Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions
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Abstract
A wealth of astronomical data indicate the presence of mass discrepancies in the Universe. The motions observed in a variety of classes of extragalactic systems exceed what can be explained by the mass visible in stars and gas. Either (i) there is a vast amount of unseen mass in some novel form — dark matter — or (ii) the data indicate a breakdown of our understanding of dynamics on the relevant scales, or (iii) both. Here, we first review a few outstanding challenges for the dark matter interpretation of mass discrepancies in galaxies, purely based on observations and independently of any alternative theoretical framework. We then show that many of these puzzling observations are predicted by one single relation — Milgrom’s law — involving an acceleration constant a_{0} (or a characteristic surface density Σ_{†} = a_{0}/G) on the order of the squareroot of the cosmological constant in natural units. This relation can at present most easily be interpreted as the effect of a single universal force law resulting from a modification of Newtonian dynamics (MOND) on galactic scales. We exhaustively review the current observational successes and problems of this alternative paradigm at all astrophysical scales, and summarize the various theoretical attempts (TeVeS, GEA, BIMOND, and others) made to effectively embed this modification of Newtonian dynamics within a relativistic theory of gravity.
1 Introduction
Two of the most tantalizing mysteries of modern astrophysics are known as the dark matter and dark energy problems. These problems come from the discrepancies between, on one side, the observations of galactic and extragalactic systems (as well as the observable Universe itself in the case of dark energy) by astronomical means, and on the other side, the predictions of general relativity from the observed amount of matterenergy in these systems. In short, what astronomical observations are telling us is that the dynamics of galactic and extragalactic systems, as well as the expansion of the Universe itself, do not correspond to the observed massenergy as they should if our understanding of gravity is complete. Thus, this indicates either (i) the presence of unseen (and yet unknown) massenergy, or (ii) a failure of our theory of gravity, or (iii) both.
The third case is a priori the most plausible, as there are good reasons for there being more particles than those of the standard model of particle physics [257] (actually, even in the case of baryons, we suspect that a lot of them have not yet been seen and, thus, literally make up unseen mass, in the form of “missing baryons”), and as there is a priori no reason that general relativity should be valid over a wide range of scales, where it has never been tested [45], and where the need for a dark sector actually prevents the theory from being tested until this sector has been detected by other means than gravity itself^{1}. However, either of the first two cases could be the dominant explanation of the discrepancies in a given class of astronomical systems (or even in all astronomical systems), and this is actually testable.
For instance, as far as (ii) is concerned, if the mass discrepancies in a class of systems are mostly caused by some subtle change in gravitational physics, then there should be a clear signature of a single, universal force law at work in this whole class of systems. If instead there is a distinct dark matter component in these, the kinematics of any given system should then depend on the particular distribution of both dark and luminous mass. This distribution would vary from system to system, depending on their environment and past history of formation, and should, in principle, not result in anything like an apparent universal force law^{2}.
Over the years, there have been a large variety of such attempts to alter the theory of gravity in order to remove the need for dark matter and/or dark energy. In the case of dark energy, there is some wiggle room, but in the case of dark matter, most of these alternative gravity attempts fail very quickly, and for a simple reason: once a force law is specified, it must fit all relevant kinematic data in a given class of systems, with the mass distribution specified by the visible matter only. This is a tall order with essentially zero wiggle room: at most one particular force law can work. However, among all these attempts, there is one survivor: the Modified Newtonian Dynamics (MOND) hypothesized by Milgrom almost 30 years ago [294, 295, 293] seems to come close to satisfying the criterion of a universal force law in a whole class of systems, namely galaxies. This success implies a unique relationship between the distribution of baryons and the gravitational field in galaxies and is extremely hard to understand within the present dominant paradigm of the concordance cosmological model, hypothesizing that general relativity is correct on every relevant scale in cosmology including galactic scales, and that the dark sector in galaxies is made of nonbaryonic dissipationless and collisionless particles. Even if such particles are detected directly in the near to far future, the success of MOND on galaxy scales as a phenomenological law, as well as the associated appearance of a universal critical acceleration constant a_{0} ≃ 10^{−10} m s^{−2} in various, seemingly unrelated, aspects of galaxy dynamics, will still have to be explained and understood by any successful model of galaxy formation and evolution. Previous reviews of various aspects of MOND, at an observational and theoretical level, can be found in [34, 81, 100, 151, 279, 311, 318, 401, 407, 429]. A website dedicated to this topic is also maintained, with all the relevant literature as well as introductory level articles [263] (see also [238]).
Here, we first review the basics of the dark matter problem (Section 2) as well as the basic ingredients of the presentday concordance model of cosmology (Section 3). We then point out a few outstanding challenges for this model (Section 4), both from the point of view of unobserved predictions of the model, and from the point of view of unpredicted observations (all uncannily involving a common acceleration constant a_{0}). Up to that point, the challenges presented are purely based on observations, and are fully independent of any alternative theoretical framework^{3}. We then show that, surprisingly, many of these puzzling observations can be summarized within one single empirical law, Milgrom’s law (Section 5), which can be most easily (although not necessarily uniquely) interpreted as the effect of a single universal force law resulting from a modification of Newtonian dynamics (MOND) in the weakacceleration regime a < a_{0}, for which we present the current observational successes and problems (Section 6). We then summarize the various attempts currently made to embed this modification in a generallycovariant relativistic theory of gravity (Section 7) and how such theories allow new predictions on gravitational lensing (Section 8) and cosmology (Section 9). We finally draw conclusions in Section 10.
2 The Missing Mass Problem in a Nutshell
The data leave no doubt that when the law of gravity as currently known is applied to extragalactic systems, it fails if only the observed stars and gas are included as sources in the stressenergy tensor. This leads to a stark choice: either the Universe is pervaded by some unseen form of mass — dark matter — or the dynamical laws that lead to this inference require revision. Though the mass discrepancy problem is now well established [394, 465], such a dramatic assertion warrants a brief review of the evidence.
Historically, the first indications of the modern missing mass problem came in the 1930s shortly after galaxies were recognized to be extragalactic in nature. Oort [342] noted that the sum of the observed stars in the vicinity of the sun fell short of explaining the vertical motions of stars in the disk of the Milky Way. The luminous matter did not provide a sufficient restoring force for the observed stellar vertical oscillations. This became known as the Oort discrepancy. Around the same time, Zwicky [518] reported that the velocity dispersion of galaxies in clusters of galaxies was far too high for these objects to remain bound for a substantial fraction of cosmic time. The Oort discrepancy was approximately a factor of two in amplitude, and confined to the Galactic disk — it required local dark matter, not necessarily the quasispherical halo we now envision. It was long considered a serious problem, but has now largely (though perhaps not fully) gone away [194, 240]. The discrepancy Zwicky reported was less subtle, as the required dark mass outweighed the visible stars by a factor of at least 100. This result was apparently not taken seriously at the time.
One of the first indications of the need for dark matter in modern times came from the stability of galactic disks. Stars in spiral galaxies like the Milky Way are predominantly on approximately circular orbits, with relatively few on highly eccentric orbits [132]. The small velocity dispersion of stars relative to their circular velocities makes galactic disks dynamically cold. Early simulations [343] revealed that cold, selfgravitating disks were subject to severe instabilities. In order to prevent the rapid, selfdestructive growth of these instabilities, and hence preserve the existence of spiral galaxies over a sizable fraction of a Hubble time, it was found to be necessary to embed the disk in a quasispherical potential well — a role that could be played by a halo of dark matter, as first proposed in 1973 by Ostriker & Peebles [343].
Perhaps the most persuasive piece of evidence was then provided, notably through the seminal works of Bosma and Rubin, by establishing that the rotation curves of spiral galaxies are approximately flat [67, 370]. A system obeying Newton’s law of gravity should have a rotation curve that, like the Solar system, declines in a Keplerian manner once the bulk of the mass is enclosed: V_{ c } ∝ r^{−1/2}. Instead, observations indicated that spiral galaxy rotation curves tended to remain approximately flat with increasing radius: V_{ c } ∼ constant. This was shown to happen over and over and over again [370] with the approximate flatness of the rotation curve persisting to the largest radii observable [67], well beyond where the details of each galaxy’s mass distribution mattered, so that Keplerian behavior should have been observed. Again, a quasispherical halo of dark matter as proposed by Ostriker and Peebles was implicated.
Other types of galaxies exhibit mass discrepancies as well. Perhaps most notable are the dwarf spheroidal galaxies that are satellites of the Milky Way [427, 477] and of Andromeda [217]. These satellites are tiny by galaxy standards, possessing only millions, or in the case of the ultrafaint dwarfs, thousands, of individual stars. They are close enough that the lineofsight velocities of individual stars can be measured, providing for a precise measurement of the system’s velocity dispersion. The mass inferred from these motions (roughly, M ∼ rσ^{2}/G) greatly exceeds the mass visible in luminous stars. Indeed, these dim satellite galaxies exhibit some of the largest mass discrepancies observed. In contrast, bright giant elliptical galaxies (often composed of much more than the ∼ 10^{11} stars of the Milky Way) exhibit remarkably modest and hard to detect mass discrepancies [367]. Thus, it is inferred that fainter galaxies are progressively more darkmatter dominated than bright ones. However, as we shall expand on in Section 4.3, the primary correlation is not with luminosity, but with surface brightness: the lower the surface brightness of a system, the larger its mass discrepancy [279].
On larger scales, groups and clusters of galaxies also show mass discrepancies, just as individual galaxies do. One of the earliest lines of evidence comes from the “timing argument” in the Local Group [213]. Presumably the material that was to become the Milky Way and Andromeda (M31) was initially expanding apart with the general Hubble expansion. Currently they are approaching one another at ∼ 100 km s^{−1}. In order for the Milky Way and M31 to have overcome the initial expansion and fallen back towards one another, there must be a greaterthanaverage gravitating mass between the two. To arrive at their present separation with the observed blueshifted line of sight velocity after a Hubble time requires a dynamical masstolight ratio M/L > 80. This greatly exceeds the masstolight ratio of the stars themselves, which is of order unity in Solar units [42] (the Sun is a fairly average star, so averaged over many stars each Solar mass produces roughly one Solar luminosity).
Rich clusters of galaxies are rare structures containing dozens or even hundreds of bright galaxies. These objects exhibit mass discrepancies in several distinct ways. Measurements of the redshifts of individual cluster members give velocity dispersions in the vicinity of 1,000 km s^{−1} typically implying dynamical masstolight ratios in excess of 100 [24]. The actual mass discrepancy is not this large, as most of the detected baryonic mass in clusters is in a diffuse intracluster gas rather than in the stars in the galaxies (something Zwicky was not aware of back in 1933). This gas is heated to the virial temperature and emits Xrays. Mapping the temperature and emission of this Xray gas provides another probe of the cluster mass through the equation of hydrostatic equilibrium. In order to hold the gas in the clusters at the observed temperatures, the dark matter must outweigh the gas by a factor of ∼ 8 [175]. Furthermore, some clusters are observed to gravitationally lens background galaxies (Figure 1). Once again, mass above and beyond that observed is required to explain this phenomenon [227]. Thus, three independent methods all imply the need for about the same amount of dark matter in clusters of galaxies.
In addition to the abundant evidence for mass discrepancies in the dynamics of extragalactic systems, there are also strong motivations for dark matter in cosmology. Two observations are particularly important: (i) the small baryonic mass density Ω_{ b } inferred from BigBang nucleosynthesis (BBN) (and from the measured Hubble parameter), and (ii) the growth of large scale structure by a factor of ∼ 10^{5} from the surface of last scattering of the cosmic microwave background at redshift z ∼ 1000 until presentday z = 0, implying Ω_{m} > Ω_{ b }. Together, these observations imply not only the need for dark matter, but for some exotic new form of nonbaryonic cold dark matter. Indeed, observational estimates of the gravitating mass density of the Universe Ω_{m}, measured, for instance, from peculiar galaxy (or largescale) velocity fields, have, for several decades, persistently returned values in the range 1/4 < Ω_{m} < 1/3 [116]. While shy of the value needed for a flat Universe, this mass density is well in excess of the baryon density inferred from BBN. The observed abundances of the light isotopes deuterium, helium, and lithium are consistent with having been produced in the first few minutes after the Big Bang if the baryon density is just a few percent of the critical value: Ω_{ b } < 0.05 [480, 107]. Thus, Ω_{ m } > Ω_{ b }. Consequently, we do not just need dark matter, we need the dark matter to be nonbaryonic.
Another early Universe constraint is provided by the Cosmic Microwave Background (CMB). The small (microKelvin) amplitude of the temperature fluctuations at the time of baryonphoton decoupling (z ∼ 1000) indicates that the Universe was initially very homogeneous, roughly to one part in 10^{5}. The Universe today (z = 0) is very inhomogeneous, at least on “small” scales of less than ∼ 100 Mpc (∼ 3 × 10^{8} ly), with huge density contrasts between planets, stars, galaxies, clusters, and empty intergalactic space. The only attractive longrange force acting on the entire Universe, that can make such structures, is gravity. In a richgetricher while the poorgetpoorer process, the small initial overdensities attract more mass and grow into structures like galaxies while underdense regions become less dense, leading to voids. The catch is that gravity is rather weak, so this process takes a long time. If the baryon density from BBN is all we have to work with, we can only obtain a growth factor of ∼ 10^{2} in a Hubble time [424], orders of magnitude short of the observed 10^{5}. The solution is to boost the growth rate with extra invisible mass displaying larger density fluctuations: dark matter. In order not to make the same mark on the CMB that baryons would, this dark matter must not interact with the photons. So, in effect, the density fluctuations in the dark matter can already be very large at the epoch of baryonphoton decoupling, especially if the dark matter is cold (i.e., with effectively zero Jeans length). The baryons fall into the already deep dark matter potential wells only after that, once released from their electromagnetic link to the photon bath. Before decoupling, the fluctuations in the baryonphoton fluid did not grow but were oscillating in the form of acoustic waves, maintaining the same amplitude as when they entered the horizon; actually they were even slightly diffusiondamped. In principle, at baryonphoton decoupling, CMB fluctuations on smaller angular scales, having entered the horizon earlier, would have been damped with respect to those on larger scales (Silk damping). Nevertheless, the presence of decoupled nonbaryonic dark matter would provide a net forcing term countering the damping of the oscillations at recombination, meaning that the second and third acoustic peaks of the CMB could then be of equal amplitude rather than exhibiting a damping tail. The actual observation of a high thirdpeak in the CMB angular power spectrum is another piece of compelling evidence for nonbaryonic dark matter (see, e.g., [229]). Both BBN and the CMB thus drive us to consider a form of mass that is nonbaryonic and which does not interact electromagnetically. Moreover, in order to form structure (see Section 3.2), the mass must be dynamically cold (i.e., moving much slower than the speed of light when it decouples from the photon bath), and is known as cold dark matter (CDM).
Now, in addition to CDM, modern cosmology also requires something even more mysterious, dubbed dark energy. The fact that the baryon fraction in clusters of galaxies was such that Ω_{ m } was implied to be much smaller than 1 — the value needed for a flat Euclidean Universe favored by inflationary models —, as well as tensions between the measured Hubble parameter and independent estimates of the age of the Universe, led Ostriker & Steinhardt [344] to propose in 1995 a “concordance model of cosmology” or ΛCDM model, where a cosmological constant Λ — supposed to represent vacuum energy or dark energy — provided the major contribution to the Universe’s energy density. Three years later, the observations of SNIa [351, 365] indicating latetime acceleration of the Universe’s expansion, led most people to accept this model. This concordance model has since been refined and calibrated through subsequent largescale observations of the CMB and of the matter power spectrum, to lead to the favored cosmological model prevailing today (see Section 3). However, as we shall see, curious coincidences of scales between the dark matter and dark energy sectors (see Section 4.1) have prompted the question of whether these two sectors are really physically independent, and the existence of dark energy itself has led to a renewed interest in modified gravity theories as a possible alternative to this exotic fluid [100].
3 A Brief Overview of the ΛCDM Cosmological Model
General relativity provides a clear and compelling cosmology, the FriedmannLemaîtreRobertsonWalker (FLRW) model. The expansion of the Universe discovered by Hubble and Slipher found a natural explanation^{4} in this context. The picture of a hot BigBang cosmology that emerged from this model famously predicted the existence of the 3 degree CMB and the abundances of the light isotopes via BBN.
Within the FLRW framework, we are inexorably driven to infer the existence of both nonbaryonic cold dark matter and a nonzero cosmological constant as discussed in Section 2. The resulting concordance ΛCDM model — first proposed in 1995 by Ostriker and Steinhardt [344] — is encouraged by a wealth of observations: the consistency of the Hubble parameter with the ages of the oldest stars [344], the consistency between the dynamical mass density of the Universe, that of baryons from BBN (see also discussion in Section 9.2), and the baryon fraction of clusters [486], as well as the power spectrum of density perturbations [103, 452]. A prediction of the concordance model is that the expansion rate of the Universe should be accelerating; this was confirmed by observations of high redshift Type Ia supernovae [351, 365]. Another successful prediction was the scale of the baryonic acoustic oscillation [134]. Perhaps the most emphatic support for ACDM comes from fits to the acoustic power spectrum of temperature fluctuations in the CMB [229].
For a brief review of the basics and successes of the concordance cosmological model we refer the reader to, e.g., [87, 349] and all references therein. We note that, while most of the cosmological probes in the above list are not uniquely fit by the ΛCDM model on their own, when they are taken together they provide a remarkably tight set of constraints. The success of this now favoured cosmological model on large scales is, thus, remarkable indeed, as there was a priori no reason that such a parameterized cosmology could explain all these completely independent data sets with such outstanding consistency.
In this model, the Hubble constant is H_{0} = 70 km s^{−1} Mpc^{−1} (i.e., h = 0.7), the amplitude of density fluctuations within a tophat sphere of 8h^{−1} Mpc is σ_{8} = 0.8, the optical depth to reionization is τ = 0.08, the spectral index measuring how fluctuations change with scale is n_{ s } =0. 97, and the price we pay for the outstanding success of the model is new physics in the form of a dark sector. This dark sector is making up 95% of the massenergy content of the Universe in ΛCDM: it is composed separately of a dark energy sector and a cold dark matter sector, which we briefly describe below.
3.1 Dark Energy (Λ)
In ΛCDM, dark energy is a nonvanishing vacuum energy represented by the cosmological constant Λ in the field equations of general relativity. Einstein’s cosmological constant is equivalent to vacuum energy with equation of state p/ρ = w = −1. In principle, the equation of state could be merely close to, but not exactly w = −1. In this case, the dark energy could evolve and clump, depending on the value of w and its evolution ẇ. However, to date, there is no compelling observational reason to require any form of dark energy more complex than the simple cosmological constant introduced by Einstein.
The various observational datasets discussed above constrain the ratio of the dark energy density to the critical density to be \({\Omega _\Lambda} = \Lambda/3H_0^2 = 0.73\) where H_{0} is Hubble’s constant and ι is expressed in s^{−2}. This value, together with the matter density Ω_{ m } (see below), leads to a total Ω = Ω_{ι}+Ω_{ m } = 1, i.e., a spatiallyflat Euclidean geometry in the RobertsonWalker sense that is nicely consistent with the expectations of inflation. It is important to stress that this model relies on the cosmological principle, i.e., that our observational location in the Universe is not special, and on the fact that on large scales, the Universe is isotropic and homogeneous. For possible challenges to these assumptions and their consequences, we refer the reader to, e.g., [83, 487, 488].
3.2 Cold Dark Matter (CDM)
In ΛCDM, dark matter is assumed to be made of nonbaryonic dissipationless massive particles [48], the “cold dark matter” (CDM). This dark matter outweighs the baryons that participate in BBN by about 5:1. The density of baryons from the CMB is Ω_{ b } = 0.046, grossly consistent with BBN [229]. This is a small fraction of the critical density; with the nonbaryonic dark matter the total matter density is Ω_{ m } = Ωcdm + Ω_{ b } = 0.27.
