies. As a modified gravity theory (at the classical level), it makes some predictions that are both unique and challenging to reproduce in the context of the ΛCDM paradigm. However, MOND faces sharp challenges, particularly with cosmology and in rich clusters of galaxies, which will not be conclusively addressed without a viable parent theory (Section 7), based on first principles and underlying the MOND paradigm (if such a theory exists at all). In any case, in his series of papers introducing the idea in 1983, Milgrom [294] made a few very explicit predictions, which we quote hereafter, and compare with modern observational data (see also the Kepler-like laws of galactic dynamics in Section 5.2):

The original predictions listed above cover many situations, but not all. Indeed, once one writes a specific force law, its application must be completely general. Such a hypothesis is readily subject to falsification, provided sufficiently accurate data to test it – a perpetual challenge for astronomy. Table 2 summarizes the tests discussed here. By and large, tests of MOND involving rotationally-supported disk galaxies are quite positive, as largely detailed above (see Section 6.5). By construction, there is no cusp problem (solution to challenge no. 6 of Section 4.2), and no missing baryons problem (solution to challenge no. 10 of Section 4.2), as the way the dynamical mass-to-light ratio systematically varies with the circular velocity is a direct consequence of Milgrom’s law (Kepler-like law no. 4 of Section 5.2). There does appear to be a relation between the quality of the data and the ease with which a MOND fit to the rotation curve is obtained, in the sense that fits are most readily obtained with the best data [28]. As the quality of the data decline [384], one begins to notice small disparities. These are sometimes attributable to external disturbances that invalidate the assumption of equilibrium [403]. 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 of LSB galaxies (bear in mind that all velocities require a sin(i) correction to project the observed velocity into the plane of the disk, and mass in MOND scales as the fourth power of velocity). Given the intrinsic difficulties of astronomical observations, it is remarkable that the success rate of MOND fits is as high as it is: of the 78 galaxies that have been studied in detail (see Section 6.5.1), only a few cases (most notably NGC 3198 [68, 166]) appear to pose challenges. Given the predictive and quantitative success of the majority of the fits, it would seem unwise to ignore the forest and focus only on the outlying trees.

One rotationally-supported system that is very familiar to us is the solar system (see Section 6.4). The solar system is many orders of magnitude removed from the MOND regime (Figure 11View Image), so no strong effects are predicted. However, it is, of course, possible to obtain exquisitely precise data in the solar system, so it is conceivable that some subtle effect may be observable [391]. Indeed, the lack of such effects on the inner planets already appears to exclude some slowly-varying interpolation functions [62]. Other tests may yet prove possible [37, 314], but, as they are strong-field gravity tests by nature, they all depend strongly on the parent relativistic theory (Section 7) and how it converges towards GR [22]. So, in Table 2, we list the status of solar-system tests as unclear, depending on the parent relativistic theory.

An important aspect of galactic disks is their stability (see Section 6.5.3). Indeed, the need to stabilize disks was one of the early motivations for invoking dark matter [343]. MOND appears able to provide the requisite stability [77Jump To The Next Citation Point]. Indeed, it gives good reason [299] for the observed maximum in the distribution of disk galaxy surface densities at ∼ Σ† = a0∕G (Freeman’s limit: Figure 8View Image and Kepler-like law no. 6 in Section 5.2). Disks with surface densities below this threshold are in the low acceleration limit and can be stabilized by MOND. Higher-surface-density disks would be purely in the Newtonian regime and subject to the usual instabilities. Going beyond the amount of stability required for existence, another positive aspect of MOND is that it does not over-stabilize disks. Features like bars and spiral arms are a natural result of disk self-gravity. Conventionally, large halo-to-disk mass ratios suppress the growth of such features, especially in LSB galaxies [291]. Yet such features are present70. The suppression is not as great in MOND [77], and numerical simulations appear to do a good job of reproducing the range of observed morphologies of spiral galaxies (solution to challenge no. 9 of Section 4.2, see [458]). Bars tend to appear more quickly and are fast, while warps can also be naturally produced (Section 6.5.3). There appears to be no reason why this should not extend to thin and bulgeless disks, whose ubiquity poses a challenge to galaxy formation models in ΛCDM. This particular point of creating large bulgeless disks (challenge no. 8 of Section 4.2) can actually be solved thanks to early structure formation followed by a low galaxy-interaction rate in MONDian cosmology (see Section 9.2), but this definitely warrants further investigation, so we mark this case as merely promising in Table 2.