The “cold” in cold dark matter means that CDM moves slowly so that it is nonrelativistic when it decouples from photons. This allows it to condense and begin to form structure, while the baryons are still electromagnetically coupled to the photon fluid. After recombination, when protons and electrons first combine to form neutral atoms so that the crosssection for interaction with the photon bath suddenly drops, the baryons can fall into the potential wells already established by the dark matter, leading to a hierarchical scenario of structure formation with the repeated merger of smaller CDM clumps to form ever larger clumps.
Particle candidates for the CDM must be massive, nonbaryonic, and immune to electromagnetic interactions. The currently preferred CDM candidates are Weakly Interacting Massive Particles (WIMPs, [46, 47, 48]) that condensed from the thermal bath of the early Universe. These should have masses on the order of about 100 GeV so that (i) the freestreaming length is small enough to create smallscale structures as observed (e.g., dwarf galaxies), and (ii) that thermal relics with crosssections typical for weak nuclear reactions account for the right amount of matter density Ω_{ m } (see, e.g., Eq. 28 of [48]). This last point is known as the WIMP miracle^{5}.
For lighter particle candidates (e.g., ordinary neutrinos or light sterile neutrinos), the damping scale becomes too large. For instance, a hot dark matter (HDM) particle candidate with mass of a few to 15 eV would have a freestreaming length of about ∼ 100 Mpc, leading to too little power at the smallscale end of the matter power spectrum. The existence of galaxies at redshift z ∼ 6 implies that the coherence length should have been smaller than 100 kpc or so, meaning that even warm dark matter (WDM) particles with masses between 1 and 10 keV are close to being ruled out as well (see, e.g., [348]). Thus, ΛCDM presently remains the stateoftheart in cosmology, although some of the challenges listed in Section 4 are leading to a slow drift of the standard concordance model from CDM to WDM [252], but this drift brings along its own problems, and fails to address most of the current observational challenges summarized in the following Section 4, which might perhaps point to a more radical alternative to the model.
4 Some Challenges for the ΛCDM Model
The great concordance of independent cosmological observables from Gpc to Mpc scales lends a certain air of inevitability to the ΛCDM model. If we accept these observables as sufficient to prove the model, then any discrepancy appears as trivia that will inevitably be explained away. If instead we require a higher standard, such as positive laboratory evidence for the dark sectors, then ΛCDM appears as a yet unproven hypothesis that relies heavily on two potentially fictitious invisible entities. Thus, an important test of ΛCDM as a scientific hypothesis is the existence of dark matter. By this we mean not just unseen mass, but specifically CDM: some novel form of particle with the right microscopic properties and correct cosmic mass density. Searches for WIMPs are now rather mature and not particularly encouraging. Direct detection experiments have as yet no positive detections, and have now excluded [19] the bulk of the parameter space (interaction crosssection and particle mass) where WIMPs were expected to reside. Indirect detection through the observation of γrays produced by the selfannihilation^{6} of WIMPs in the galactic halo and in nearby satellite galaxies have similarly returned null results [6, 84, 172] at interestingly restrictive levels. For the mostplausible minimallysupersymmetric models, particle colliders should already have produced evidence for WIMPs [2, 1, 23]. The right model need not be minimal. It is always possible to construct a more complicated model that manages to evade all experimental constraints. Indeed, it is readily possible to imagine dark matter candidates that do not interact at all with the rest of the Universe except through gravity. Though logically possible, such dark matter candidates are profoundly unsatisfactory in that they could not be detected in the laboratory: their hypothesized existence could neither be confirmed nor falsified.
Apart from this current nondetection of CDM candidates, there also exists prominent observational challenges for the ΛCDM model, which might point towards the necessity of an alternative model (or, at the very least, an improved one). These challenges are that (i) some of the parameters of the model appear finetuned (Section 4.1), and that (ii) as far as galaxy formation and evolution are concerned (mainly processes happening on kpc scales so that the predictions are more difficult to make because the baryon physics should play a more prominent role), many predictions that have been made were not successful (Section 4.2); (iii) what is more, a number of observations on these galactic scales do exhibit regularities that are fully unexpected in any CDM context without a substantial amount of finetuning in terms of baryon feedback (Section 4.3).
4.1 Coincidences
What is generally considered as the biggest problem for the ΛCDM model is that it requires a large and still unexplained finetuning to reduce by 120 orders of magnitude the theoretical expectation of the vacuum energy to yield the observed cosmologicalconstant value, and, even more importantly, that it faces a coincidence problem to explain why the dark energy density Ω_{Λ} is precisely of the same order of magnitude as the other cosmological components today^{7}. This uncanny coincidence is generally seen as evidence for some yettobediscovered underlying cosmological mechanism ruling the evolution of dark energy (such as quintessence or generalized additional fluid components, see, e.g., [106]). But it could also indicate that the effect attributed to dark energy is rather due to a breakdown of general relativity (GR) on the largest scales [158].
Finally, let us note that the existence of the a_{0}scale is actually not the only darkmatterrelated coincidence, as there is also, in principle, absolutely no reason why the mechanism leading to the baryon asymmetry (between baryonic matter and antimatter) would simultaneously leave both the baryon and dark matter densities with a similar order of magnitude (Ω_{dm}/Ω_{ b } = 5). If the effects we attribute to dark matter are actually also due to a breakdown of GR on cosmological scales, then such a coincidence might perhaps appear more natural as the baryons would then be the actual source of the effect attributed to the dark matter sector.
4.2 Unobserved predictions
 1.
The bulk flow challenge. Peculiar velocities of galaxy clusters are predicted to be on the order of 200 km/s in the ΛCDM model. These can actually be measured by studying the fluctuations in the CMB generated by the scattering of the CMB photons by the hot Xrayemitting gas inside clusters (the kinematic SZ effect). This yields an observed coherent bulk flow of order 1000 km/s (5 times more than predicted) on scales out to at least 400 Mpc [221]. This bulk flow challenge appears not only in SZ studies but also in galaxy studies [483]. A related problem is the collision velocity larger than 3100 km/s for the merging bullet cluster 1E065756 at z = 0.3, much too high to be accounted for by ΛCDM [249, 455]. These observations would seem to indicate that the attractive force between DM particles is enhanced compared to what ΛCDM predicts, and changing CDM into WDM would not solve the problem.
 2.
The highz clusters challenge. Observation of even a single massive cluster at high redshift can falsify ΛCDM [331]. In this respect the existence of the galaxy cluster XMMU J2235.32557 [368] with a mass of of ∼ 4 × 10^{14} M_{⊙} at z = 1.4, even though not sufficient to rule out the model, is very surprising and could indicate that structure formation is actually taking place earlier and faster than in ΛCDM (see also [420] on the Shapley supercluster and the Sloan Great Wall).
 3.
The Local Void challenge. The Local Volume is composed of 562 known galaxies at distances smaller than 8 Mpc from the center of the Local Group, and the region known as the “Local Void” hosts only 3 of them. This is much less than the expected ∼ 20 for a typical similar void in ΛCDM [350]. What is more, in the Local Volume, large luminous galaxies are overrepresented by a factor of 6 in the underdense regions, exactly opposite to what is expected from ΛCDM. This could mean that the Local Volume is just a statistical anomaly, but it could also point, in line with the two previous challenges, towards more rapid structure formation, allowing sparse regions to more quickly form large galaxies cleaning their environment, making the galaxies larger and the voids emptier at early times [350].
 4.
The missing satellites challenge. It has long been known that the model predicts an overabundance of dark subhalos orbiting MilkyWaysized galaxies compared to the observed number of satellite galaxies around the Milky Way [329]. This is a different problem from the abovepredicted overabundance of small galaxies in voids. It has subsequently been suggested that stellar feedback and heating processes limit baryonic growth, that reionisation prevents lowmass dark halos from forming stars, and that tidal forces from the host halo limit growth of the darkmatter subhalos and lead to their truncation. This important theoretical effort has led recent semianalytic models to predict a reduced number of ∼ 100 to 600 faint satellites rather than the original thousands. Moreover, during the past 15 years 13 “new” and mostly ultrafaint satellite galaxies have been found in addition to the 11 previouslyknown classical bright ones. Since these new galaxies have been largely discovered with the Sloan Digital Sky Survey (SDSS), and since this survey covered only one fifth of the sky, it has been argued that the problem was solved. However, there are actually still missing satellites on the low mass and high mass end of the mass function predicted by “ΛCDM+reinoisation” semianalytic models. This is best illustrated on Figure 2 of [239] showing the cumulative distribution for the predicted and observationallyderived masses within the central 300 pc of Milky Way satellites. A lot of lowmass satellites are still missing, and the most massive predicted subhaloes are also incompatible with hosting any of the known Milky Way satellites [73, 75, 74]. This is the modern version of the missing satellites challenge. An obvious but rather discomforting wayout would be to simply state that the Milky Way must be a statistical outlier, but this is contradicted by the study of [447] on the abundance of bright satellites around Milky Waylike galaxies in SDSS. Another solution would be to change from CDM to WDM [252] (it is actually one of the only listed challenges that such a change would probably immediately solve).
 5.
The satellites phasespace correlation challenge. In addition to the above challenge, the distribution of dark subhalos around the Galaxy is also predicted by ΛCDM to be isotropic, or quasiisotropic. However, the Milky Way satellites are currently observed to be correlated in phasespace: they lie within a seemingly rotationsupported disk [239]. Young halo globular clusters define the same disk, and streams of stars and gas, tracing the orbits of the objects from which they are stripped, preferentially lie in this disk, too [347]. Since SDSS covered only one fifth of the sky, it will be interesting to see whether future surveys such as PanStarrs will confirm this state of affairs. Whether or not this phasespace correlation would be unique to the Milky Way should also be carefully checked, the evidence in M31 being currently much less convincing, with a richer and more complex satellite population [289]. But in any case, the current distribution of satellites around the Milky Way is statistically incompatible with the predictions of ΛCDM at a very high level of confidence, even when taking into account the observational bias from SDSS [239]. While this might perhaps have been explained by the infall of a small group of galaxies that would have retained correlated orbits, this solution is ruled out by the fact that no nearby groups are observed to be anywhere near as spatially small as the disk of satellites [290]. Another solution might be that most Milky Way satellites are actually not primordial galaxies but old tidal dwarf galaxies created in an early major merger event, accounting for their presentlycorrelated phasespace distribution [346]. Note in passing that if only one or two longlived tidal dwarfs are created in each gasdissipational galaxy encounter, they could probably account for most of the dwarf galaxy population in the Universe, leaving no room for small CDM subhalos to create galaxies, which would transform the missing satellites challenge into a missing satellites catastrophe [239].
 6.
The cuspcore challenge. Another longstanding problem of ΛCDM is the fact that the simulations of the collapse of CDM halos lead to a density distribution as a function of radius, ρ(r), which is well fitted by a smooth function asymptoting to a central cusp with slope d ln ρ/d ln r = −1 in the central parts [126, 332], while observations clearly point towards large constant density cores in the central parts [118, 169, 479]. Even though the latest simulations [333] rather point towards Einasto [133] profiles with d ln ρ/d ln r ∝ − r^{(1/n)} (with n slightly varying with halo mass, and n ∼ 6 for a Milky Waysized halo, meaning that the slope is zero only very close to the nucleus [177], and is still ∼ −1 at 200 pc from the center), fitting such profiles to observed galactic kinematical data such as rotation curves [88] leads to values of n that are much smaller than simulated values (meaning that they have much larger cores), which is another way of reassessing the old cusp problem of ΛCDM. Note that a change from CDM to WDM could solve the problem in dwarf galaxies, by leading to the formation of small cores, but certainly not in large galaxies where large cores are needed from observations. Thus, one has to rely on baryon feedback to erase the cusp from all galaxies. But this is not easily done, as the adiabatic cooling of baryons in the center of dark matter halos should lead to an even more concentrated dark matter distribution. A possibility would be that angular momentum transfer from a rotating stellar bar destroys darkmatter cusps: however, significant cusp destruction requires substantially more angular momentum than is realistically available in stellar bars [89, 286]. Note also that not all galaxies are barred (e.g., M33 is not). The stateoftheart solution nowadays is to enforce strong supernovae outflows that move large amounts of lowangularmomentum gas from the central parts and that “pull” on the central dark matter concentration to create a core [176], but this is still a highly finetuned process, which fails to address the baryon fraction problem (see challenge 10 below).
 7.
The angular momentum challenge. As a consequence of the merger history of galaxy disks in a hierarchical formation scenario, as well as of the associated transfer of angular momentum from the baryonic disk to the dark halo, the specific angular momentum of the baryons ends up being much too small in simulated disks, which in turn end up much smaller than the observed ones [4]. Similarly, elliptical systems end up too concentrated as well. Addressing this challenge within the standard paradigm essentially relies on forming disks through latetime quiescent gas accretion from largescale filaments, with much less latetime mergers than presently predicted in ΛCDM.
 8.
The pure disk challenge. Related to the previous challenge, large bulgeless thin disk galaxies are extremely difficult to produce in simulations. This is because major mergers, at any time in the galaxy formation process, typically create bulges, so bulgeless galaxies would represent the quiescent tail of a distribution of merger histories for galaxies of the Local Volume. However, these bulgeless disk galaxies represent more than half of large galaxies (with V_{ c } > 150 km/s) in the Local Volume [178, 231]. Solving this problem would rely, e.g., on suppressing central spheroid formation for mergers with mass ratios lower than 30% [228].
 9.
The stability challenge. Round CDM halos tend to stabilize very low surface density disks against the formation of bars and spirals, due to a lack of disk selfgravity [291]. The observation [282] of Low Surface Brightness (LSB) disk galaxies with strong bars and spirals is thus challenging in the absence of a significant disk component of dark matter. What is more, in the absence of such a disk DM component, the lack of disk selfgravity prevents the creation of verylarge razorthin LSB disks, but these are observed [222, 260]. In the standard context, these observations would tend to point towards an additional disk DM component, either a CDMone linked to inplane accretion of satellites or a baryonic one in the form of molecular gas.
 10.
The missing baryons challenge(s). As mentioned above, constraints from the CMB imply Ω_{ m } = 0.27 and Ω_{ b } = 0.046. However, our inventory of known baryons in the local Universe, summing over all observed stars, gas, etc., comes up short of the total. For example, [42] estimate that the sum of stars and cold gas is only ∼ 5% of Ω_{ b } While there now seems to be a good chance that many of the missing baryons are in the form of highly ionized gas in the warmhot intergalactic medium (WHIM), we are still far from being able to give a confident account of where all the baryons reside. Indeed, there could be multiple distinct reservoirs in addition to the WHIM, each comparable to the mass in stars, within the current uncertainties. But there is another missing baryons challenge, namely the halobyhalo missing baryons. Indeed, each CDM halo can, to a first approximation, be thought of as a microcosm of the whole. As such, one would naively expect each halo to have the same baryon fraction as the whole Universe, f_{ b } = Ω_{ b }/Ω_{ m } = 0.17. On the scale of clusters of galaxies, this is approximately true (but still systematically low), but for individual galaxies, observations depart from this in a systematic way which we have yet to understand, and which has nothing to do with the truncation radius. The ratio of the galaxydetected baryon fraction over the cosmological one, fd, is plotted as a function of the potential well of the systems in Figure 2 [284]. There is a clear correlation, less massive objects being much more darkmatter dominated than massive ones. This correlation is a priori not predicted at all by ΛCDM, at least not with the correct shape [273]. This missing baryons challenge is actually closely related to the baryonic TullyFisher relation, which we expand on in Section 4.3.1.
However, let us note that, while challenges 1 to 3 are not real smoking guns yet for the ΛCDM model, challenges 4 to 10 are concerned with processes happening on kpc scales, for which it is fair to consider that the model is not very predictive because the baryon physics should play a more important role, and this is hard to take into account rigorously. However, it is not sufficient to qualitatively invoke handwavy baryon physics to avoid confronting predictions of ΛCDM with observations. It is also mandatory to show that the feedback from the baryons, which is needed to solve the observational problems, is what would quantitatively happen in a physical galaxy. This, presently, is not yet the case for the aforementioned challenges. However, these challenges are “modeldependent problems”, in the sense of being failed predictions of a given model, but would not have appeared a priori surprising without the standard concordance model at hand. This means that subtly changing some parameters of the model (like, e.g., swapping CDM for WDM, making DM more selfinteracting, etc.) might help solving at least a few of them. But what is even more challenging is a set of observations that appear surprising independently of any specific dark matter model, as they involve a finetuned relation between the distribution of visible and dark matter. These are what we call hereafter “unpredicted observations”.
4.3 Unpredicted observations
There are several important examples of systematic relations between the dynamics of galaxies (in theory presumed to be dominated by dark matter) and their baryonic content. These relations are fully empirical, and as such must be explained by any viable theory. As we shall see, they inevitably involve a critical acceleration scale, or equivalently, a critical surface density of baryonic matter.
4.3.1 Baryonic TullyFisher relation
One of the strongest correlations in extragalactic astronomy is the TullyFisher relation [467]. Originally identified as an empirical relation between a galaxy’s luminosity and its HI linewidth, it has been widely employed as a distance indicator. Though extensively studied for decades, the physical basis of the relation remains unclear.
Luminosity and linewidth are readily accessible observational quantities. The optical luminosity of a galaxy is a proxy for its stellar mass, and the HI linewidth is a proxy for its rotation velocity. The quality of the correlation improves as more accurate indicators of these quantities are employed. For example, resolved rotation curves, where the flat portion of the rotation curve V_{ f } or the maximum peak velocity V_{ p } can be measured, give relations that are tighter than those utilizing only linewidth information [108]. Similarly, the scatter declines as we shift from optical luminosities to those in the nearinfrared [475] as the latter are expected to give a more reliable mapping of starlight to stellar mass [42].
A BTFR of the observed form does not arise naturally in ΛCDM. The naive expectation is \(\alpha = 3\) and \(\beta = 10f_V^3G{H_0}\) [446]^{11} where H_{0} is the Hubble constant and f_{ V } is a factor of order unity (currently estimated to be ≈ 1.3 [361]) that relates the observed V_{ f } to the circular velocity of the potential at the virial radius^{12}. This modest fudge factor is necessary because ΛCDM does not explicitly predict either axis of the observed BTFR. Rather, there is a relationship between total (baryonic plus dark) mass and rotation velocity at very large radii. This simple scaling fails (dashed line in Figure 3), obliging us to introduce an additional fudge factor f_{ d } [273, 284] that relates the detected baryonic mass to the total mass of baryons available in a halo. This mismatch drives the variation in the detected baryon fraction f_{ d } seen in Figure 2. A constant f_{ d } is excluded by the difference between the observed and predicted slopes; f_{ d } must vary with V_{ f }, or M, or the gravitational potential Φ
This brings us to the first finetuning problem posed by the data. There is essentially zero intrinsic scatter in the BTFR [276], while the detected baryon fraction f_{ d } could, in principle, obtain any value between zero and unity. Somehow galaxies must “know” what the circular velocity of the halo they reside in is so that they can make observable the correct fraction of baryons.
One might be able to avoid finetuning if all galaxies are darkmatter dominated [109]. In the limit Σ _{ dm } ≫_{b}, the dynamics are entirely darkmatter dominated and the distribution of the baryons is irrelevant. There is some systematic uncertainty in the masstolight ratios of stellar populations [42], making such an approach a priori tenable. In effect, we return to the interpretation of Σ ∼ constant originally made by [3] in the context of Freeman’s Law, but now we invoke a constant surface density of CDM rather than of baryons. But as we will see, such an interpretation, i.e., that Σ_{b} ≪Σ_{ dm } in all disk galaxies, is flatly contradicted by other observations (e.g., Figure 9 and Figure 13).
The TullyFisher relation is remarkably persistent. Originally posited for bright spirals, it applies to galaxies that one would naively expected to deviate from it. This includes lowluminosity, gasdominated irregular galaxies [445, 462, 276], LSB galaxies of all luminosities [517, 443], and even tidal dwarfs formed in the collision of larger galaxies [165]. Such tidal dwarfs may be especially important in this context (see also Section 6.5.4). Galactic collisions should be very effective at segregating dark and baryonic matter. The rotating gas disks of galaxies that provide the fodder for tidal tails and the tidal dwarfs that form within them initially have nearly circular, coplanar orbits. In contrast, the darkmatter particles are on predominantly radial orbits in a quasispherical distribution. This difference in phase space leads to tidal tails that themselves contain very little dark matter [72]. When tidal dwarfs form from tidal debris, they should be largely devoid^{13} of dark matter. Nevertheless, tidal dwarfs do appear to contain dark matter [72] and obey the BTFR [165].
The acceleration scale a_{0} is clearly imprinted on the data for local galaxies. This is an empirical statement that might not hold at all times, perhaps evolving over cosmic time or evaporating altogether. Substantial efforts have been made to investigate the TullyFisher relation to high redshift. To date, there is no persuasive evidence of evolution in the zero point of the BTFR out to z = 0.6 [356, 357] and perhaps even to z = 1 [485]. One must exercise caution in interpreting such results given the difficulty inherent in peering many Gyr back in cosmic time. Nonetheless, it appears that the scale a_{0} remains present in the data and has not obviously changed over the more recent half of the age of the Universe.
4.3.2 The role of surface density
An upper limit to the surface brightness distribution is interesting in the context of disk stability. Recall that dynamically cold, purely Newtonian disks are subject to potentiallyselfdestructive instabilities, one cure being to embed them in the potential wells of spherical darkmatter halos [343]. While the proper criterion for stability is much debated [131, 415], it is clear that the dark matter halo moderates the growth of instabilities and that the ratio of halo to disk self gravity is a relevant quantity. The more selfgravitating a disk is, the more likely it is to suffer undamped growth of instabilities. But, in principle, galaxies with a baryonic disk and a dark matter halo are totally scalable: if a galaxy model has a certain dynamics, and one multiplies all densities by any (positive) constant (and also scales the velocities appropriately) one gets another galaxy with exactly the same dynamics (with scaled time scales). So if one is stable, so is the other. In turn, the mere fact that there might be an upper limit to Σ_{ b } is a priori surprising, and even more so that there might be a coincidence of this upper limit with the acceleration scale a_{0} identified dynamically.
The scale Σ_{†} = a_{0}/G is clearly present in the data (Figure 8). Selection effects make highsurfacebrightness (HSB) galaxies easy to detect and hence discover, but their intrinsic numbers appear to decline exponentially when the central surface density of the stellar disk Σ_{0} > Σ_{†} [264]. It seems natural to associate the dynamical scale a_{0} with the disk stability scale since they are numerically indistinguishable and both arise in the context of the mass discrepancy. However, there is no reason to expect this in ΛCDM, which predicts denser dark matter halos than observed [280, 169, 167, 241, 243, 478, 118]. Such dense dark matter halos could stabilize much higher density disks than are observed to exist. Lacking a clear mechanism to specify this scale, it is introduced into models by hand [115].
4.3.3 Mass discrepancyacceleration relation
So far we have discussed total quantities. For the BTFR, we use the total observed mass of a galaxy and its characteristic rotation velocity. Similarly, the dynamical accelerationbaryonic surface density relation uses a single characteristic value for each galaxy. These are not the only ways in which the “magical” acceleration constant a_{0} appears in the data. In general, the mass discrepancy only appears at very low accelerations a < a_{0} and not (much) above a_{0}. Equivalently, the need for dark matter only becomes clear at very low baryonic surface densities Σ < Σ_{†} = a_{0}/G. Indeed, the amplitude of the mass discrepancy in galaxies anticorrelates with acceleration [270].
There is no reason in the dark matter picture why the mass discrepancy should correlate with any physical scale. Some systems might happen to contain lots of dark matter; others very little. In order to make a prediction with a dark matter model, it is necessary to model the formation of the dark matter halo, the condensation of gas within it, the formation of stars therefrom, and any feedback processes whereby the formation of some stars either enables or suppresses the formation of further stars. This complicated sequence of events is challenging to model. Baryonic “gastrophysics” is particularly difficult, and has thus far precluded the emergence of a clear prediction for galaxy dynamics from ΛCDM.
ΛCDM does make a prediction for the distribution of mass in baryonless dark matter halos: the NFW halo [332, 333]. These are remarkable for being scale free. Small halos have a profile similar to large halos. No feature stands out that marks a unique physical scale as observed. Galaxies do not resemble pure NFW halos [416], even when dark matter dominates the dynamics as in LSB galaxies [241, 243, 118]. The inference in ΛCDM is that gastrophysics, especially the energetic feedback from stellar winds and supernova explosions, plays a critical role in sculpting observed galaxies. This role is not restricted to the minority baryonic constituents; it must also affect the majority dark matter [176]. Simulations incorporating these effects in a quasirealistic way are extremely expensive computationally, so a comprehensive survey of the plausible parameter space occupied by such models has yet to be made. We have no reason to expect that a particular physical scale will generically emerge as the result of baryonic gastrophysics. Indeed, feedback from star formation is inherently a random process. While it is certainly possible for simple laws to emerge from complicated physics (e.g., the fact that SNIa are standard candles despite the complicated physics involved), the more common situation is for chaos to beget chaos. Therefore, it seems unnatural to imagine feedback processes leading to the orderly behavior that is observed (Figure 10); nor is it obvious how they would implicate any particular physical scale. Indeed, the dark matter halos formed in ΛCDM simulations [332, 333] provide an initial condition with greater scatter than the final observed one [280, 478], so we must imagine that the chaotic processes of feedback not only impart order, but do so in a way that cancels out some of the scatter in the initial conditions.
In any case, and whatever the reason for it, a physical scale is clearly observationally present in the data: a_{0} (Eq. 1). At high accelerations a ≫ a_{0}, there is no indication of the need for dark matter. Below this acceleration, the mass discrepancy appears. It cannot be emphasized enough that the role played by a_{0} in the BTFR and this role as a transition acceleration have strictly no intrinsic link with each other, they are fully independent of each other. There is nothing in ΛCDM that stipulates that these two relations (the existence of a transition acceleration and the BTFR) should exist at all, and even less that these should harbour an identical acceleration scale.
The systematic increase in the amplitude of the mass discrepancy with decreasing acceleration and baryonic surface density has a remarkable implication. Even though the observed velocity is not correctly predicted by the observed baryons, it is predictable from them. Independent of any theory, we can simply fit a function D(GΣ) to describe the variation of the discrepancy (V/V_{ b })^{2} with baryonic surface density [270]. We can then apply it to any new system we encounter to predict V = D^{1/2}V_{ b }. In effect, D boosts the velocity already predicted by the observed baryons. While this is a purely empirical exercise with no underlying theory, it is quite remarkable that the distribution of dark matter required in a galaxy is entirely predictable from the distribution of its luminous mass (see also [167]). In the conventional picture, dark matter outweighs baryonic matter by a factor of five, and more in individual galaxies given the halobyhalo missing baryon problem (Figure 2), but apparently the baryonic tail wags the dark matter dog. And it does so again through the acceleration scale a_{0}. Indeed, at very low accelerations, the mass discrepancy is precisely defined by the inverse of the squareroot of the gravitational acceleration generated by the baryons in units of a_{0}. This actually asymptotically leads to the BTFR.
So, up to now, we have seen five roles of a_{0} in galaxy dynamics. (i) It defines the zero point of the TullyFisher relation, (ii) it appears as the characteristic acceleration at the effective radius of spheroidal systems, (iii) it defines the Freeman limit for the maximum surface density of pure disks, (iv) it appears as a transitionacceleration above which no dark matter is needed, and below which it appears, and (v) it defines the amplitude of the massdiscrepancy in the weakfield regime (this last point is not a fully independent role as it leads to the TullyFisher relation). Let us eventually note that there is yet a final role played by a_{0}, which is that it defines the central surface density of all dark matter halos as being on the order of a_{0}/(2πG) [129, 167, 313].
4.3.4 Renzo’s rule
The relation between dynamical and baryonic surface densities appears as a global scaling relation in disk galaxies (Figure 9) and as a local correspondence within each galaxy (Figure 10). When all galaxies are plotted together as in Figure 10, this connection appears as a single smooth function D(a). This does not suffice to illustrate that individual galaxies have features in their baryon distribution that are reflected in their dynamics. While the above correlations could be interpreted as a sort of repulsion between dark and baryonic matter, the following rather indicates closerthannatural attraction.
Mass models (bottom panels of Figure 13) are constructed from the surface density profiles by numerical solution of the Poisson equation [52, 472]. No approximations (like sphericity or an exponential disk) are made at this step. The disks are assumed to be thin, with radial scale length exceeding their vertical scale by 8:1, as is typical of edgeon disks [236]. Consequently, the computed rotation curves (various broken lines in Figure 13) are not smooth, but reflect the observed variations in the observed surface density profiles of the various components. The sum (in quadrature) leads to the total baryonic rotation curve V_{ b }(r) (the solid lines in Figure 13): this is what would be observed if no dark matter were implicated. Instead, the observed rotation (data points in Figure 13) exceeds that predicted by V_{ b },(r): this is the mass discrepancy.
It is often merely stated that flat rotation curves require dark matter. But there is considerably more information in rotation curve data than asymptotic flatness. For example, it is common that the rotation curve in the inner parts of HSB galaxies like NGC 6946 is well described by the baryons alone. The data are often consistent with a very low density of dark matter at small radii with baryons providing the bulk of the gravitating mass. This condition is referred to as maximum disk [471], and also runs contrary to our inferences of dark matter dominance from Figure 5 [414]. More generally, features in the baryonic rotation curve V_{ b } (r) often correspond to features in the total rotation V_{ c }(r).
Perhaps the most succinct empirical statement of the detailed connection between baryons and dynamics has been given by Renzo Sancisi, and known as Renzo’s rule [379]: “For any feature in the luminosity profile there is a corresponding feature in the rotation curve.” Both galaxies shown in Figure 13 illustrate this statement. In the inner region of NGC 6946, the small but compact bulge component causes a sharp feature in V_{ b }(r) that declines rapidly before the rotation curve rises again, as mass from the disk begins to contribute. The updownup morphology predicted by the observed distribution of the baryons is observed in high resolution observations [54, 114]. A dark matter halo with a monotonicallyvarying density profile cannot produce such a morphology; the stellar bulge must be the dominant mass component at small radii in this galaxy.
A surprising aspect of Renzo’s rule is that it applies to LSB galaxies as well as those of high surface brightness. That the baryons should have some dynamical impact where their surface density is highest is natural, though there is no reason to demand that they become competitive with dark matter. What is distinctly unnatural is for the baryons to have a perceptible impact where dark matter must clearly dominate. NGC 1560 provides an example where they appear to do just that. The gas distribution in this galaxy shows a substantial kink in its surface density profile [28] (recently confirmed by [163]) that has a distinct impact on V_{ b }(r). This occurs at a radius where V ≫ V_{ b }, so dark matter should be dominant. A sphericaldarkmatter halo with particles on randomly oriented, highly radial orbits cannot support the same sort of structure as seen in the gas disk, and the spherical geometry, unlike a disk geometry, would smear the effect on the local acceleration. And yet the wiggle in the baryonic rotation curve is reflected in the total, as per Renzo’s rule.^{16}
One inference that might be made from these observations is that the dark matter is baryonic. This is unacceptable from a cosmological perspective, but it is possible to have a multiplicity of dark matter components. That is, we could have baryonic dark matter in the disks of galaxies in addition to a halo of nonbaryonic cold dark matter. It is often possible to scale up the atomic gas component to fit the total rotation [193]. That implies a component of mass that is traced by the atomic gas — presumably some other dynamically cold gas component — that outweighs the observed hydrogen by a factor of six to ten [193]. One hypothesis for such a component is very cold molecular gas [352]. It is difficult to exclude such a possibility, though it also appears to be hard to sustain in LSB galaxies[292]. Dynamically, one might expect the extra mass to destabilize the LSB disk. One also returns to a finetuning between baryonic surface density and masstolight ratio. In order to maintain the balance observed in Figure 5, relatively more dark molecular gas will be required in LSB galaxies so as to maintain a constant surface density of gravitating mass, but given the interactions at hand, this might be at least a bit more promising than explaining it with CDM halos.
As a matter of fact, LSB galaxies play a critical role in testing many of the existing models for dark matter. This happens in part because they were appreciated as an important population of galaxies only after many relevant hypotheses were established, and thus provide good tests of their a priori expectations. Observationally, we infer that LSB disks exhibit large mass discrepancies down to small radii [119]. Conventionally, this means that dark matter completely dominates their dynamics: the surface density of baryons in these systems is never high enough to be relevant. Nevertheless, the observed distribution of baryons suffices to predict the total rotation [279, 120]. Once again, the baryonic tail wags the dark matter dog, with the observations of the minority baryonic component sufficing to predict the distribution of the dominant dark matter. Note that, conversely, nothing is “observable” about the dark matter, in presentday simulations, that predicts the distribution of baryons.
Thus, we see that there are many observations, mostly on galaxy scales, that are unpredicted, and perhaps unpredictable, in the standard dark matter context. They mostly involve a unique relationship between the distribution of baryons and the gravitational field, as well as an acceleration constant a_{0} on the order of the squareroot of the cosmological constant, and they represent the most significant challenges to the current ΛCDM model.
5 Milgrom’s Empirical Law and “Kepler Laws” of Galactic Dynamics
Up to this point in this review, the challenges that we have presented have been purely based on observations, and fully independent of any alternative theoretical framework. However, at this point, it would obviously be a step forward if at least some of these puzzling observations could be summarized and empirically unified in some way, as such a unifying process is largely what physics is concerned with, rather than simply exposing a jigsaw of apparently unrelated empirical observations. And such an empirical unification is actually feasible for many of the unpredicted observations presented in the previous Section 4.3, and goes back to a rather old idea of the Israeli physicist Mordehai Milgrom.
Almost 30 years ago, back in 1983 (and thus before most of the aforementioned observations had been carried out), simply prompted by the question of whether the missing mass problem could perhaps reflect a breakdown of Newtonian dynamics in galaxies, Milgrom [293] devised a formula linking the Newtonian gravitational acceleration g_{ N } to the true gravitational acceleration g in galaxies. Such attempts to rectify the mass discrepancy by gravitational means often begin by noting that galaxies are much larger than the solar system. It is easy to imagine that at some suitably large scale, let’s say on the order of 1 kpc, there is a transition from the usual dynamics applicable in the comparativelytiny solar system to some more general theory that applies on the scale of galaxies in order to explain the mass discrepancy problem. If so, we would expect the mass discrepancy to manifest itself at a particular length scale in all systems. However, as already noted, there is no universal length scale apparent in the data (Figure 10) [382, 266, 406, 279, 270]. The mass discrepancy appears already at small radii in some galaxies; in others there is no apparent need for dark matter until very large radii. This now observationally excludes all hypotheses that simply alter the force law at a linear lengthscale.
5.1 Milgrom’s law and the dielectric analogy
It was suggested, for instance, [218] that such a relation might arise naturally in the CDM context, if halos possess a oneparameter density profile that leads to a characteristic acceleration profile that is only weakly dependent upon the mass of the halo. Then, with a fixed collapse factor for the baryonic material, the transition from dominance of dark over baryonic occurs at a universal acceleration, which, by numerical coincidence, is on the order of cH_{0} and thus of a_{0} (see also [411]). While, still today, it remains to be seen whether this scenario would quantitatively hold in numerical simulations, it was noted by Milgrom [306] that this scenario only explained the role of a_{0} as a transition radius between baryon and dark matter dominance in HSB galaxies, precluding altogether the existence of LSB galaxies where dark matter dominates everywhere. The real challenge for ΛCDM is rather to explain all the different roles played by a_{0} in galaxy dynamics, different roles that can all be summarized within the single law proposed by Milgrom, just like Kepler’s laws are unified under Newton’s law. We list these Keplerlike laws of galactic dynamics hereafter, and relate each of them with the unpredicted observations of Section 4, keeping in mind that these were mostly a priori predictions of Milgrom’s law, made before the data were as good as today, not “postdictions” like we are used to in modern cosmology.
5.2 Galactic Keplerlike laws of motion
 1.
Asymptotic flatness of rotation curves. The rotation curves of galaxies are asymptotically flat, even though this flatness is not always attained at the last observed point (see point hereafter about the shapes of rotation curves as a function of baryonic surface density). What is more, Milgrom’s law can be thought of as including the total acceleration with respect to a preferred frame, which can lead to the prediction of asymptoticallyfalling rotation curves for a galaxy embedded in a large external gravitational field (see Section 6.3).
 2.
Ga_{ 0 } defining the zeropoint of the baryonic TullyFisher relation. The plateau of a rotation curve is V_{ f } = (GMa_{0})^{1/4}. The true TullyFisher relation is predicted to be a relation between this asymptotic velocity and baryonic mass, not luminosity. Milgrom’s law yields immediately the slope (precisely 4) and zeropoint of this baryonic TullyFisher law. The observational baryonic TullyFisher relation should thus be consistent with zero scatter around this prediction of Milgrom’s law (the dotted line of Figure 3). And indeed it is. All rotationallysupported systems in the weak acceleration limit should fall on this relation, irrespective of their formation mechanism and history, meaning that completely isolated galaxies or tidal dwarf galaxies formed in interaction events all behave as every other galaxy in this respect.
 3.
Ga_{ 0 } defining the zeropoint of the FaberJackson relation. For quasiisothermal systems [296], such as elliptical galaxies, the bulk velocity dispersion depends only on the total baryonic mass via σ^{4} ∼ GMa_{0}. Indeed, since the equation of hydrostatic equilibrium for an isotropic isothermal system in the weak field regime reads d(σ^{2}ρ)/dr = −ρ(GMa_{0})^{1/2}/r, one has σ^{4} = α^{−2} × GMa_{0} where α = d ln ρ/d ln r. This underlies the FaberJackson relation for elliptical galaxies (Figure 7), which is, however, not predicted by Milgrom’s law to be as tight and precise (because it relies, e.g., on isothermality and on the slope of the density distribution) as the BTFR.
 4.
Mass discrepancy defined by the inverse of the acceleration in units of a_{ 0 }. Or alternatively, defined by the inverse of the squareroot of the gravitational acceleration generated by the baryons in units of a_{0}. The mass discrepancy is precisely equal to this in the verylowacceleration regime, and leads to the baryonic TullyFisher relation. In the lowacceleration limit, g_{ N }/g = g/a_{0}, so in the CDM language, inside the virial radius of any system whose virial radius is in the weak acceleration regime (well below a_{0}), the baryon fraction is given by the acceleration in units of a_{0}. If we adopt a rough relation \({M_{500}} \simeq 1.5 \times {10^5}{M_ \odot} \times V_c^3{({\rm{km/s)}}^{ 3}}\), we get that the acceleration at R_{500}, and thus the system baryon fraction predicted by Milgrom’s formula, is M_{ b }/M_{500} = a_{500}/a_{0} ≃ 4 × 10^{−4} × V_{ c } (km/s)^{−1}. Divided by the cosmological baryon fraction, this explains the trend for f_{ d } = M_{ b }/(0.17 M_{500}) with potential \((\Phi = V_c^2)\) in Figure 2, thereby naturally explaining the halobyhalo missing baryon challenge in galaxies. No baryons are actually missing; rather, we infer their existence because the natural scaling between mass and circular velocity \({M_{500}} \propto V_c^3\) in ΛCDM differs by a factor of V_{ c } from the observed scaling \({M_b} \propto V_c^4\).