Interacting galaxies are, by definition, non-stationary systems in which the customary assumption of equilibrium does not generally hold. This renders direct tests of MOND difficult. However, it is worth investigating whether commonly observed morphologies (e.g., tidal tails) are even possible in MOND. Initially, this seemed to pose a fundamental difficulty [279], as dark matter halos play a critical role in absorbing the orbital energy and angular momentum that it is necessary to shed if passing galaxies are to not only collide, but stick and merge. Nevertheless, recent numerical simulations appear to do a nice job of reproducing observed morphologies [459]. This is no trivial feat. While it is well established that dark matter 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 [130]. Narrow tidal tails appear to be natural in MOND, as well as more extended resulting galaxies, thanks to the absence of angular momentum transfer to the dark halo (solution to challenge no. 7 of Section 4.2). Additionally, tidal dwarfs that form in these tails clearly have characteristics closer to those observed (see Section 6.5.4) than those from dark matter simulations [165, 309].

Spheroidal systems also provide tests of MOND (Section 6.6). Unlike the case of disk galaxies, where orbits are coplanar and nearly circular so that the centripetal acceleration can be equated with the gravitational force, the orbits in spheroidal systems are generally eccentric and randomly oriented. This introduces an unknown geometrical factor usually subsumed into a parameter that characterizes the anisotropy of the orbits. Accepting this, MOND appears to perform well in the classical dwarf spheroidal galaxies, but implies that the ultrafaint dwarfs are out of equilibrium (see Section 6.6.2). For small systems like the ultrafaint dwarfs and star clusters (Section 6.6.3) within the Milky Way, the external field effect (Section 6.3) can be quite important. This means that star clusters generally exhibit Newtonian behavior by virtue of being embedded in the larger galaxy. Deviations from purely Newtonian behavior are predicted to be subtle and are fodder for considerable debate [199, 397], rendering the present status unclear (Table 2). At the opposite extreme of giant elliptical galaxies (Section 6.6.1), the data accord well with MOND [323]. Indeed, bright elliptical galaxies are sufficiently dense that their inner regions are well into the Newtonian regime. In the MONDian context, this is the reason that it has historically been difficult to find clear evidence for mass discrepancies in these systems. The apparent need for dark matter does not occur until radii where the accelerations become low. That only spheroidal stellar systems appear to exist at surface densities in excess of Σ † is the corollary of Freeman’s limit: such dense systems could not exist as stable disks, so must perforce become elliptical galaxies, regardless of the formation mechanism that made them so dense. That populations of elliptical galaxies should obey the Faber–Jackson relation (Kepler-like law no. 3 in Section 5.2, Figure 7View Image) is also very natural to MOND [383, 395].

The largest gravitationally-bound systems are also spheroidal systems: rich clusters of galaxies. The situation here is quite problematic for MOND (Section 6.6.4). Applying MOND to ascertain the dynamical mass routinely exceeds the observed baryonic mass by a factor of 2 to 3. In effect, MOND requires additional dark matter in galaxy clusters. The need to invoke unseen mass is most unpleasant for a theory that otherwise appears to be a viable alternative to the existence of unseen mass. However, one should remember that the present-day motivation for studying MOND is driven by the observed phenomenology on galaxy scales, summarized above, and not by an aesthetic wish of getting rid of DM. What is more, parent relativistic theories of MOND might well involve additional degrees of freedom in the form of “dark fields”. But in any case, one must be careful not to conflate the rather limited missing mass problem that MOND suffers in clusters with the non-baryonic collisionless cold dark matter required by cosmology. There is really nothing about the cluster data that requires the excess mass to be non-baryonic, as long as it behaves in a collisionless way. There could for instance be baryonic mass in some compact non-luminous form (see Section 6.6.4 for an extensive discussion). This might seem to us unlikely, but it does have historical precedent. When Zwicky [518] first identified the dark matter problem in clusters, the mass discrepancy was of order ∼ 100. That is, unseen mass outweighed the visible stars by two orders of magnitude. It was only decades later that it was recognized that baryons residing in a hot intracluster gas greatly outweighed those in stars. In effect, there were [at least] two missing mass problems in clusters. One was the hot gas, which reduces the conventional discrepancy from a factor of ∼ 100 to a factor of ∼ 8 [175] in Newtonian gravity. From this perspective, the remaining factor of two in MOND seems modest. Rich clusters of galaxies are rare objects, so the total required mass density can readily be accommodated within the baryon budget of BBN. Indeed, according to BBN, there must still be a lot of unidentified baryons lurking somewhere in the universe. But the excess dark mass in clusters need not be baryonic, even in MOND. Massive ordinary neutrinos [389, 392] and light sterile neutrinos [9Jump To The Next Citation Point, 13] have been suggested as possible forms of dark matter that might provide an explanation for the missing mass in clusters. Both are non-baryonic, but as they are hot DM particle candidates, neither can constitute the cosmological non-baryonic cold dark matter. At this juncture, all we can say for certain is that we do not know what the composition of the unseen mass is. It could even just be evidence for the effect of additional “dark fields” in the parent relativistic formulation of MOND, such as massive scalar fields, vector fields, dipolar dark matter, or even subtle non-local effects (see Section 7).