 5.
a_{ 0 } as the characteristic acceleration at the effective radius of isothermal spheres. As a corollary to the FaberJackson relation for isothermal spheres, let us note that the baryonic isothermal sphere would not require any dark matter up to the point where the internal gravity falls below a_{0}, and would thus resemble a purely baryonic Newtonian isothermal sphere up to that point. But at larger distances, in the presence of the added force due to Milgrom’s law, the baryonic isothermal sphere would fall [296] as r^{−4}, thereby making the radius at which the gravitational acceleration is a_{0} the effective baryonic radius of the system, thereby explaining why, at this radius R in quasiisothermal systems, the typical acceleration σ^{2}/R is almost always observed to be on the order of a_{0}. Of course, this is valid for systems where such a transition radius does exist, but going to veryLSB systems, if the internal gravity is everywhere below a_{0}, one can then have typical accelerations as low as one wishes.
 6.
a_{ 0 }/G as a critical mean surface density for stability. Disks with mean surface density 〈Σ〉 ≤ Σ_{†} = a_{0}/G have added stability. Most of the disk is then in the weakacceleration regime, where accelerations scale as \(a \propto \sqrt M\), instead of a ∝ M. Thus, δa/a = (1/2)δM/M instead of δa/a = δM/M, leading to a weaker response to small mass perturbations [299]. This explains the Freeman limit (Figure 8).
 7.
a_{ 0 } as a transition acceleration. The mass discrepancy in galaxies always appears (transition from baryon dominance to dark matter dominance) when \(V_c^2/R \sim {a_0}\), yielding a clear massdiscrepancy acceleration relation (Figure 10). This, again, is the case for every single rotationallysupported system irrespective of its formation mechanism and history. For HSB galaxies, where there exist two distinct regions where \(V_c^2/R > {a_0}\) in the inner parts and \(V_c^2/R < {a_0}\) in the outer parts, locally measured masstolight ratios should show no indication of hidden mass in the inner parts, but rise beyond the radius where \(V_c^2/R \approx {a_0}\) (Figure 14). Note that this is the only role of a_{0} that the scenario of [218] was poorly trying to address (forgetting, e.g., about the existence of LSB galaxies).
 8.
a_{ 0 }/G as a transition central surface density. The acceleration a_{0} defines the transition from HSB galaxies to LSB galaxies: baryons dominate in the inner parts of galaxies whose central surface density is higher than some critical value on the order of Σ_{†} = a_{0}/G, while in galaxies whose central surface density is much smaller (LSB galaxies), DM dominates everywhere, and the magnitude of the mass discrepancy is given by the inverse of the acceleration in units of a_{0}; see (5). Thus, the mass discrepancy appears at smaller radii and is more severe in galaxies of lower baryonic surface densities (Figure 14). The shapes of rotation curves are predicted to depend on surface density: HSB galaxies are predicted to have rotation curves that rise steeply, then become flat, or even fall somewhat to the notyetreached asymptotic flat velocity, while LSB galaxies are supposed to have rotation curves that rise slowly to the asymptotic flat velocity. This is precisely what is observed (Figure 15), and is in accordance [162] with the more complex empirical parametrization of observed rotation curves that has been proposed in [376]. Finally, the total (baryons+DM) acceleration is predicted to decline with the mean baryonic surface density of galaxies, exactly as observed (Figure 16), in the form \(a \propto \Sigma _b^{1/2}\) (see also Figure 9).
 9.
a_{ 0 }/2πG as the central surface density of dark halos. Provided they are mostly in the Newtonian regime, galaxies are predicted to be embedded in dark halos (whether real or virtual, i.e., “phantom” dark matter) with a central surface density on the order of a_{0}/(2πG) as observed^{19}. LSBs should have a halo surface density scaling as the squareroot of the baryonic surface density, in a much more compressed range than for the HSB ones, explaining the consistency of observed data with a constant central surface density of dark matter [167, 313].
 10.
Features in the baryonic distribution imply features in the rotation curve. Because a small variation in g_{ N } will be directly translated into a similar one in g, Renzo’s rule (Section 4.3.4) is explained naturally.
As a conclusion, all the apparently independent roles that the characteristic acceleration a_{0} plays in the unpredicted observations of Section 4.3 (see end of Section 4.3.3 for a summary), as well as Renzo’s rule (Section 4.3.4), have been elegantly unified by the single law proposed by Milgrom [293] in 1983 as a unique scaling relation between the gravitational field generated by observed baryons and the total observed gravitational force in galaxies.
6 Milgrom’s Law as a Modification of Classical Dynamics: MOND
6.1 Modified inertia or modified gravity: Nonrelativistic actions
If one wants to modify dynamics in order to reproduce Milgrom’s heuristic law while still benefiting from usual conservation laws such as the conservation of momentum, one can start from the action at the classical level. Clearly such theories are only toymodels until they become the weakfield limit of a relativistic theory (see Section 7), but they are useful both as targets for such relativistic theories, and as internally consistent models allowing one to make predictions at the classical level (i.e., neither in the relativistic or quantum regime).
6.1.1 Modified inertia
6.1.2 BekensteinMilgrom MOND
The idea of modified gravity is, on the one hand, to preserve the particle equation of motion by preserving the kinetic action, but, on the other hand, to change the gravitational action, and thus modify the Poisson equation. In that case, all the usual conservation laws will be preserved by construction.
An important point, demonstrated by Bekenstein & Milgrom [38], is that a system with a low centerofmass acceleration, with respect to a larger (more massive) system, sees the motion of its constituents combine to give a MOND motion for the centerofmass even if it is made up of constituents whose internal accelerations are above a_{0} (for instance a compact globular cluster moving in an outer galaxy). The centerofmass acceleration is independent of the internal structure of the system (if the mass of the system is small), namely the Weak Equivalence Principle is satisfied.

ρi,j,k is the density discretized on a grid of step h,

Φi,j,k is the MOND potential discretized on the same grid of step h,

μM_{1}, and μ_{L1}, are the values of μ(x) at points M_{1} and L_{1} corresponding to (i + 1/2, j, k) and (i − 1/2, j, k) respectively (Figure 17).
In [457] the GaussSeidel relaxation with red and black ordering is used to solve this discretized equation, with the boundary condition for the Dirichlet problem given by Eq. 20 at large radii. It is obvious that subsequently devising an evolving Nbody code for this theory can only be done using particlemesh techniques rather than the gridless multipole expansion treecode schemes widely used in standard gravity.
6.1.3 QUMOND
6.2 The interpolating function
6.3 The external field effect
The above return to a rescaled Newtonian behavior at very large radii and in the central parts of isolated systems, in order to avoid theoretical problems with the interpolating function, would happen anyway, even with the interpolating function going to zero, for any nonisolated system in the universe (and this return to Newtonian behavior could actually happen at much lower radii) because of a very peculiar aspect of MOND: the external field effect, which appeared in its full significance already in the pristine formulation of MOND [293].
For the exact behavior of the MOND gravitational field in the regime where g and g_{ e } are of the same order of magnitude, one again resorts to a numerical solver, both for the BM equation case and for the QUMOND case (see Eq. 25 and Eq. 35). For the BM case, one adds the three components of the external field (no longer assumed to be in the zdirection only) in the argument of μ_{M1} which becomes {[(Φ(B) − Φ(A))/h − g_{ ex }]^{2} + [(Φ(I) + Φ(H) − Φ(K) − Φ(J))/(4h) − g_{ ey }]^{2} + [(Φ(C) + Φ(D) − Φ(E) − Φ(F))/(4h) − g_{ ez }]^{2}}^{1/2}, and similarly for the other Mi and Li points on the grid (Figure 17). One also adds the respective component of the external field to the term estimating the force at the Mi and Li points in Eq. 25. With M_{1}, for instance, one changes (Φ_{ i }+1,j,k − Φi,j,k) → (Φ_{ i }+1,j,k − Φi,j,k − hg_{ ex }) in the first term of Eq. 25. One then solves this discretized equation with the large radius boundary condition for the Dirichlet problem given by Eq. 61 instead of Eq. 20. Exactly the same is applicable to calculating the phantom dark matter component of QUMOND with Eq. 35, except that now the Newtonian external field is added to the terms of the equation in exactly the same way.
This external field effect (EFE) is a remarkable property of MONDian theories, and because this breaks the strong equivalence principle, it allows us to derive properties of the gravitational field in which a system is embedded from its internal dynamics (and not only from tides). For instance, the return to a Newtonian (Eq. 61 or Eq. 63) instead of a logarithmic (Eq. 20) potential at large radii is what defines 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; see discussion in Section 6.5.2), one can estimate the amplitude of the external field in which the system is embedded, and by measuring the shape of its isopotential contours at large radii, one can determine the direction of that external field, without resorting to tidal effects. It is also noticeable that the phantom dark matter has a tendency to become negative in “conoidal” regions perpendicular to the external field direction (see Figure 3 of [490]): with accurateenough weaklensing data, detecting these pockets of negative phantom densities could, in principle, be a smoking gun for MOND [490], but such an effect would be extremely sensitive to the detailed distribution of the baryonic matter. A final important remark about the EFE is that it prevents most possible MOND effects in Galactic disk open clusters or in wide binaries, apart from a possible rescaling of the gravitational constant. Indeed, for wide binaries located in the solar neighborhood, the galactic EFE (coming from the distribution of mass in our galaxy) is about 1.5 × a_{0}. The corresponding rescaling of the gravitational constant then depends on the choice of the μfunction, but could typically account for up to a 50% increase of the effective gravitational constant. Although this is not, properly speaking, a MOND effect, it could still perhaps imply a systematic offset of mass for verylongperiod binaries. However, any effect of the type claimed to be observed by [188] would not be expected in MOND due to the external field effect.
6.4 MOND in the solar system
The primary place to test modified gravity theories is, of course, the solar system, where general relativity has, until now, passed all the proposed tests. Detecting a deviation from Einsteinian gravity in our backyard would actually be the holy grail of modified gravity theories, in the same sense as direct detection in the lab is the holy grail of the CDM paradigm. However, MOND anomalies typically manifest themselves only in the weakgravity regime, several orders of magnitudes below the typical gravitational field exerted by the sun on, e.g., the inner planets. But in the case of modified inertia (Section 6.1.1), the anomalous acceleration at any location depends on properties of the whole orbit (nonlocality), so that anomalies may appear in the motion of Solar system bodies that are on highlyeccentric trajectories taking them to large distances (e.g., long period comets or the Pioneer spacecraft), where accelerations are low [314]. Such MOND effects have been proposed as a possible mechanism for generating the Pioneer anomaly [314, 469], without affecting the motions of planets, whose orbits are fully in the high acceleration regime. On the other hand, in classical, nonrelativistic modified gravity theories (Sections 6.1.2 and 6.1.3), small effects could still be observable and would primarily probe two aspects of the theory: (i) the shape of the interpolating function (Section 6.2) in the regime x ≫ 1, and (ii) the external Galactic gravitational field (Section 6.3) acting on the solar system, testing the interpolating function in the regime x ≪ 1.
If, as a first approximation, one considers the solar system as isolated, and the Sun as a point mass, the MOND effect in the inner solar system appears as an anomalous acceleration field in addition to the Newtonian one. In units of 0, the amplitude of the anomalous acceleration is given by x[1 − μ(x)], which can be constrained from the motion of the inner planets, typically their perihelion precession and the (non)variation of Kepler’s constant [293, 391, 417]. These constraints typically exclude the wholefamily of interpolating functions (Eq. 46) that are natural for multifield theories such as TeVeS (see Section 6.2 and Section 7) because they yield x[1 − μ(x)] > 1 for x ≫ 1 while it must be smaller than 0.04 at the orbit of Mars [391]^{35}. Of course, this does not mean that the μfunction cannot be represented by the αfamily in the intermediate gravity regime characterizing galaxies, but it must be modified in the strong gravity regime^{36}. Another potential effect of MOND is anomalously strong tidal stresses in the vicinity of saddle points of the Newtonian potential, which might be tested with the LISA pathfinder [37, 49, 255, 464]. The MOND bubble can be quite big and clearly detectable, or the effect could be small and undetectable, depending on the interpolating function [255, 161].
The approximation of an isolated Solar system being incorrect, it is also important to add the effect of the external field from the galaxy. Its amplitude is typically on the order of ∼ 1.5 × a_{0}. From there, Milgrom [314] has predicted (both analytically and numerically) a subtle anomaly in the form of a quadrupole field that may be detected in planetary and spacecraft motions (as subsequently confirmed by [62, 185]). This has been used to constrain the form of the interpolating function in the weak acceleration regime characteristic of the external field itself. Constraints have essentially been set on the nfamily of μfunctions from the perihelion precession of Saturn [63, 154], namely that one must have n > 8 in order to fit these data^{37}.
However, it should be noted that it is slightly inconsistant to compare the classical predictions of MOND with observational constraints obtained by a global fit of solar system orbits using a fullyrelativistic firstpostNewtonian model. Although the above constraints on classical MOND models are useful guides, proper constraints can only truly be set on the various relativistic theories presented in Section 7, the firstorder constraints on these theories coming from their own postNewtonian parameters [65, 99, 173, 372, 391, 450]. What is more, and makes all these tests perhaps unnecessary, it has recently been shown that it was possible to cancel any deviation from general relativity at small distances in most of these relativistic theories, independently of the form of the μfunction [22].
6.5 MOND in rotationallysupported 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. 7). 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 “Keplerlike” 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 [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 μfunctions.
The procedure is then the following (see Section 4.3.4 for more detail). One usually assumes that light traces stellar mass (constant masstolight ratio, but see the counterexample 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 a_{0} by fitting, all at once, a sample of highquality rotation curves with small distance uncertainties and no obvious noncircular motions. Then, all individual rotation curve fits can be performed with the masstolight ratio of the disk as the single free parameter of the fit^{39}. It turns out that using the simple interpolating function (α= n = 1, see Eqs. 46 and 49) yields a value of a_{0} = 1.2 × 10^{−10} m s^{−2}, and excellent fits to galaxy rotation curves [166]. 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. 52 and 53 (solid blue line on Figure 19), very similar to the simple interpolating function in the intermediate to weak gravity regime.
6.5.2 The Milky Way
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 BekensteinMilgrom MOND, QUMOND, or multifield theories, the predictions being anyway difficult to make in modified inertia versions of MOND when noncircular orbits are considered). Most of the predictions made until today and reviewed hereafter have been using the BekensteinMilgrom version of MOND (Eq. 17).
 (i)
By measuring the force perpendicular to the galactic plane: at the solar radius, MOND predicts a 60 percent enhancement of the dynamic surface density at 1.1 kpc above the plane compared to the baryonic surface density, a value in agreement with current data (Table 1, see also [339]). The enhancement would become more apparent at large galactocentric radii where the stellar disk mass density becomes negligible.
 (ii)
By determining dynamically the scale length of the disk mass density distribution. This scale length is a factor ∼ 1.25 larger than the scale length of the visible stellar disk if BekensteinMilgrom MOND applies. Such a test could be applied with existing RAVE data [423], but the accuracy of available proper motions still limits the possibility to explore the gravitational forces too far from the solar neighborhood.
 (iii)
By measuring the velocity ellipsoid tilt angle within the meridional galactic plane. This tilt is different within the MOND and Newton+dark halo cases in the inner part of the Galactic disk. The tilt of about 6 degrees at z =1 kpc at the solar radius is in agreement with the recent determination of 7.3 ± 1.8 degrees obtained by [422]. The difference between MOND and a Newtonian model with a spherical halo becomes significant at z =2 kpc. Interestingly, recent data [328] on the tilt of the velocity ellipsoid at these heights clearly favor the MOND prediction [50].
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 [50]. Predictions for a round dark halo without a dark disk are also compatible with the current constraints, though [194, 422]. The tilt at z =2 kpc should be more discriminating.
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 Besancon 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 [378] that BekensteinMilgrom 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 scaleheight 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 nongravitational phenomena, namely ordered and smallscale 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 nonlinearity of MOND. However, combining a MOND Nbody 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. 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 [144] that with an external field of 0.01a_{0}, 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 BekensteinMilgrom MOND and with the Besancon 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.
6.5.3 Disk stability and interacting galaxies
A lot of questions in galaxy dynamics require using Nbody codes. This is notably necessary for studying stability of galaxy disks, the formation of bars and spirals, or highly timevarying 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 particlemesh Nbody codes. More general codes based on QUMOND (Section 6.1.3) are currently under development.
 (i)
LSB disks are more unstable regarding bar and spiral instabilities in MOND than in the Newton+spherical halo equivalent case,
 (ii)
Bars always tend to appear more quickly in MOND than in the Newton+spherical halo equivalent, and are not slowed down by dynamical friction, leading to fast bars,
 (iii)
LSB disks can be both very thin and extended in MOND thanks to the effect of the “phantom disk”, and vertical velocity dispersions level off at 8 km/s, instead of 2 km/s for Newtonian disks,
 (iv)
Warps can be created in apparently isolated galaxies from the external field effect of large scale structure in MOND,
 (v)
Merging timescales are longer in MOND for interacting galaxies,
 (vi)
Reproducing interacting systems such as the Antennae require relatively finetuned initial conditions in MOND, but the resulting galaxy is more extended and thus closer to observations, thanks to the absence of angular momentum transfer to the dark halo.
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 Σ = 1 M_{⊙} pc^{−2} (depending on the thickness of the disk). However, the observed value is usually attributed to nongravitational phenomena. Note that [279] utilized this fact to predict that conventional analyses of LSB disks would infer abnormally high masstolight 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 [497], even though, as said, nongravitational effects such as ordered and smallscale 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 [79]. 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 [104]. 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 [95] 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 timescales being much larger in MOND [338, 459]. Concerning bars, Nipoti [335] found that they were indeed slowed down more in MOND, as predicted analytically [95], 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 [95], is in reality insufficient to affect the bar pattern speed in MOND. Concerning long merging timescales, 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 timescales 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 [239]. 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 [104].
6.5.4 Tidal dwarf galaxies
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 ΛCDM model, these objects are difficult to form, and require very extended dark matter distribution [71]. 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.
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 statisticallysignificant 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 [239], as a possible solution to the satellites phasespace 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 diskofsatellites. 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).
6.6 MOND in pressuresupported stellar systems
We have already outlined (Section 5.2) how Milgrom’s formula accounts for general scaling relations of pressuresupported systems such as the FaberJackson relation (Figure 7 and see [395]), and that isothermal systems have a finite mass in MOND with the density at large radii falling approximately as r^{−4} [296]. Note also that, in order to match the observed fundamental plane, MOND models must actually deviate somewhat from being strictly isothermal and isotropic: a radial orbit anisotropy in the outer regions is needed [388, 86]. Here we concentrate on slightly more detailed predictions and scaling relations. In general, these detailed predictions are less obvious to make than in rotationallysupported systems, precisely because of the new degree of freedom introduced by the anisotropy of the velocity distribution, very difficult to constrain observationally (as higherorder moments than the velocity dispersions would be needed to constrain it). As we shall see, the successes of MOND are in general a bit less impressive in pressuresupported systems than in rotationallysupported ones, and even in some cases really problematic (e.g., in the case of galaxy clusters, see Section 6.6.4). Whether this is due to the fact that predictions are less obvious to make, or whether this truly reflects a breakdown of Milgrom’s formula for these objects (or the fact that certain theoretical versions of MOND would explicitly deviate from Milgrom’s formula in pressuresupported systems, see Section 6.1.1) remains unclear.
6.6.1 Elliptical galaxies
Luminous elliptical galaxies are dense bodies of old stars with very little gas and typically large internal accelerations. The age of the stellar populations suggest they formed early and all the gas has been used to form stars. To form early, one might expect the presence of a massive darkmatter halo, but the study of, e.g., [367] showed that actually, there is very little evidence for dark matter within the effective radius, and even several effective radii, in ellipticals. On the other hand, these are veryHSB objects and would thus not be expected to show a large mass discrepancy within the bright optical object in MOND. And indeed, the results of [367] were shown to be in perfect agreement with MOND predictions, assuming very reasonable anisotropy profiles [323]. On the theoretical side, it was also importantly shown that triaxial elliptical galaxies can be reproduced using the Schwarzschild orbit superposition technique [482], and that these models are stable [493]^{41}.
On the other hand, [225] used satellite galaxies of ellipticals to test MOND at distances of several 100 kpcs. They used the stacked SDSS satellites to generate a pair of mock galaxy groups with reasonably precise lineofsight velocity dispersions as a function of radius across the group. When these systems were first analysed by [225] they claimed that MOND was excluded by 10σ, but this was only for models that had constant velocity anisotropy. It was then found [14] that with varying anisotropy profiles similar to those found in simulations of the formation of ellipticals by dissipationless collapse in MOND [337], excellent fits to the lineofsight velocity dispersions of both mock galaxies could be found. This can be taken as strong evidence that MOND describes the dynamics in the surroundings of relatively isolated ellipticals very well.
Finally, let us note an intriguing possibility in a MONDian universe (see also Section 9.2). While massive ellipticals would form at z ≈ 10 [393] from monolithic dissipationless collapse [337], dwarf ellipticals could be more difficult to form. A possibility to form those would then be that tidal dwarf galaxies would be formed and survive more easily (see Section 6.5.4) in major mergers, and could then evolve to lead to the population of dwarf ellipticals seen today, thereby providing a natural explanation for the observed densitymorphology relation [239] (more dwarf ellipticals in denser environments).
6.6.2 Dwarf spheroidal galaxies
Dwarf spheroidal (dSph) satellites of the Milky Way [427, 477] exhibit some of the largest mass discrepancies observed in the universe. In this sense, they are extremely interesting objects in which to test MOND. Observationally, let us note that there are essentially two classes of objects in the galactic stellar halo: globular clusters (see Section 6.6.3) and dSph galaxies. These overlap in baryonic mass, but not in surface brightness, nor in age or uniformity of the stellar populations. The globular clusters are generally composed of old stellar populations, they are HSB objects and mostly exhibit no mass discrepancy problem, as expected for HSB objects in MOND. The dSphs, on the contrary, generally contain slightly younger stellar populations covering a range of ages, they are extreme LSB objects and exhibit, as said before, an extreme mass discrepancy, as generically expected from MOND. So, contrary to the case of ΛCDM where different formation scenarios have to be invoked (see Section 6.6.3), the different mass discrepancies in these objects find a natural explanation in MOND.
At a more detailed level, MOND should also be able to fit the whole velocity dispersion profiles, and not only give the right ballpark prediction. This analysis has recently been possible for the eight “classical” dSph around the Milky Way [477]. Solving Jeans equation (Eq. 65), it was found [8] that the four most massive and distant dwarf galaxies (Fornax, Sculptor, Leo I and Leo II) have typical stellar masstolight ratios, exactly within the expected range. Assuming equilibrium, two of the other four (smallest and most nearby) dSphs have masstolight ratios that are a bit higher than expected (Carina and Ursa Minor), and two have very high ones (Sextans and Draco). For all these dSphs, there is a remarkable correlation between the stellar M/L inferred from MOND and the ages of their stellar populations [189]. Concerning the high inferred stellar M/L, note that it has been shown [78] that a dSph will begin to suffer tidal disruption at distances from the Milky Way that are 4–7 times larger in MOND than in CDM, Sextans and Draco could thus actually be partly tidally disrupted in MOND. And indeed, after subjecting the five dSphs with published data to an interloper removal algorithm [418], it was found that Sextans was probably littered with unbound stars, which inflated the computed M/L, while Draco’s projected distancel.o.s. velocity diagram actually looks as outofequilibrium as Sextans’ one. Ursa Minor, on the other hand, is the typical example of an outofequilibrium system, elongated and showing evidence of tidal tails. In the end, only Carina has a suspiciously high M/L (> 4; see [418]).
6.6.3 Star clusters
Star clusters come in two types: open clusters and globular clusters. Most observed open clusters are in the inner parts of the Milky Way disk, and for that reason, the prediction of MOND is that their internal dynamics is Newtonian [293] with, perhaps, a slightly renormalized gravitational constant and slightly squashed isopotentials, due to the external field effect (Section 6.3). Therefore, the possibility of distinguishing Newtonian dynamics from MOND in these objects would require extreme precision. On the other hand, globular clusters are mostly HSB halo objects (see Section 6.6.2), and are consequently predicted to be Newtonian, and most of those that are fluffy enough to display MONDian behavior are close enough to the Galactic disk to be affected by the external field effect (Section 6.3), and so are Newtonian, too. Interestingly, MOND thus provides a natural explanation for the dichotomy between dwarf spheroidals and globular clusters. In ΛCDM, this dichotomy is rather explained by the formation history [235, 397]: globular clusters are supposedly formed in primordial diskbound supermassive molecular clouds with high baryontodark matter ratio, and later become more spheroidal due to subsequent mergers. In MOND, it is, of course, not implied that the two classes of objects have necessarily the same formation history, but the different dynamics are qualitatively explained by MOND itself, not by the different formation scenarios.
However, there exist a few globular clusters (roughly, less than ∼ 10 compared to the total number of ∼ 150) both fluffy enough to display typical internal accelerations well below a_{0}, and far away enough from the galactic plane to be more or less immune from the external field effect [27, 182, 181, 436]. Thus, these should, in principle, display a MONDian mass discrepancy. They include, e.g., Pal 14 and Pal 3, or the large fluffy globular cluster NGC 2419. Pal 3 is interesting, because it indeed tends to display a largerthanNewtonian global velocity dispersion, broadly in agreement with the MOND prediction (Baumgardt & Kroupa, private communication). However, it is difficult to draw too strong a conclusion from this (e.g., on excluding Newtonian dynamics), since there are not many stars observed, and one or two outliers would be sufficient to make the dispersion grow artificially, while a slightlyhigherthanusual masstolight ratio could reconcile Newtonian dynamics with the data. Other clusters such as NGC 1851 and NGC 1904 apparently display the same MONDian behavior [408] (see also [187]). On the other hand, Pal 14 displays exactly the opposite behavior: the measured velocity dispersion is Newtonian [212], but again the number of observed stars is too small to draw a statistically significant conclusion [164], and it is still possible to reconcile the data with MOND assuming a slightly low stellar masstolight ratio [437]. Note that if the cluster is on a highly eccentric orbit, the external gravitational field could vary very rapidly both in amplitude and direction, and it is possible that the cluster could take some time to accomodate this by still displaying a Newtonian signature in its kinematics after a sudden decrease of the external field.
NGC 2419 is an interesting case, because it allows not only for a measure of the global velocity dispersion, but also of the detailed velocity dispersion profile [199]. And, again, like in the case of Pal 14 (but contrary to Pal 3), it displays Newtonian behavior. More precisely, it was found, solving Jeans equations (Eq. 65), that the best MOND fit, although not extremely bad in itself, was 350 times less likely than the best Newtonian fit without DM [199, 200]. However, the stability [336] of this best MOND fit has not been checked in detail. These results are heavily debated as they rely on the small quoted measurement errors on the surface density, and even a slight rotation of only the outer parts of this system near the plane of the sky (which would not show up in th velocity data) would make a considerable difference in the right direction for MOND [398]. However, these observations, together with the results on Pal 14, although not ruling out any theory, are not a resounding success for MOND. However, it could perhaps indicate that globular clusters are generically on highly eccentric orbits, and out of equilibrium due to this (however, the effect would have to be opposite to that prevailing in ultrafaint dwarfs, where the departure from equilibrium would boost the velocity dispersion instead of decreasing it). A stronger view on these results could indicate that MOND as formulated today is an incomplete paradigm (see, e.g., Eq. 27), or that MOND is an effect due to the fundamental nature of the DM fluid in galaxies (see Sections 7.6 and 7.9), which is absent from globular clusters. Concerning NGC 2419, it is perhaps useful to remind oneself that it is very plausibly not a globular cluster. It is part of the Virgo stream and is thus most probably the remaining nucleus of a disrupting satellite galaxy in the halo of the Milky Way, on a genericallyhighlyeccentric orbit. Detailed Nbody simulations of such an event, and of the internal dynamics of the remaining nucleus, would thus be the key to confront MOND with observations in this object. All in all, the situation regarding MOND and the internal dynamics of globular clusters remains unclear.
On the other hand, it has been noted that MOND seems to overpredict the Roche lobe volume of globular clusters [499, 500, 512]. Again, the fact that globular clusters could generically be on highly eccentric orbits could come to the rescue here. What is more, it was shown that, in MOND, globular clusters can have a cutoff radius, which is unrelated to the tidal radius when nonisothermal [397]. In general, the cutoff radii of dwarf spheroidals, which have comparable baryonic masses, are larger than those of the globular clusters, meaning that those may well extend to their tidal radii because of a possibly different formation history than globular clusters.
Finally, a last issue for MOND related to globular clusters [335, 377] is the existence of five such objects surrounding the Fornax dwarf spheroidal galaxy. Indeed, under similar environmental conditions, dynamical friction occurs on significantly shorter timescales in MOND than standard dynamics [95], which could cause the globular clusters to spiral in and merge within at most 2 Gyrs [377]. However, this strongly depends on the orbits of the globular clusters, and, in particular, on their initial radius [10], which can allow for a Hubble time survival of the orbits in MOND.
6.6.4 Galaxy groups and clusters
So, interestingly, the data are still reasonably consistent with the slope predicted by MOND [383], but not with the normalization. There is roughly a factor of two of residual missing mass in these objects [170, 354, 387, 389, 392, 453]. This conclusion, reached from applying the hydrostatic equilibrium equation to the temperature profile of the Xray emitting gas of these objects, has also been reached for low mass Xray emitting groups [12]. This is essentially because, contrary to the case of galaxies, there is observationally a need for “Newtonian” missing mass in the central parts^{42} of clusters, where the observed acceleration is usually slightly larger than a_{0}, meaning that the MOND prescription is not enough to explain the observed discrepancy between visible and dynamical mass there. For this reason, the residual missing mass in MOND is essentially concentrated in the central parts of clusters, where the ratio of MOND dynamical mass to observed baryonic mass reaches a value of 10, to then only decrease to a value of roughly ∼ 2 in the very outer parts, where almost no residual mass is present. Thus, the profile of this residual mass would thus consist of a large constant density core of about 100–200 kpc in size (depending on the size of the group/cluster in question), followed by a sharp cutoff.
 (i)
Practical falsification of MOND,
 (ii)
Evidence for missing baryons in the central parts of clusters,
 (iii)
Evidence for nonbaryonic dark matter (existing or exotic),
 (iv)
Evidence that MOND is an incomplete paradigm,
 (v)
Evidence for the effect of additional fields in the parent relativistic theories of MOND, not included in Milgrom’s formula.
Yet another possibility (iv) would be that MOND is incomplete, and that a new scale should be introduced, in order to effectively enhance the value of a_{0} in galaxy clusters, while lowering it to its preferred value in galaxies. There are several ways to implement such an idea. For instance, Bekenstein [36] proposed adding a second scale in order to allow for effective variations of the acceleration constant as a function of the deepness of the potential (Eq. 27). This idea should be investigated more in the future, but it is not clear that such a simple rescaling of a_{0} would account for the exact spatial distribution of the residual missing mass in MOND clusters, especially in cases where it is displaced from the baryonic distribution (see Section 8.3). However, as even Gauss’ theorem would not be valid anymore in spherical symmetry, the high nonlinearity might provide nonintuitive results, and it would thus clearly be worth investigating this suggestion in more detail, as well as developing similar ideas with other additional scales in the future (such as, for instance, the baryonic matter density; see [82, 143] and Section 7.6).
Finally, as we shall see in Section 7, parent relativistic theories of MOND often require additional degrees of freedom in the form of “dark fields”, which can nevertheless be globally subdominant to the baryon density, and thus do not necessarily act precisely as true “dark matter”. Thus, the last possibility (v) is that these fields, which are obviously not included in Milgrom’s formula, are responsible for the cluster missing mass in MOND. An example of such fields are the vector fields of TeVeS (Section 7.4) and Generalized EinsteinAether theories (Section 7.7). It has been shown (see Section 9.2) that the growth of the spatial part of the vector perturbation in the course of cosmological evolution can successfully seed the growth of baryonic structures, just as dark matter does. If these seeds persist, it was shown [112] that they could behave in very much the same way as a dark matter halo in relatively unrelaxed galaxy clusters. However, it remains to be seen whether the spatiallyconcentrated distribution of missing mass in MOND would be naturally reproduced in all clusters. In other relativistic versions of MOND (see, e.g., Sections 7.6 and 7.9), the “dark fields” are truly massive and can be thought of as true dark matter (although more complex than simple collisionless dark matter), whose energy density outweighs the baryonic one, and could provide the missing mass in clusters. However, again, it is not obvious that the centrallyconcentrated distribution of residual missing mass in clusters would be naturally reproduced. All in all, there is no obviously satisfactory explanation for the problem of residual missing mass in the center of galaxy clusters, which remains one of the most serious problems facing MOND.
7 Relativistic MOND Theories
In Section 6, we have considered the classical theories of MOND and their predictions in a vast number of astrophysical systems. However, as already stated at the beginning of Section 6, these classical theories are only toymodels until they become the weakfield limit of a relativistic theory (with invariant physical laws under differentiable coordinate transformations), i.e., an extension of general relativity (GR) rather than an extension of Newtonian dynamics. Here, we list the various existing relativistic theories boiling down to MOND in the quasistatic weakfield limit. It is useful to restate here that the motivation for developing such theories is not to get rid of dark matter but to explain the Keplerlike laws of galactic dynamics predicted by Milgrom’s law (see Section 5). As we shall see, many of these theories include new fields, so that dark matter is often effectively replaced by “dark fields” (although, contrary to dark matter, their energy density can be subdominant to the baryonic one; note that, even more importantly, in a static configuration these dark fields are fully determined by the baryons, contrary to the traditional dark matter particles, which may, in principle, be present independent of baryons).
These theories are great advances because they enable us to calculate the effects of gravitational lensing and the cosmological evolution of the universe in MOND, which are beyond the capabilities of classical theories. However, as we shall see, many of these relativistic theories still have their limitations, ranging from true theoretical or observational problems to more aesthetic problems, such as the arbitrary introduction of an interpolating function (Section 6.2) or the absence of an understanding of the \(\Lambda \sim a_0^2\) coincidence. What is more, the new fields introduced in these theories have no counterpart yet in microphysics, meaning that these theories are, at best, only effective. So, despite the existing effective relativistic theories presented here, the quest for a more profound relativistic formulation of MOND continues. Excellent reviews of existing theories can also be found in, e.g., [34, 35, 81, 100, 136, 183, 318, 429, 431].
7.1 Scalartensor kessence
MOND is an accelerationbased modification of gravity in the ultraweakfield limit, but since the Christoffel symbol, playing the role of acceleration in GR, is not a tensor, it is, in principle, not possible to make a general relativistic theory depend on it. Another natural way to account for the departure from Newtonian gravity in the weakfield limit and to account for the violation of the SEP inherent to the external field effect is to resort to a scalartensor theory, as first proposed by [38]. The added scalar field can play the role of an auxiliary potential, and its gradient then has the dimensions of acceleration and can be used to enforce the accelerationbased modification of MOND.
However, another problem was immediately realized at an observational level [38, 40]. Because of the conformal transformation of Eq. 75, one has that Ψ ≠ −Φ in the RAQUAL equivalent of Eq. 73. In other words, as it is wellknown that gravitational lensing is insensitive to conformal rescalings of the metric, apart from the contribution of the stressenergy of the scalar field to the source of the Einstein metric [40, 81], the “nonNewtonian” effects of the theory respectively on lensing and dynamics do not at all correspond to similar amounts of “missing mass”. This is also considered a generic problem with any local pure metric formulation of MOND [441].
7.2 Stratified theory
7.3 Original TensorVectorScalar theory
The static weakfield limit equation for the scalar field is precisely the same as Eq. 77, and the scalar field enters the static weak field metric Eq. 73 as Φ = −Ψ = ΞΦ_{ N } + c^{2}ϕ meaning that lensing and dynamics are compatible, with Ξ being a factor depending on K and on the cosmological value of the scalar field (see Eq. 58 of [33]). This can be normalized to yield Ξ = 1 at redshift zero. Again, all the relations between the free function and Milgrom’s μfunction can be found in Section 6.2 (see also [145, 431]).
This theory has played a true historical role as a proof of concept that it was possible to construct a fully relativistic theory both enhancing dynamics and lensing in a coherent way and reproducing the MOND phenomenology for static configurations with the dynamical 4vector pointing in the time direction. However, the question remained whether these static configurations would be stable. What is more, although a classical Hamiltonian^{49} unbounded from below in flat spacetime would not necessarily be a concern at the classical level (and even less if the model is only “phenomenological”), it would inevitably become a worry for the existence of a stable quantum vacuum (see however [196]). And indeed, it was shown in [98] that models with such “Maxwellian” vector fields having a TeVeSlike Lagrange multiplier constraint in their action have a corresponding Hamiltonian density that can be made arbitrarily large and negative (see also Section IV.A of [81]). What is more, even at the classical level, it has been shown that sphericallysymmetric solutions of TeVeS are heavily unstable [412, 413], and that this type of vector field causes caustic singularities [105], in the sense that the integral curves of the vector are timelike geodesics meeting each other when falling into gravity potential wells. Thus, another form was needed for the action of the TeVeS vector field.
7.4 Generalized TensorVectorScalar theory
Thus, this generalized version is the current “working version” of what is now called TeVeS: a tensorvectorscalar theory with an Einsteinlike metric, an EinsteinAetherlike unitnorm vector field, and a kessencelike scalar field, all related to the physical metric through Eq. 83. It has been extensively studied, both in its original and generalized form. It has for instance been shown that, contrary to many gravity theories with a scalar sector, the theory evidences no cosmological evolution of the Newtonian gravitational constant and only minor evolution of Milgrom’s constant a_{0} [145, 39]. However, the fact that the latter is still put in by hand through the lengthscale of the theory l ∼ c^{2}/a_{0}, and has no dynamical connection with the Hubble or cosmological constant is perhaps a serious conceptual shortcoming, together with the free function put by hand in the action of the scalar field (but see [22] for a possible solution to the latter shortcoming). The relations between this free function and Milgrom’s μ can be found in [145, 431] (see also Section 6.2), the detailed structure of null and timelike geodesics of the theory in [431], the analysis of the parametrized postNewtonian coefficients (including the preferredframe parameters quantifying the local breaking of Lorentz invariance) in [173, 372, 391, 450], solutions for black holes and neutron stars in [244, 245, 247, 246, 374, 438, 439], and gravitational waves in [216, 214, 215, 373]. It is important to remember that TeVeS is not equivalent to GR in the strong regime, which is why it can be tested there, e.g., with binary pulsars or with the atomic spectral lines from the surface of stars [122], or other very strong field effects^{50}. However, these effects can always generically be suppressed (at the price of introducing a Galileon type term in the action [22]), and such tests would never test MOND as a paradigm. It is by testing gravity in the weak field regime that MOND can really be put to the test.
Finally, let us note that TeVeS (and its generalization) has been shown to be expressible (in the “matter frame”) only in terms of the physical metric g_{ μv }, and the vector field U_{ μ } [513], the scalar field being eliminated from the equations through the “unitnorm” constraint in terms of the Einstein metric \({{\tilde g}^{\mu \nu}}{U_\mu}{U_\nu} =  1\), leading to g^{ μv }U_{ μ }U_{ v } = −e^{−2ϕ}. In this form, TeVeS is sometimes thought of as GR with an additional “dark fluid” described by a vector field [503].