There are other aspects of cluster observations that are more in line with MOND’s predictions. Clusters obey a mass–temperature relation that parallels the M ∝ T2 ∝ V 4 prediction of MOND (Figures 39View Image and 48View Image) more closely than the conventional prediction of M ∝ T 3∕2 expectation in ΛCDM, without the need to invoke preheating (a need that may arise as an artifact of the mismatch in slopes). Indeed, Figure 48View Image shows clearly both the failing of MOND in the offset in characteristic acceleration between clusters and lower mass systems, and its successful prediction of the slope (a horizontal line in this figure). A further test, which may be important is the peculiar and bulk velocity of clusters. For example, the collision velocity of the bullet cluster is so large71 as to be highly improbable in ΛCDM (occurring with a probability of ∼ 10 −10 [249]). In contrast, large collision velocities are natural to MOND [16Jump To The Next Citation Point]. Similarly, the large scale peculiar velocity of clusters is observed to be ∼ 1000 km ∕s [221], well in excess of the expected ∼ 200 km s−2. Ongoing simulations with MOND [11Jump To The Next Citation Point] show some promise to produce large peculiar velocities for clusters. In general, one would expect high speed collisions to be more ubiquitous in MOND than ΛCDM.

An important line of evidence for mass discrepancies in the universe is gravitational lensing in excess of that expected from the observed mass of lens systems. Lensing is an intrinsically relativistic effect that requires a generally covariant theory to properly address. This necessarily goes beyond MOND itself into specific hypotheses for its parent theory (Section 7), so is somewhat different than the tests discussed above. Broadly speaking, tests involving strong gravitational lensing fare tolerably well (Section 8.1), whereas weak lensing tests, that are sensitive to larger-scale mass distributions, are more problematic (Sections 8.2, 8.3, and 8.4) or simply crash into the usual missing mass problem of MOND in clusters. Note that weak lensing in relativistic MOND theories produces the same amount of lensing as required from dynamics, so this is not the problem. The problematic fact is just that some tests seem to require more dark matter than the effect of MOND provides.

On larger (cosmological) scales, MOND, as a modification of classical (non-covariant) dynamics, is simply unsatisfactory or mute. MOND itself has no cosmology, providing analogs for neither the Friedmann equation for the dynamics of the universe, nor the Robertson–Walker metric for its geometry. For these, one must appeal to specific hypotheses for the relativistic parent theory of MOND (Section 7), which is far from unique, and theoretically not really satisfactory, as none of the present candidates emerges from first principles. At this juncture, it is not clear whether a compelling candidate cosmology will ever emerge. But on the other hand, there is nothing about MOND as a paradigm that contradicts per se the empirical pillars of the hot big bang: Hubble expansion, BBN, and the relic radiation field (Section 9). The formation of large scale structure is one of the strengths of conventional theory, which can be approached with linear perturbation theory. This leads to good fits of the power spectrum, both at early times (z ≈ 1000 in the cosmic microwave background) and at late times (the z = 0 galaxy power spectrum [452]). In contrast, the formation of structure in MOND is intrinsically non-linear. Therefore, it is unclear whether MOND-motivated relativistic theories will inevitably match the observed galaxy power spectrum, a possible problem being how to damp the baryon acoustic oscillations [127, 430Jump To The Next Citation Point]. At this stage, a unique prediction does not exist. Nevertheless, there are two aspects of structure formation in MOND that appear to be fairly generic and distinct from ΛCDM. The stronger effective long range force in MOND speeds the growth rate, but has less mass to operate with as a source. Consequently, radiation domination persists longer and structure formation is initially inhibited (at redshifts of hundreds). Once structure begins to form, the non-linearity of MOND causes it to proceed more rapidly than in GR with CDM. Three observable consequences would be (i) the earlier emergence of large objects like galaxies and clusters in the cosmic web (as well as the associated low interaction rate at smaller redshifts) providing a possible solution to challenge no. 2 of Section 4.2 [11], (ii) the more efficient evacuation of large voids (possible solution to challenge no. 3 of Section 4.2), and (iii) larger peculiar (and collisional [16]) velocities of galaxy clusters (solution to challenge no. 1 of Section 4.2). However, the potential downside to rapid structure formation in MOND is that it may overproduce structure by redshift zero [341, 250].