7.5 BiScalarTensorVector theory
7.6 Nonminimal scalartensor formalism
7.7 Generalized EinsteinAether theories
The remarkable feature of GEA theories allowing for the desired enhancing of gravitational lensing without any on the form of the physical metric is that, writing the metric as in Eq. 73, it can be shown [431] that in the limit X_{gea} → 0 the action of Eq. 93 is only a function of ϒ = Φ + Ψ and is thus invariant under disformal transformations [Φ → Φ + β(r); Ψ → Ψ − β(r)], of the type of Eq. 83. These GEA theories are currently extensively studied, mostly in a cosmological context (see Section 9), but also for their parametrized postNewtonian coefficients in the solar system [65] or for black hole solutions [451].
Interestingly, it has been shown that all these vector field theories (TeVeS, BSTV, GEA) are all part of a broad class of theories studied in [183]. Yet other phenomenologicallyinteresting theories exist among this class, such as, for instance, the VΛ models considered by Zhao & Li [502, 506, 510] with a dynamical norm vector field, whose norm obeys a potential (giving it a mass) and has a nonquadratic kinetic term àlaRAQUAL, in order to try reproducing both the MOND phenomenology and the accelerated expansion of the universe, while interpreting the vector field as a fluid of neutrinos with varying mass [504, 505]. This has the advantage of giving a microphysics meaning to the vector field. Such vector fields have also been argued to arise naturally from dimensional reduction of higherdimensional gravity theories [34, 261], or, more generally, to be necessary from the fact that quantum gravity could need a preferred rest frame [206] in order to protect the theory against instabilities when allowing for higher derivatives to make the theory renormalizable (e.g., in Horava gravity [64, 195]). Inspired by this possible need of a preferred rest frame in quantum gravity, relativistic MOND theories boiling down to particular cases of GEA theories in which the vector field is hypersurfaceorthogonal have, for instance, been proposed in [61, 396].
7.8 Bimetric theories
This promising broad class of theories should be carefully theoretically investigated in the future, notably against the existence of ghost modes [69]. At a more speculative level, this class of theories can be interpreted as a modification of gravity arising from the interaction between a pair of membranes: matter lives on one membrane, twin matter on the other, each membrane having its own standard elasticity but coupled to the other one. The way the shape of the membrane is affected by matter then depends on the combined elasticity properties of the double membrane, but matter response depends only on the shape of its home membrane. Interestingly, bimetric theories have also been advocated [256] to be a useful ingredient for the renormalizability of quantum gravity (although they currently considered theories with only metric interactions, not derivatives like in BIMOND).
7.9 Dipolar dark matter
As we have seen, many relativistic MOND theories do invoke the existence of new “dark fields” (scalar or vector fields), which, if massive, can even sometimes truly be thought of as “dark matter” enjoying nonstandard interactions with baryons^{52} (Section 7.6 and [82]). The bimetric version of MOND (Section 7.8) also invokes the existence of a new type of matter, the “twin matter”. This clearly shows that, contrary to common misconceptions, MOND is not necessarily about “getting rid of dark matter” but rather about reproducing the success of Milgrom’s law in galaxies. It might require adding new fields, but the key point is that these fields, very massive or not, would not behave simply as collisionless particles.
This model has many advantages. The monopolar density of the dipolar atoms ρ will play the role of CDM in the early universe, while the minimum of the potential W(P) naturally adds a cosmological constant term, thus making the theory precisely equivalent to the ΛCDM model for expansion and large scale structure formation. The dark matter fluid behaves like a perfect fluid with zero pressure at firstorder cosmological perturbation around a FLRW background and thus reproduces CMB anisotropies. Let us also note that, if the potential W(P) defining the internal force of the dipolar medium is to come from a fundamental theory at the microscopic level, one expects that the dimensionless coefficients in the expansion all be of order unity after rescaling by \(a_0^2\), thus naturally leading to the coincidence \(\Lambda \sim a_0^2\).
However, while the weak clustering hypothesis and stationarity of the dark matter fluid in galaxies are suppported by an exact and stable solution in spherical symmetry [58], it remains to be seen whether such a configuration would be a natural outcome of structure formation within this model. The presence of this stationary DM fluid being necessary to reproduce Milgrom’s law in stellar systems, this theory loses a bit of the initial predictability of MOND, and inherits a bit of the flexibility of CDM, inherent to invoking the presence of a DM fluid. This DM fluid could, e.g., be absent from some systems such as the globular clusters Pal 14 or NGC 2419 (see Section 6.6.3), thereby naturally explaining their apparent Newtonian behavior. However, the weak clustering hypothesis in itself might be problematic for explaining the missing mass in galaxy clusters, due to the fact that the MOND missing mass is essentially concentrated in the central parts of these objects (see Section 6.6.4).
7.10 Nonlocal theories and other ideas
All the models so far somehow invoke the existence of new “dark fields”, notably because for local pure metric theories, the Hamiltonian is generically unbounded from below if the action depends on a finite number of derivatives [81, 136, 441]. A somewhat provocative solution would thus be to consider nonlocal theories. A nonlocal action could, e.g., arise as an effective action due to quantum corrections from superhorizon gravitons [440]. Deffayet, EspositoFarèse & Woodard [123] have notably exhibited the form that a pure metric theory of MOND could take in order to yield MONDian dynamics and MONDian lensing for a static, sphericallysymmetric baryonic source.
8 Gravitational Lensing in Relativistic MOND
The viable MOND theories from Section 7, although still mostly effective, have the great advantage of proving that constructing relativistic MOND theories is possible, and that it is thus possible to calculate from them the effects of gravitational lensing. But the nonuniqueness of the theories means that there is not really a unique prediction for gravitational lensing, especially in heavilytimedependent configurations, or when the predictions of the theories for the expansion history of the universe deviate from the concordance model. As we have seen, some theories also deviate slightly from classical MOND predictions for dynamics of quasistatic systems, due to the presence of massive dark fields, and the same would of course happen for gravitational lensing. However, at the zeroth order, and in static weakfield configurations, we can make predictions for all theories whose expansion history would be similar to that of ΛCDM (see Section 9.1) and whose static weakfield limit is represented by a physical metric^{56} with Ψ = −Φ in Eq. 73 (Φ obeying Eq. 17). In this case, the way the light propagates on the null geodesics of this metric is exactly the same in all these theories once Φ is known. What differs from GR is only the relation between Φ and the underlying mass distribution of the lens.
8.1 Strong lensing by galaxies
As outlined above, what differs from GR in all the relativistic MOND theories is the relation between the nonrelativistic potential Φ and the underlying mass distribution of the lens ρ. However, different theories yield slightly different relations between Φ and ρ in the weakfield limit (see especially Sections 6.1 and 6.2). For instance, while GEA theories (Section 7.7) boil down to Eq. 17 in the static weakfield limit, TeVeS (Section 7.4) leads to the situation of Eq. 40, and BIMOND (Section 7.8) to Eq. 30. However, like in the case of rotation curves (see Figure 20), the differences are only minor outside of spherical symmetry (and null in spherical symmetry), and the global picture can be obtained by assuming a relation given by the BM equation (Eq. 17).
Due to the fact that all the above models were using the Bekenstein μfunction (α = 0 in Eq. 46), and that this function has a tendency of slightly underpredicting stellar masstolight ratios in galaxy rotation curve fits [145], it was claimed that this was a sign for a MOND missing mass problem in galaxy lenses [152, 153, 262]. While such a missing mass is indeed possible, and even corroborated by some dynamical studies [364] of galaxies residing inside clusters (i.e., the smallscale equivalent of the problem of MOND in clusters), for isolated systems with wellconstrained stellar masstolight ratio, the use of the simple μfunction (α = 1 in Eq. 46) has, on the contrary, been shown to yield perfectly acceptable fits [94] in accordance with the lensing fundamental plane [400].
Finally, the probability distribution of the angular separation of the two images in a sample of lensed quasars has been investigated by Chen [90, 91]. This important question has proven somewhat troublesome for the ΛCDM paradigm, but is well explained by relativistic MOND theories [90].
8.2 Weak lensing by galaxies
A gravitational lens not only produces multiple images close to caustics, but also weakly distorted images (arclets) of other background sources. The weak and noisy signals from several individual arclets (not necessarily detected by eye, but rather numerically exploited with the help of image analysis) can be averaged by statistical techniques to get the shear components γ_{1} and γ_{2} in Eq. 114 from the mean ellipticity of the images. One can then get the convergence κ from the azimuthal average of the tangential component of the shear. This is what is known as weak lensing. In the case of galaxygalaxy weak lensing, since the gravitational distortions induced by an individual lens are too small to be detected, one has to resort to the study of the ensemble averaged signal around a large number of lenses. This has been investigated in the context of MOND for a sample of relativelyisolated galaxylenses, stacked by luminosity ranges [456]. The derived MOND masses were obtained by fitting a point mass model to the lensing data within a distance of 200 kpc from the lens. While the MOND masses are perfectly compatible with the baryonic masses in all galaxies less luminous than 10^{11} L_{⊙}, it was found that the required MOND masstolight ratios tended to be slightly too high (M/L ≃ 10) for the most massive and luminous galaxies (L > 10^{11} L_{⊙}). However, this whole result is dictated by only one data point, which “pulls up” the result and make all the data points lie below the “best fit”, and the curve is “pulled up” strongly by only the first point. Thus, the masstolight ratios could easily be scaled down by a factor of two, making these galaxies in perfect agreement with MOND. But it is also worth noting that due to the very large distances probed, the presence of some weaklyclustering residual mass (hot dark matter, or some sort of “dark field” in the relativistic MOND theories) could start playing a role at these distances. While ordinary neutrinos are still too weakly clustering, a slightly more massive fermion such as a 10 eVscale sterile neutrino could cluster on these scales, and, of course, the presence of baryonic dark matter in the form of dense molecular gas clouds could also be present around these very massive objects (see Section 6.6.4).
Also related to weak lensing, it is important to recall that the “phantom dark matter” of MOND (Eq. 33) can sometimes become negative in cones perpendicular to the direction of the external gravitational field in which a system is embedded: with accurate enough weaklensing data, detecting these pockets of negative phantom densities around a sample of nonisolated galaxies could, in principle, be a smoking gun for MOND [490], but such an effect would be extremely sensitive to the detailed distribution of the baryonic matter, and finding a sample of galaxies with similar gravitational environments would also be extremely difficult.
8.3 Strong and weak lensing by galaxy clusters
For TeVeS (Section 7.4) and GEA (Section 7.7), the growth of the spatial part of the vector perturbation in the course of cosmological evolution can successfully seed the growth of baryonic structures, just as dark matter does, and it is possible to reconstruct the gravitational field of the bullet cluster without any extra matter but with a substantial contribution from the vector field. However, why the dynamical evolution of the vector field perturbations would lead to precisely such a configuration remains unclear. Similarly, the massive scalar field of Section 7.6 or the monopolar part of the dipolar DM of Section 7.9 could, in principle, provide the offcentered missing mass too, but again, why they would appear distributed as they do remains unclear, especially in the case of dipolar DM, which is supposed to cluster only very weakly, and, in principle, not to appear as densely clustered. Whether the twin matter of BIMOND (Section 7.8) could help providing the right convergence map also remains to be seen, while for nonlocal models (Section 7.10), there is a strong dependence on the past lightcone, meaning that recentlydisturbed systems, such as the Bullet, may be far from the static MOND limit (but in that case, it would not be clear why all the other clusters from Section 6.6.4 exhibit the same amount of residual missing mass). So, while the bullet cluster clearly does not represent the MONDkiller that it was supposed to be, explaining its convergence map remains an outstanding challenge for all MOND theories. However, the bullet cluster also represents an outstanding challenge to ΛCDM (see Section 4.2), due to its high collision speed [249]. In that respect, MOND is much more promising [16].
On the other hand, a comprehensive weak lensing mass reconstruction of the rich galaxy cluster Cl0024+17 at z = 0.4 [211] has been argued to have revealed the first dark matter structure that is offset from both the gas and galaxies in a cluster. This structure is ringlike, located between r ∼ 60″ and r ∼ 85′. It was, again, argued to be the result of a collision of two massive clusters 1–2 Gyr in the past, but this time along the lineofsight. It has also been argued [211] that this offset was hard to explain in MOND. Assuming that this ringlike structure is real and not caused by instrumental bias or spurious effects in the weak lensing analysis (due, e.g., to the unification of strong and weaklensing or to the use of spherical/circular priors), and that cluster stars and galaxies do not make up a high fraction of the mass in the ring (which would be too faint to observe anyway), it has been shown that, for certain interpolating functions with a sharp transition, this is actually natural in MOND [325]. A peak in the phantom dark matter distribution generically appears close to the transition radius of MOND r_{ t } = (GM/a_{0})^{1/2}, especially when most of the mass of the system is wellcontained inside this radius (which is the case for the cluster Cl0024+17). This means that the ring in Cl0024+17 could be the first manifestation of this pure MOND phenomenon, and thus be a resounding success for MOND in galaxy clusters. However, the sharpness of this phantom dark matter peak strongly depends on the choice of the μfunction, and for some popular ones (such as the “simple” μfunction) the ring cannot be adequately reproduced by this pure MOND phenomenon. In this case, a collisional scenario would be needed in MOND too, in order to explain the feature as a peak of cluster dark matter. Indeed, we already know that there is a mass discrepancy in MOND clusters, and we know that this dark matter must be in collisionless form (e.g., neutrinos or dense clumps of cold gas). So the results of the simulation with purely collisionless dark particles [211] would surely be very similar in MOND gravity. Again, it was shown that the density of missing mass was compatible with 2 eV ordinary neutrinos, like in most clusters with T > 4 keV [139]. Finally, let us note that strong lensing was also recently used as a robust probe of the matter distribution on scales of 100 kpc in galaxy clusters, especially in the cluster Abell 2390 [149]. A residual missing mass was again found, compatible with the densities provided by fermionic hot dark matter candidates only for masses of ∼ 10 eV and heavier. All in all, the problem posed by gravitational lensing from galaxy clusters is thus very similar to the one posed by the temperature profiles of their Xray emitting gas (Section 6.6.4), and remains one of the two main current problems of MOND, together with its problem at reproducing the CMB anisotropies (see Section 9.2).
Finally, let us note in passing that another (nonlensing) test of relativistic MOND theories in galaxy clusters has been performed by analysing the gravitational redshifts of galaxies in 7800 galaxy clusters [489], which were originally found to be difficult to reconcile with MOND: however, this original analysis assumed a distribution of residual missing mass in MOND by simply scaling down the Newtonian dynamical mass represented by a NFW halo by a factor 0.8, and the analysis confused the interpolating functions μ(x) and \(\tilde \mu (x)\) (see Section 6.2). A subsequent analysis [41] showed that these gravitational redshifts were in accordance with relativistic MOND when the correct residual mass and acceptable μfunctions were used.
8.4 Weak lensing by largescale structure
Like in the static case, weak gravitational lensing from largescale structure will actually depend on Φ–Ψ, whereas galaxy clustering will arise only from the nonrelativistic potential Φ. By combining information on the matter overdensity at a given redshift (obtained by measuring the peculiar velocity field) and on the weak lensing maps, Zhang et al. [498] proposed a clever method to observationally estimate Φ — Ψ This allowed Reyes et al. [362] to use luminous red galaxies in the SDSS survey in to exclude one model from the original TeVeS theory (Section 7.3) with the original f(X) function of [33], thus explicitly showing how such measurements could be a possible future smokinggun for all theories based on dynamical vector fields. But note that other MOND theories such as BIMOND would not be affected by such measurements.
However, let us finally note a caveat in the interpretation of the weak lensing shear map in the context of relativistic MOND. While intercluster filaments negligibly contribute to the weak lensing signal in GR, a single filament inclined by π/4 from the line of sight can cause substantial distortion of background sources pointing toward the filament’s axis in relativistic MOND theories [148]. Since galaxies are generally embedded in filaments or are projected on such structures, this contribution should be taken into account when interpreting weak lensing data. This additional difficulty for interpreting weaklensing data in MOND is not only true for filaments, but more generally for all lowdensity structure such as sheets and voids.
9 MOND and Cosmology
9.1 Expansion history
A viable theory of modified gravity, including dark fields or not, should not only be able to reproduce observations in quasistationary galactic and extragalactic systems, but also to reproduce all of the major probes of observational cosmology, including (i) the Hubble diagram out to large z, (ii) the anisotropies in the cosmic microwave background (CMB), and (iii) the matter power spectrum on large scales. The first requires a detailed knowledge of FLRW cosmology, and the last two a knowledge of cosmological perturbations on a FLRW background.
Concerning the first point, the FLRW solutions have been extensively studied for TeVeS (Sections 7.3 and 7.4, see, e.g., [70]) and GEA (Section 7.7, see, e.g., [515]) theories, for BIMOND (Section 7.8, see, e.g., [101]), and for theories based on dipolar dark matter (Section 7.9, see, e.g., [60]). In the latter case, the theory [58, 60] has been shown to be strictly equivalent to ΛCDM out to firstorder cosmological perturbations (but very different in the galaxy formation regime), together with a natural explanation for \(\Lambda \sim a_0^2\). For the other theories, it has been shown that the contribution of the extra fields to the overall expansion is subdominant to the baryonic mass and does not affect the overall expansion [151]. Such theories can predict an extremely wide range of cosmological behavior, ranging from accelerated expansion to contraction on a finite time scale [70]. The key point is that the expansion history mainly depends on the form of the “MOND function” f(X) for the unconstrained domain X < 0 in any of these theories.
All in all, with the additional freedom of a hypothetical dark component in the matter sector, in the form of, e.g., ordinary or sterile neutrinos, playing with the form of f(X) for X < 0 in TeVeS, GEA and BIMOND always allows one to reproduce an expansion history and a Hubble diagram almost precisely identical to ΛCDM, justifying the assumption made in Section 8 of an expansion history for gravitational lensing in relativistic MOND. However, it is important to note that MOND theories are not providing a unique prediction on this.
9.2 Largescale structure and Cosmic Microwave Background
Modified gravity theories should, of course, not only produce a reasonable Hubble expansion but also reproduce the observed anisotropies in the CMB, and the matter power spectrum. Taken at face value, these require not only dark matter, but nonbaryonic cold dark matter. Any alternative theory must account for these, just as dark matter models need to explain galaxy scale phenomenology.
Using the hypothesis that the universe is filled with some form of cold dark matter, it is possible to simultaneously fit observations of the CMB [229] and provide an elegant picture for the growth of large scale structure [444]. Thus, an obvious question is how MOND fares with these subjects. Of course, as we have seen, there is no unique existing MOND theory (Section 7), and the basic theory underlying MOND as a paradigm is probably yet to be found. Nevertheless, we can make a few general considerations about how any MOND theory should behave, and then look in more details at specific predictions from existing relativistic theories. The general picture is that, in some ways MOND does surprisingly well, in others it clearly gives no real unique prediction by now, and in still others it appears to fail outright.
If one alters the force law as envisioned by MOND, the effective longrange force becomes stronger. Though details will, of course, depend on the specific relativistic theory, we can speculate about the consequences of a MONDlike force in cosmology. Note, however, that most of what follows cannot be rigorously justified at the moment for lack of a compelling unique underlying theory. But, obviously, because of the stronger force, dynamical measures of the cosmic mass density will be overestimated, just as in galaxies. Applying MOND to the peculiar motions of galaxies yields Ω_{ m } ≈ Ω_{ b } [279]. There are large uncertainties in estimating the extragalactic peculiar acceleration field, so this merely shows that MOND might alleviate the need for nonbaryonic dark matter inferred conventionally from Ω_{ m } > Ω_{ b }.