The final entries in Table 2 regard the cosmic microwave background, discussed in more detail in Section 9.2. The third peak of the acoustic power spectrum of the CMB poses perhaps the most severe challenge to a MONDian interpretation of cosmology. The amplitude of the third peak measured by WMAP is larger than expected in a universe composed solely of baryons [442]. This implies some substance that does not oscillate with the baryons. Cold dark matter fits this bill nicely. In the context of MOND, we must invoke some other massive substance (i.e., non-baryonic dark matter such as, e.g., light sterile neutrinos [9Jump To The Next Citation Point]) that plays the role of CDM, or rely on additional degrees of freedom in the relativistic parent theory of MOND (see Section 7) that would have the same net result (see the extensive discussion in Section 9.2), or even combine non-baryonic dark matter with these additional degrees of freedom [430]. While these are real possibilities, neither are particularly appealing, any more than it is to invoke CDM with complex fine-tuned feedback to explain rotation curves that apparently require only baryons as a source.

The missing baryon problem that MOND suffers in rich clusters of galaxies and the third peak of the acoustic power spectrum of the CMB are thus the most serious challenges presently facing MOND. But even so, the interpretation of the acoustic power spectrum is not entirely clear cut. Though there is no detailed fit to the power spectrum in MOND (unless we invoke 10 eV-scale sterile neutrinos [9]), MOND did motivate the prediction [265Jump To The Next Citation Point] of two aspects of the CMB that were surprising in ΛCDM (see Section 9.2). The amplitude ratio of the first-to-second peak in the acoustic power spectrum was outside the bounds expected ahead of time by ΛCDM for Ωb from BBN as it was then known (see Section 9.2). In contrast, the first:second acoustic peak ratio that is now well measured agrees well with the quantitative value predicted in advance for the case of the absence of cold dark matter [268, 269Jump To The Next Citation Point]. Similarly, the rapid formation of structure expected in MOND leads naturally to an earlier epoch of re-ionization than had been anticipated in ΛCDM [265, 269]. Thus, while the amplitude of the third peak is clearly problematic and poses a severe challenge to any MOND-inspired theories, the overall interpretation of the CMB is debatable. While the existence of non-baryonic cold dark matter is the most obvious explanation of the third peak indeed, it is not at all obvious that straightforward CDM – in the form of rather simple massive inert collisionless particles – is uniquely required.

Science is, in principle, about theories or models that are falsifiable, and thus that are presently either falsified or not. But in practice it does not (and cannot) really work that way: if a model that was making good predictions up to a certain point suddenly does not work anymore (i.e., does not fit some new data), one obviously first tries to adjust it to make it fit the observations rather than throwing it away immediately. This is what one calls the requisite “compensatory adjustments” of the theory (or of the model): Popper himself drew attention to these limitations of falsification in The Logic of Scientific Discovery [355]. In the case of the ΛCDM model of cosmology, which is mostly valid on large scales, the current main trend is to find the “compensatory adjustments” to the model to make it fit in galaxies, mainly by changing (or mixing) the mass(es) of the dark matter particles, and/or through artificially fine-tuned baryonic feedback in order to reproduce the success of MOND. Incidentally, exactly the same is true for MOND, but for the opposite scales: MOND works remarkably well in galaxies but apparently needs compensatory adjustments on larger scales to effectively replace CDM. Now does that mean that falsification is impossible? That all models are equal? Surely not. In the end, a theory or a model is really falsified once there are too many compensatory adjustments (needed in order to fit too many discrepant data), or once these become too twisted (like Tycho Brahe’s geocentric model for the solar system). But there is obviously no truly quantitative way of ascertaining such global falsification. How one chooses to weigh the evidence presented in this review necessarily informs one’s opinion of the relative merits of ΛCDM and MOND. If one is most familiar with cosmology and large scale structure, ΛCDM is the obvious choice, and it must seem rather odd that anyone would consider an alternative as peculiar as MOND, needing rather bizarre adjustments to match observations on large scales. But if one is more concerned with precision dynamics and the observed phenomenology in a wide swath of galaxy data, it seems just as strange to invoke non-baryonic cold dark matter together with fine-tuned feedback to explain the appearance of a single effective force law that appears to act with only the observed baryons as a source. Perhaps the most important aspect before one throws away any model is to have a “simpler” model at hand, that still reproduces the successes of the earlier favored model but also naturally explains the discrepant data. In that sense, right now, it is absolutely fair to say that there is no alternative, which really does better overall than ΛCDM, and in favor of which Ockham’s razor would be. However, it would probably be a mistake to persistently ignore the fine-tuning problems for dark matter and the related uncanny successes of the MOND paradigm on galaxy scales, as they could very plausibly point at a hypothetical better new theory. It is also important to bear in mind that MOND, as a paradigm or as a modification of Newtonian dynamics, is not itself generally covariant. Attempts to construct relativistic theories that contain MOND in the appropriate limit (Section 7) are correlated but distinct efforts, and one must be careful not to conflate the two. For example, some theories, like TeVeS (Section 7.4), might make predictions that are distinct from GR in the strong-field regime. Should future tests falsify these distinctive predictions of TeVeS while confirming those of GR, this would perhaps falsify TeVeS as a viable parent theory for MOND, but would have no bearing on the MONDian phenomenology observed in the weak-field regime, nor indeed on the viability of MOND itself. It would perhaps simply indicate the need to continue to search for a deeper theory. It would, for instance, be extremely alluring if one could manage to find a physical connection between the dark energy sector and the possible breakdown of standard dynamics in the weak-field limit, since both phenomena would then simply reflect discrepancies with the predictions of GR when Λ ∼ a20 is set to zero (see, e.g., Section 7.10). Of course, it is perfectly conceivable that such a deep theory does not exist, and that the apparent MONDian behavior of galaxies will be explained through small compensatory adjustments of the current ΛCDM paradigm, but one has yet to demonstrate how this will occur, and it will inevitably involve a substantial amount of fine-tuning that will have to be explained naturally. In any case, the existence of a characteristic acceleration a0 (Figure 48View Image) playing various different roles in many seemingly-independent galactic scaling relations (see Sections 4.3 and 5.2) is by now an empirically established fact, and it is thus mandatory for any successful model of galaxy formation and evolution to explain it. The future of this field of research might thus still be full of exciting surprises for astronomers, cosmologists, and theoretical physicists.

Table 2: Observational tests of MOND.
Observational Test Successful Promising Unclear Problematic
Rotating Systems
solar system X
galaxy rotation curve shapes X
surface brightness ∝ Σ ∝ a2 X
galaxy rotation curve fits X
fitted M*/L X
Tully–Fisher Relation
baryon based X
slope X
normalization X
no size nor Σ dependence X
no intrinsic scatter X
Galaxy Disk Stability
maximum surface density X
spiral structure in LSBGs X
thin & bulgeless disks X
Interacting Galaxies
tidal tail morphology X
dynamical friction X
tidal dwarfs X
Spheroidal Systems
star clusters X
ultrafaint dwarfs X
dwarf Spheroidals X
ellipticals X
Faber–Jackson relation X
Clusters of Galaxies
dynamical mass X
mass–temperature slope X
velocity (bulk & collisional) X
Gravitational Lensing
strong lensing X
weak lensing (clusters & LSS) X
expansion history X
geometry X
Big-Bang nucleosynthesis X
Structure Formation
galaxy power spectrum X
empty voids X
early structure X
Background Radiation
first:second acoustic peak X
second:third acoustic peak X
detailed fit X
early re-ionization X

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