The stronger effective gravitational attraction of MOND would change the growth rate of perturbations. Instead of adding dark mass to speed the growth of structure, we now rely on the modified force law to do the work. While it is obvious that MOND will form structures more rapidly than conventional gravity with the same source perturbation, we immediately encounter a challenge posed by the nonlinear nature of the theory, precluding an easy linear perturbation analysis. One can nevertheless sketch a naive overview of how structure might form under the influence of MOND. The following picture emerges from numerical calculations of particles interacting under MOND in an assumed background [386, 341, 226, 250], and is thus obviously slightly (or very) different from the various relativistic MOND theories of Section 7 and from those yet to be found, especially from those MONDian theories involving the existence of some form of dark matter (twin matter, dipolar dark matter, etc.). In the early universe, perturbations cannot grow because the baryons are coupled to the photon fluid. The mass density is lower, so matter domination occurs later than in ΛCDM. Consequently, MOND structure formation initially has to lag behind ΛCDM at very high redshift (z > 200). However, as the influence of the photon field declines and perturbations begin to enter the MOND regime, structure formation rapidly speeds up. Large galaxies may form by z ≈ 10 and clusters by z ≈ 2 [11, 386], considerably earlier than in ΛCDM. By z = 0, the voids have become more empty than in ΛCDM, but otherwise simulations (of collisionless particles, which is, of course, not the best representation of the baryon fluid) show the same qualitative features of the cosmic web [226, 250]. This similarity is not surprising since MOND is a subtle alteration of the force law. The chief difference is in the timing of when structures of a given mass appear, it being easier to assemble a large mass early in MOND. This means that MOND is promising in addressing many of the challenges of Section 4.2, namely the highz clusters challenge [11] and Local Void challenge, as well as the bulk flow challenge and high collisional velocity of the bullet cluster [16, 251], again due to the muchlargerthanNewtonian MOND force in the structure formation context. What is more, it could allow large massive galaxies to form early (z ≈ 10) from monolithic dissipationless collapse [393], with welldefined relationships between the mass, radius and velocity dispersion. Consequently, there would be less mergers than in ΛCDM at intermediate redshifts, in accordance with constraints from interacting galaxies (see Section 6.5.3), which could explain the observed abundance of large thin bulgeless disks unaffected by major mergers (see Section 4.2), and in those rare mergers between large spirals, tidal dwarf galaxies would be formed and survive more easily (see Section 6.5.4). This could lead to the intriguing possibility that most dwarf galaxies are not primordial but have been formed tidally in these encounters [239]. These populations of satellite galaxies, associated with globular clusters that formed along with them, would naturally appear in (more than one) closely related planes (because a gasrich galaxy pair undergoes many close encounters in MOND before merging, see Section 6.5.3), thereby perhaps providing a natural solution to the Milky Way satellites phasespace correlation problem of Section 4.2. What is more, the densitymorphology relation for dwarf ellipticals (more dE galaxies in denser environments [239]), observed in the field, in galaxy groups and in galaxy clusters could also find a natural explanation.
Actually, the chief problem seems not to be forming structure in MOND, but the danger of overproducing it [341, 401]. The amplitude of the power spectrum is well measured at z = 1091 in the CMB and at z ≈ 0 by surveys like the Sloan Digital Sky Survey. Simulations normalized to the CMB overproduce the structure at z = 0 by a factor of ∼ 2. Given the uncertainty in the parent relativistic theory and hence the appropriate form of the expansion history, this seems remarkably close. Given the nonlinear nature of the theory, MOND could easily have been wrong by many orders of magnitude in this context. Nevertheless, it may be necessary to somehow damp the growth of structure at late times [401]. In this regard, a laboratory measurement of the ordinary neutrino mass might be relevant. Conventional structure cannot form in ΛCDM if m_{ ν } > 0.2 eV [229]. In contrast, some modest damping from a nontrivial neutrino mass might be desirable in MOND, and is also relevant to the CMB and clusters of galaxies (see Section 6.6.4).
In addition to mapping the growth factor as a function of redshift, one would also like to predict the power spectrum of mass fluctuations as a function of scale at a given epoch. It is certainly possible to match the power spectrum of galaxies at z = 0 [401], but because of MOND’s nonlinearity and the uncertainty in the background cosmology, it is rather harder to know if such a match faithfully represents a viable theory. Indeed, a natural prediction of baryondominated cosmologies is the presence of strong baryon acoustic oscillations in the matter power spectrum at z = 0 [267, 127]. Dodelson [127] portrays this as a problem, but as already pointed out in [267], the nonlinearity of MOND can lead to mode mixing that washes out the initially strong signal by z= 0. A more interesting test would be provided by the galaxy power spectrum at high redshift (z ∼ 5). This is a challenging observation, as one needs both a large survey volume and high resolution inspace. The latter requirement arises because the predicted features in the power spectrum are very sharp. The window functions necessarily employed in the analysis of large scale structure data are typically wider than the predicted features. Convolution of the predicted power spectrum with the SDSS analysis procedure [326] shows that essentially all the predicted features wash out, with the possible exception of the strongest feature on the largest scale. This means that the BAO signal detected by SDSS and consistent with ΛCDM [135] could also be interpreted as a confirmation of the prediction [267] of such features^{60} in MOND. However, there is no definitive requirement that the BAO appears at the same scale as observed, or that it survives at all. In relativistic theories such as TeVeS (Section 7.3 and 7.4), damping of the baryonic oscillations can be taken care of by parameters of the theory such as in original TeVeS (Eq. 84, see Figure 3 of [430]) or the c_{ i } coefficients in generalized TeVeS (Eq. 87). In any case, as in standard cosmology, the angular power spectrum of the CMB should be a cleaner probe.
The density of both the baryons and the nonbaryonic cold dark matter are critical to the shape of the acoustic power spectrum. For a given baryon density, models with CDM will have a larger second peak than models without it. Similarly, the third peak is always lower than the second in purely baryonic models, while it can be either higher or lower in CDM models, depending on the mix of each type of mass. Moreover, both parameters were well constrained prior to observation of the CMB [468]: Ω_{ b } from BBN [480] and Ω_{ m } from a variety of methods [116]. Therefore, it seemed like a straightforward exercise to predict the difference one should observe. The most robust prediction that could be made was the ratio of the amplitude of the first to second acoustic peak [265]. For the range of baryon and dark matter densities allowed at the time, ΛCDM predicted a range in this ratio anywhere from 1.5 to 1.9. That is, the first peak should be almost but not quite twice as large as the second, with the precise value containing the information necessary to much better constrain both density parameters. For the same baryon densities allowed by BBN but no dark matter, the models fell in a distinct and much narrower range: 2.2 to 2.6, with the most plausible value being 2.4. The second peak is smaller (so the ratio of first to second higher) because there is no driving term to counteract baryonic damping. In this limit, the small range of relative peak heights follows directly from the narrow range in Ω_{ b } from BBN.
The BOOMERanG experiment [117] provided the first data capable of testing this prediction, and was in good agreement with the noCDM prediction [268]. This result was subsequently confirmed by WMAP, which measured a ratio 2.34 ± 0.09 [345]. This is in good quantitative agreement with the prediction of the noCDM ansatz, and outside the range first expected in ΛCDM. ΛCDM can nevertheless provide a good fit to the CMB power spectrum. The chief parameter adjustment required to obtain a fit is the baryon density, which must be increased: this is the reason for the near doubling of the longstanding value Ω_{ b }h^{2} = 0.0125 [480] to the more recent Ω_{ b }h^{2} = 0.02249 [229].
However, the noCDM ansatz must fail at some point. It could fail outright if the parent MOND theory deviates substantially from GR in the early universe. However, the more obvious [265] points of failure are rather due to the anticipated early structure formation in MOND discussed above. This should lead, in a true MOND theory, to early reionization of the universe and an enhancement of the integrated SachsWolfe effect. Evidence for both these effects are present in the WMAP data [269]. Indeed, it turns out to be rather easy, and perhaps too easy, to enhance the integrated SachsWolfe (ISW) effect in theories like TeVeS or GEA [430, 516]. Nevertheless, early reinioniaztion is an especially natural consequence of MOND structure formation that was predicted a priori [265]. In contrast, structure is expected to build up more slowly in ΛCDM such that obtaining the observed early reionization implies that the earliest objects to collapse were ∼ 50 times as efficient at converting mass to ionizing photons as are collapsed objects at the present time [435].
One prediction of the noCDM anzatz that should not obviously fail is that the third peak should be smaller than the second peak of the acoustic power spectrum of the CMB. In a universe governed by MOND rather than cold dark matter, there is no obvious nonbaryonic mass that is decoupled from the photonbaryon fluid. Therefore, it is a strong expectation that we observe only baryonic damping in the power spectrum, and each peak should be smaller in amplitude than the previous one. Contrary to this expectation, WMAP observes the third peak to be nearly equal in amplitude to the second [442, 229]. This approximate equality of the second and third peaks falsifies the simple noCDM anzatz.
The PLANCK mission should soon report a new and much higher resolution measurement of the CMB acoustic power spectrum. It is conceivable^{62} that improved data will reveal a different power spectrum. A third peak as low as that expected in the noCDM anzatz would be one of the few observations capable of clearly falsifying the existence of cosmic nonbaryonic dark matter. A more likely result is basic confirmation of existing observations with only minor tweaks to the exact power spectrum. Such a result would have little impact on the discussion here as it would simply confirm the need for some degrees of freedom in relativistic MOND theories that can play a role analogous to CDM. However, the uncertainties on the best fit cosmological parameters may become negligibly small. Precise as current data are, cosmology (with the exception of BBN) is still far from being overconstrained. Hopefully, PLANCK data will be sufficiently accurate that they either agree or clearly do not agree^{63} with a host of other observations.
 (i)
Practical falsification of MOND,
 (ii)
Proof of the existence of some form of nonbaryonic matter particles,
 (iii)
An indication of some necessary additional freedom in relativistic parent theories of MOND, playing the role of the nonbaryonic mass in the CMB^{64}.
Perhaps the most intriguing possibility is (iii), that the height of the third peak is providing a glimpse of some new aspect of modified gravity theories. As we have seen, generalizations of GR seeking to incorporate MONDian phenomenology must, per force, introduce either nonlocality (Section 7.10), or new degrees of freedom in local theories. It is at least conceivable that these new degrees of freedom result in the net driving of the acoustic oscillations that is implied by the departure from pure baryonic damping. For instance, Dodelson & Liguori [128] have shown that in TeVeS (Sections 7.3 and 7.4) or GEA (Section 7.7) theories, based on unitnorm vector fields, the growth of the spatial part of the vector perturbation in the course of cosmological evolution is acting as an additional seed akin to nonbaryonic dark matter^{65} (but unlike dark matter, its energy density is subdominant to the baryonic mass). Actually, it has been shown that, with the help of this effect prior to baryonphoton decoupling, it is actually possible^{66} to produce as high a third peak as the second one in TeVeS and GEA theories without nonbaryonic dark matter, but at the cost of leading to unacceptably high temperature anisotropies in the CMB on large angular scales, due to an overenhanced ISW effect [430, 516]. Indeed, when making the effect of the growth of the perturbed vector modes large, one also generates [151, 409, 498] a large gravitational slip (see Section 8.4) in the perturbed FLRW metric (Eq. 116), which in turn leads to enhanced ISW^{67}. For this reason, acceptable fits to the CMB in TeVeS or GEA still need to appeal to nonbaryonic mass [430]. In this case, ordinary neutrinos within their modelindependent masslimit [234] are sufficient, though^{68}. However, the gravitational slip could be able to soon exclude at least some of these models from combined information on the matter overdensity and weak lensing [362, 498]. However, an important caveat is that all of the above arguments are based on adiabatic initial conditions^{69}. While initial isocurvature perturbations are basically ruled out in the GR context, this is not necessarily true for modified gravity theories, so that correlated mixtures of adiabatic and isocurvature modes could perhaps lower the ISW effect and/or raise the third peak [429].
Of course, when the additional “dark fields” of relativistic MOND theories are truly massive (as is the case in some theories), they can be thought of as true “dark matter”, whose energy density outweighs the baryonic one in the early universe: this is the case for the second scalar field of BSTV (Section 7.5), the scalar field of Section 7.6, and of course the dipolar dark matter of Section 7.9. In all these cases, reproducing the acoustic peaks of the CMB is, by construction, not a problem at all (nor erasing the baryon acoustic oscillations in the matter power spectrum contrary to [127]), while the MOND phenomenology is still nicely recovered in galaxies. In the case of BIMOND (Section 7.8), the possible appeal to twin matter could also have important consequences on the growth of structure [316] and, of course, on the CMB acoustic peaks too, although the latter analysis is still lacking. In an initially mattertwin matter symmetric universe, if the initial quantum fluctuations are not identical in the two sectors, matter and twin matter would still segregate efficiently, since density differences grow much faster that the sum [316]. The inhomogeneities of the two matter types would then develop, eventually, into mutually avoiding cosmic webs, and the tensors coming from the variation of the interaction term between the two metrics with respect to the matter metric can then act precisely as the energymomentum tensor of cosmological dark matter [316], besides its contribution to the cosmological constant (see Section 9.1). Finally, the most thoughtprovoking and interesting possibility would perhaps be to explain all these cosmological observations through nonlocal effects (Section 7.10). In any case, it is likely that MOND will not be making truly clear predictions regarding cosmology until a more profound theory, based on first principles and underlying the MOND paradigm, is found.
10 Summary and Discussion
With inert, collisionless and dissipationless DM, making Milgrom’s law emerge requires a huge, and perhaps even unreasonable, amount of finetuning 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 historyindependent 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 a_{0} being a truly fundamental quantity, the strong equivalence principle cannot hold anymore, and local Lorentz invariance could perhaps be spontaneously violated too.

“Velocity curves calculated with the modified dynamics on the basis of the observed mass in galaxies should agree with the observed curves.”
It is now well established that MOND provides good fits to the rotation curves of galaxies (Figure 23 [401, 166]), including bumps and wiggles associated with a baryonic counterpart (Figure 21, Keplerlike law no. 10 in Section 5.2). These fits are obtained with a single free parameter per galaxy, the masstolight ratio of the stars. What makes them most impressive is that the bestfit masstolight ratios, obtained on purely dynamical grounds assuming MOND, vary with galaxy color exactly as one would expect from stellar population synthesis models [42], that are based on astronomers’ detailed understanding of stars. Note that the rotation curves of galaxies are predicted to be asymptotically flat, even though this flatness is not always attained at the last observed point (see Keplerlike law no. 1 in Section 5.2, and last explicit prediction hereafter).

“The relation between the asymptotic velocity and the mass of the galaxy is an absolute one.”
This is the Baryonic TullyFisher relation with \({M_b} = {a_0}GV_f^4\) (see Keplerlike laws no. 2 in Section 5.2). It appears to hold quite generally [272], even for galaxies that we would conventionally expect to deviate from it [165, 279, 276].

“Analysis of the zdynamics in disk galaxies using the modified dynamics should yield surface densities, which agree with the observed ones.”
This states that, in addition to the radial force giving the rotation curve, the motions of stars perpendicular to the disk must also follow from the source baryons (see Section 6.5.3). This proves to be a remarkably challenging observation, and such data for external galaxies are difficult to obtain [44]. To make matters still more difficult, the radial acceleration usually dominates the vertical \(({V^2}/r \gg \sigma _z^2/z)\). This has the consequence that the distinction between MOND and conventional dynamics is not pronounced in regions that are well observed, becoming pronounced only at rather low baryonic surface densities [279]. The vertical velocity dispersions in low surface density regions (see Section 6.5.3) is typically ∼ 8 km/s [25, 241]. This exceeds the nominal Newtonian expectation (typically ∼ 2 km/s for Σ = 1 M_{ ⊙ } pc^{−2}, depending on the thickness of the disk), and is more in accordance with MOND. However, it would require a considerably more detailed analysis to consider this a test, let alone a success, of MOND. The Milky Way (Section 6.5.2) may provide an excellent test for this prediction [50, 378] as more precision data become available.

“Effects of the modification are predicted to be particularly strong in (LSB) dwarf galaxies.”
The dwarf spheroidal satellite galaxies of the Milky Way have very low surface densities of stars, so (see Keplerlike law no. 8 in Section 5.2) are far into the MOND regime. As expected, these systems exhibit large mass discrepancies [477, 427]. Detailed fits to the better observed “classical” dwarfs [8] are satisfactory in most cases (see Section 6.6.2). The “ultrafaint” dwarfs appear more problematic [285], in the sense that their velocity dispersions are higher than expected. This might be an indication of the MONDspecific external field effect (see Section 6.3 and [78]), as the field of the Milky Way dominates the internal fields of the ultrafaint dwarfs. If so, these objects are not in dynamical equilibrium, which considerably complicates their analysis.

Locallymeasured masstolight ratios should show no indication of hidden mass when \(V_c^2/R \gg {a_0}\), but rise beyond the radius where \(V_c^2/R \approx {a_0}\).
We have paraphrased this prediction for brevity (see also Keplerlike law no. 7 in Section 5.2). The test of this prediction is shown in Figures 10, 11, and 14. The predicted effect is obvious in the data with populations synthesis masstolight ratios for the stars [42], or with dynamical masstolight ratios [279] that make no assumption about stellar mass. In HSB spirals, there is no obvious need for dark matter in the inner regions, with the mass discrepancy only appearing at large radii as the acceleration drops below a_{0} (Figure 10).

“Disk galaxies with low surface brightness provide particularly strong tests.”
Low surface brightness means low stellar surface density, which in turns means low acceleration. LSB galaxies are thus predicted to be well into the modified regime (see also Keplerlike law no. 8 in Section 5.2). This was a strong prediction, because few bonafide examples of such objects were known at the time. Indeed, in 1983, when these predictions were published, it was widely thought that nearly all disk galaxies shared a common high surface brightness. One specific consequence of MOND for LSB galaxies is that they should lie on the same BTFR, with the same normalization, as HSB spirals. This was subsequently observed to be the case [517, 443]. There is no systematic deviation from the BTFR with surface brightness (Figure 5), thus contrary to what is naturally expected in conventional dynamics [279, 109]. Another consequence of low surface density is that the acceleration is low (< a_{0}) everywhere. As a result, the mass discrepancy appears at a smaller radius in LSB galaxies, and is larger in amplitude than in HSB galaxies. This effect was subsequently observed (Figure 14 [279]).

“We predict a correlation between the value of the average surface density of a galaxy and the steepness with which the rotational velocity rises to its asymptotic value.”
MOND does not simply make rotation curves flat. It predicts that HSB galaxies have rotation curves that rise rapidly before becoming flat, and may even fall towards asymptotic flatness. In contrast, LSB galaxies should have slowly rising rotation curves that only gradually approach asymptotic flatness (see also Keplerlike law no. 8 in Section 5.2). Both morphologies are observed (Figure 15). The expected connection between dynamical acceleration and the surface density of the source baryons is illustrated in Figures 9 and 16.
Observational tests of MOND.
Observational Test  Successful  Promising  Unclear  Problematic 

Rotating Systems  
solar system  X  
galaxy rotation curve shapes  X  
surface brightness ∝ Σ ∝ a^{2}  X  
galaxy rotation curve fits  X  
fitted M_{*}/L  X  
TullyFisher 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 dynamical friction  X  X  
tidal dwarfs  X  
Spheroidal Systems  
star clusters  X  
ultrafaint dwarfs  X  
dwarf Spheroidals  X  
ellipticals  X  
FaberJackson relation  X  
Clusters of Galaxies  
dynamical mass  X  
masstemperature slope  X  
velocity (bulk & collisional)  X  
Gravitational Lensing  
strong lensing  X  
weak lensing (clusters & LSS)  X  
Cosmology  
expansion history  X  
geometry  X  
BigBang 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 reionization  X 
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 rotationallysupported 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 masstolight ratio systematically varies with the circular velocity is a direct consequence of Milgrom’s law (Keplerlike 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 rotationallysupported 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 11), 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 slowlyvarying interpolation functions [62]. Other tests may yet prove possible [37, 314], but, as they are strongfield 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 solarsystem 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 [77]. Indeed, it gives good reason [299] for the observed maximum in the distribution of disk galaxy surface densities at ∼ Σ_{†} = a_{0}/G (Freeman’s limit: Figure 8 and Keplerlike 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. Highersurfacedensity 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 overstabilize disks. Features like bars and spiral arms are a natural result of disk selfgravity. Conventionally, large halotodisk mass ratios suppress the growth of such features, especially in LSB galaxies [291]. Yet such features are present^{70}. 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 galaxyinteraction 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, nonstationary 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 FaberJackson relation (Keplerlike law no. 3 in Section 5.2, Figure 7) is also very natural to MOND [383, 395].
The largest gravitationallybound 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 presentday 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 nonbaryonic collisionless cold dark matter required by cosmology. There is really nothing about the cluster data that requires the excess mass to be nonbaryonic, as long as it behaves in a collisionless way. There could for instance be baryonic mass in some compact nonluminous 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 [9, 13] have been suggested as possible forms of dark matter that might provide an explanation for the missing mass in clusters. Both are nonbaryonic, but as they are hot DM particle candidates, neither can constitute the cosmological nonbaryonic 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 nonlocal effects (see Section 7).
There are other aspects of cluster observations that are more in line with MOND’s predictions. Clusters obey a masstemperature relation that parallels the M ∝ T^{2} ∝ V^{4} prediction of MOND (Figures 39 and 48) 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 48 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 large^{71} as to be highly improbable in ΛCDM (occurring with a probability of ∼ 10^{−10} [249]). In contrast, large collision velocities are natural to MOND [16]. 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 [11] 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 largerscale 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 (noncovariant) 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 RobertsonWalker 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 nonlinear. Therefore, it is unclear whether MONDmotivated relativistic theories will inevitably match the observed galaxy power spectrum, a possible problem being how to damp the baryon acoustic oscillations [127, 430]. 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 nonlinearity 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., nonbaryonic dark matter such as, e.g., light sterile neutrinos [9]) 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 nonbaryonic 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 finetuned 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 eVscale sterile neutrinos [9]), MOND did motivate the prediction [265] of two aspects of the CMB that were surprising in ΛCDM (see Section 9.2). The amplitude ratio of the firsttosecond peak in the acoustic power spectrum was outside the bounds expected ahead of time by ΛCDM for 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, 269]. Similarly, the rapid formation of structure expected in MOND leads naturally to an earlier epoch of reionization 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 MONDinspired theories, the overall interpretation of the CMB is debatable. While the existence of nonbaryonic 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 finetuned 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 nonbaryonic cold dark matter together with finetuned 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 finetuning 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 strongfield 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 weakfield 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 weakfield limit, since both phenomena would then simply reflect discrepancies with the predictions of GR when \(\Lambda \sim a_0^2\) 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 finetuning that will have to be explained naturally. In any case, the existence of a characteristic acceleration a_{0} (Figure 48) playing various different roles in many seeminglyindependent 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.
Footnotes
 1.
Up to now, all the darkmatterparticle candidates still elude both direct and indirect nongravitational detection.
 2.
However, a way to effectively reproduce an apparent universal force law from an exotic dark component could be to enforce an intimate connection between the distribution of baryons, the dark component, and the gravitational field through, e.g., a fifth force effect. This possibility will be extensively discussed in Section 7, notably Section 7.9
 3.
The first four sections provide the observational evidence for the MOND phenomenology through the different appearances of a_{0} in galactic dynamics, but they are actually independent of any specific theory, while the reader more specifically interested in MOND per se could go directly to Section 5.
 4.
Arguably, a nonstatic, expanding or contracting Universe was an a priori prediction of general relativity in its original form, lacking the cosmological constant.
 5.
However, the WIMP miracle seems to fade away with modern particle physics constraints [23].
 6.
The simplest WIMPs are their own antiparticle.
 7.
In addition, the timeaveraged value of the deceleration parameter q over the present age of the Universe is quite consistently 〈q〉 = 0 [473], another currently unexplained coincidence.
 8.
c = G = ħ = 1.
 9.
We have that \(\Lambda \sim a_0^2/{c^2}\) if expressed in inverse timesquared or \(\Lambda \sim a_0^2/{c^4}\) if expressed in inverse lengthsquared (more precisely, the natural scale associated with the cosmological constant is \({a_0} \approx ({c^2}/2\pi)\sqrt (\Lambda/3))\). Another way of expressing this coincidence is to say that predictions of GR from visible matter alone always break down for physics involving a lengthscale constant on the order of the Hubble radius l ∼ Λ^{−1/2} ∼ c^{2}/a_{0}. This scale l could perhaps play a similar role to the Planck scale l_{ P } [233, 43], at the other end of the ladder (as we have l ≈ 10^{60}l_{ P }). However, this is not the length at which the modification would be seen, exactly as quantum mechanics does not depart from classical physics at a given length.
 10.
As we shall see (Section 5 and 6), MOND was constructed to predict a relation \({a_0} = V_f^4/(GM)\) for a point mass M (however, note that the slope of 4 is a pure consequence of the acceleration base; it is not possible to get an arbitrary slope from such an idea). Since spiral galaxies are not point masses, but rather flattened mass distributions that rotate faster than the equivalent spherical mass distribution [52], the empirical acceleration a is close to but not identical to a_{0} in MOND. The geometric correction is about 20% so that a_{0} = 0.8a [272].
 11.
The factor 10 arises from the commonly adopted definition of the virial radius of the dark matter halo at an overdensity of 200 times the critical density of the Universe [332].
 12.
Note that [180] claimed to measure a slope of 3 for the BTFR, but they relied on unresolved linewidths from single dish 21 cm observations to estimate rotation velocity rather than measuring V_{ f } from resolved rotation curves. Linewidths give a systematically different estimate of the slope of the BTFR than V_{ f }, even for the same galaxies [277, 340, 475], and they cannot be related at all to the circular velocity of the potential at the virial radius, nor to the prediction of MOND (Section 5 and 6).
 13.
The difference in phase space between gas and dark matter also prevents the accretion of tidal gas onto any dark matter subhalos that may be present. It does not suffice for a tidal tail to intersect the location of a subhalo in coordinate space, they must also dock in velocity space. The gas is moving at the characteristic velocity of the entire system (typically ∼ 200 km s^{−1}), which, by definition, exceeds the escape speed of typical subhalos (usually < 100 km s^{−1}). Therefore, the odds of capture are effectively zero unless the tail and subhalo happen to be on very nearly the same orbit initially, which is itself very unlikely because of the initial difference in their phase space distribution.
 14.
Note that this correlation with acceleration was looked at notably because it was pointed to by Milgrom’s law (see Section 5).
 15.
The Pioneer anomaly has an amplitude on the order of ∼ 10^{−9} m s^{−2} but appears at a location in the solar system where the total gravitational acceleration is ∼ 10^{−6} m s^{−2}. Thus, the discrepancy in Figure 11 is (V/V_{ b })^{2} ≈ 1.001.
 16.
Note that such wiggles are often associated with spiral arm features (the existence of which in LSB galaxies being itself challenging in the presence of a massive dark matter halo, see Section 4.2), and hence associated with noncircular motions. It is conceivable that such observed wiggles are partly due to these, but the effect of local density contrasts due to spiral arms on the tangential velocity should be damped by the global effect of the sphericaldarkmatter halo, which is apparently not the case.
 17.
Note that many of these relations were scrutinized during the last 30 years because they were pointed to by Milgrom’s law. Thus, this law has already achieved the important role of a theoretical idea, i.e., to point and direct observations and their arrangement
 18.
Of course, there is also a natural length scale associated with this acceleration constant, I = c^{2}/a_{0}, but this length scale will enter the modification nonlinearly, and is thus not the length at which the modification would be seen in galaxies, as it is rather on the order of the Hubble radius
 19.
Note that the denominator 2πG comes from integrating the phantom dark matter density along a vertical line as per [313], which leads to a slightly smaller characteristic surface density for phantom dark matter than the defining Freeman limit in the 6th law.
 20.
Note that the main motivation for modifying dynamics is not to get rid of DM, but to explain why the observed gravitational field in galaxies is apparently mimicking a universal force law generated by the baryons alone. The simplest explanation is, of course, not that DM arranges itself by chance to mimic this force law, but rather that the force law itself is modified. Note that at a fundamental level, relativistic theories of modified gravity often will have to include new fields to reproduce this force law, so that dark matter is effectively replaced by “dark fields” in these theories, or even by dark matter exhibiting a new interaction with baryons (one could speak of “dark matter” if the stressenergy tensor of the new fields is numerically comparable to the density of baryons): this makes the confrontation between modified gravity and dark matter less clear than often believed. The actual confrontation is rather that between all sorts of theories embedding the phenomenology of Milgrom’s law vs. theories of DM made of simple selfuninteracting billiard balls assembling themselves in galactic halos under the sole influence of unmodified gravity, theories, which currently appear unable to explain the observed phenomenology of Milgrom’s law.
 21.
Generally covariant theories approaching these classical theories in the weakfield limit will then also be classified under this same MOND acronym, even if they really are Modified Einsteinian Dynamics (see Section 7)
 22.
The Newtonian mass density also satisfies the continuity equation ∂μ/∂t + ∇.(μv) = 0.
 23.
In general relativity, the first two terms ∫ ρ(v^{2}/2 − Φ_{ N })d^{3}x dt are lumped together into the matter action (also containing the rest mass contribution in GR), and the last term is generalized by the EinsteinHilbert action.
 24.
Let us note in passing that it would not be the first time that the kinetic action would be modified as special relativity does just this too, changing for a single particle mυ^{2}/2 → −mc^{2}γ^{−1} in S_{kin} (where \(\gamma (\upsilon) = 1/\sqrt {1  {{(\upsilon/c)}^2})}\), leading for a moving body to a redefinition of the effective mass as m_{eff} = mγ(υ). With this analogy in mind, a rather simplified view of the Lorentzbreaking modification of inertia needed in order to reproduce MOND would be that m_{eff} ≃ mμ(a), where a is the amplitude of the acceleration with respect to an absolute preferred inertial frame.
 25.
Such nonlocal theories, which also have to be nonlinear (like any MOND theory) are not easy to construct, and there is presently no real fullyfledged theory, which has been developed in this vein, although hints in this direction are summarized in Section 7.10.
 26.
 27.
This is similar to the Palatini formalism of GR, where the present auxiliary acceleration field is replaced by a connection
 28.
 29.
In principle, α can be slightly larger, but if α ≫ 1, then in the range of gravities of interest for galaxy dynamics (between 0.1a_{0} and a few times a_{0}) the scalar field contribution s is too small to account for the MOND effect, or said in another way, the corresponding Milgrom μfunction would deviate significantly from μ(x) = x (i.e., μ(x) > x, so that there would be less modification to the Newtonian prediction).
 30.
In principle, one could make A = α^{−1} as small as desired in the αfamily, by not limiting α to the range between 0 and 1, but passing solar system constraints would require α > 20, which would cancel the MOND effect in the range of interest for galaxy dynamics.
 31.
Note that, among the freedom of choice of that function, one could additionally even imagine that the μfunction is not a scalar function but a “tensor” μij such that the modification becomes anisotropic and the modified Poisson equation becomes something like ∂_{ i }[μ_{ ijgj }] = 4πGρ.
 32.
It is interesting to note that different MOND theories offer (very) different answers to the generic question “acceleration with respect to what?”. For instance, in the MONDfromvacuum idea (see [304] and Section 7.10), the total acceleration is measured with respect to the quantum vacuum, which is well defined. In BIMOND (Section 7.8) it is the relative acceleration between the two metrics, which is also well defined through the difference of Christoffel symbols.
 33.
Thus, a Cavendish experiment in a freely falling satellite in Earth orbit would return a Newtonian result in MOND.
 34.
For instance, using the “simple” function μ(x) = x/(1+ x) in Eq. 59 would lead to \(g = [({g_N}{a_0} + {g_N}{g_e}  2{g_e}{a_0}  g_e^2) + \sqrt {(2{g_e}{a_0} + g_e^2  {g_N}{a_0}  {g_N}{g_e}) + 4{g_N}{{({a_0} + {g_e})}^3}]}/[2({a_0} + {g_e})]\)
 35.
 36.
 37.
 38.
 39.
If one assumes that a lot of dark baryons are present in the form of molecular gas, one can add another free parameter in the form of a factor multiplying the gas mass[460]. The, good MOND fits can still be obtained but with a lower value of a_{0}.
 40.
However, the masstolight ratio is not really a constant in galaxies. Thus, Figure 22 gives an example of a rotation curve fit (to the Local Group galaxy M33), where the variation of the masstolight ratio according to the colorgradient has been included, even improving the MOND fit.
 41.
Separable models have also been investigated in [97].
 42.
The conventional baryon fraction of clusters increases monotonically with radius [426], only obtaining the cosmic value of 0.17 at or beyond the virial radius. Therefore, one might infer the presence of dark baryons in cluster cores in ΛCDM as well as in MOND.
 43.
If the action has the units of ħ, the factor in front of the gravitational action is then c^{3}/16πG. And if one wishes to include a cosmological constant Λ, the integral then rather reads \(\int {{d^4}x\sqrt { g} (R  2\Lambda)}\).
 44.
With this signature, the propertime is defined by \(d{\tau ^2} =  {g_{\mu \nu}}d{x^\mu}d{x^\nu}\).
 45.
Note that, at 1PN, this weakfield metric can also be written as \({g_{00}} =  {e^{2\Phi/{c^2}}},{g_{ij}} = {e^{2\Psi/{c^2}}}{\delta _{ij}}\). Note also that Taylor expanding Eq. 70 yields S_{pp} = ∫ m(v^{2}/2 − Φ_{ N } − c^{2}) dt, so that the sum of the classical kinetic and internal actions for a point particle (see Eq. 13) are now lumped together into the matter action.
 46.
The derived lensing and dynamical masses are typically very close to each other but the data are not yet precise enough to ascertain that they are exactly identical.
 47.
The frame associated to the Einstein metric is called the “Einstein frame” as opposed to the “matter frame” or “Jordan frame”, associated to the physical metric.
 48.
kessence fields have also recently been reintroduced as possible dark energy fluids, that could also drive inflation [20, 21, 92]. This name comes from the fact that their dynamics are dominated by their kinetic term f(X) (in the case of RAQUAL, there is no potential at all), contrary to other dark energy models such as quintessence, in which the scalar field potential plays the crucial role.
 49.
Expressed in terms of U_{ μ } and its congugate momenta \({P^\mu} = \partial L/\partial {{\dot U}_\mu}\).
 50.
It is also important to remember that some interpolation functions (Section 6.2) are already excluded by solar system tests, and thus, it is useless to exclude these over and over again.
 51.
 52.
In the case of TeVeS and GEA theories, the dark fields do not really count as dark matter because their energy density is subdominant to the baryonic one.
 53.
This is to be contrasted with the timelike nature of TeVeS and GEA vector fields in the static weakfield limit.
 54.
And the current J^{ μ } is conserved, i.e., ∇_{ μ }ρJ^{ μ } = 0
 55.
It can be shown that only the projection perpendicular to the fourvelocity enters the field equations deduced from the action of Eq. 100. Thus, the dipole moment is always fully spacelike.
 56.
 57.
However, by this we do not mean that the MOND lensing can be computed from the projected surface density on the lensplane as in GR, because the convergence parameter (Eq. 113 below) is not a measure of the projected surface density anymore. This is also sometimes referred to as the “thinlens approximation” in GR, and is not valid in MOND: two lenses with the same projected surface density can have different convergence parameters, because lensing also depends strongly on the distribution of the source mass along the lineofsight in MOND.
 58.
 59.
Note, however, that this is not always the case in colliding clusters: Abell 520 actually provides a counterexample to the bullet cluster in which the mass peaks indicated by weak lensing do not behave as collisionless matter should [210].
 60.
Even if BAO features are present at high redshift in MOND, it is not clear that low redshift structures will correlate with the ISW in the CMB as they should in conventional cosmology because of the late time nonlinearity of MOND.
 61.
Perhaps the most famous modern example of confirmation bias is in measurements of the Hubble constant [466], where over many years de Vaucouleurs persistently found H_{0} ≈ 100 km s^{−1} Mpc^{−1} while Sandage persistently found H_{0} ≈ 50 km s^{−1} Mpc^{−1}. Then, as now, there was a conflation of data with theory: the lower value of H_{0} was more widely accepted because it was required for cosmology to be consistent with the ages of the oldest stars.
 62.
At ℓ ≈ 800, the third peak is only marginally resolved by WMAP. This scale is comparable to a single (frequencydependent) beam size, and, as such, is extraordinarily sensitive to corrections for the instrumental point spread function [404].
 63.
Determining agreement between independent observations requires that we believe not just the result (e.g., the value of H_{0} from direct distance measurements) but also its uncertainty. The latter has always been challenging in astronomy, and the history of cosmology is replete with examples of results that were simply wrong. While we may have entered the era of precision cosmology, we have yet to reach an era when data are so accurate that we can hope to challenge cosmology with falsification if, for example, PLANCK data require H_{0} < 60 km s^{−1} Mpc^{−1}, while galaxy distances require H_{0} > 70 km s^{−1} Mpc^{−1}.
 64.
The third possibility actually means either nonlocal effects in nonlocal theories (Section 7.10), or the effect of additional fields in local modified gravity theories. The important difference with CDM is that these fields are not simply representative of collisionless massive particles, that their behavior is determined by the baryons in static configurations, and that they can be subdominant to the baryonic density. In theories where their energy density dominates that of baryons, these new fields then really act as dark matter in the early universe, which is also a possibility (see Section 7.6 and 7.9)
 65.
In TeVeS, the perturbations of the scalar field also play an important role in generating enhanced growth [146].
 66.
Ferreira’s talk, Alternative Gravities and Dark Matter Workshop, Edinburgh, April 2006.
 67.
The ISW effect can be cast as the integral of − (Φ +Ψ)′ + 2Φ’, thus involving both a gravitational slip part and a growth rate part.
 68.
 69.
This means that, on surfaces of constant temperature, the densities of the various components (e.g. baryons, neutrinos, additional dark fields) are uniform, and that these components share a common velocity field.
 70.
 71.
The observed shock velocity of ∼ 4700 km/s is thought to be enhanced by hydrodynamical effects. The collision velocity is improbable after a substantial (∼ 1700 km/s) correction for this [258].
Notes
Acknowledgements
The authors are grateful for stimulating conversations about the dark matter problem and MOND over the years, as well as for comments and help from: Garry Angus, JeanPhilippe Bruneton, Martin Feix, Gianfranco Gentile, HongSheng Zhao, Moti Milgrom, Rodrigo Ibata, Pavel Kroupa, Olivier Tiret, Dominique Aubert, Jacob Bekenstein, Olivier Bienaymé, James Binney, Luc Blanchet, Christian Boily, Greg Bothun, Laurent Chemin, Françoise Combes, Jörg Dabringhausen, Erwin de Blok, Gilles EspositoFarése, Filippo Fraternali, André Füzfa, Alister Graham, Hosein Haghi, Anaëlle Halle, Xavier Hernandez, Jean Heyvaerts, Alex Ignatiev, Alain Jorissen, Frans Klinkhamer, Joachim Köppen, Rachel Kuzio de Naray, Claudio Llinares, Fabian Lüghausen, João Magueijo, Mario Mateo, Chris Mihos, Ivan Minchev, Mustapha Mouhcine, Carlo Nipoti, Adi Nusser, Marcel Pawlowski, Jan PflammAltenburg, Tom Richtler, Paolo Salucci, Bob Sanders, James Schombert, Ylva Schuberth, Jerry Sellwood, Arnaud Siebert, Christos Siopis, Kristine Spekkens, Rob Swaters, Marc Verheijen, Matt Walker, Joe Wolf, Herve Wozniak, Xufen Wu, and many others. We also thank Frédéric Bournaud, Chuck Bennett, Douglas Clowe, Andrew Fruchter, Tom Jarrett, and, again, Garry Angus and Olivier Tiret, for allowing us to make use of their figures. We finally thank Clifford Will for inviting us to write this review. We acknowledge the support of the CNRS, the AvH foundation, and the NSF grant AST 0908370, and both acknowledge hospitality at Case Western Reserve University, where a substantial part of this review has been written.
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