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 matter-energy 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 mass-energy as they should if our understanding of gravity is complete. Thus, this indicates either (i) the presence of unseen (and yet unknown) mass-energy, 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  (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 , 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 itself1. 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 law2.
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 non-baryonic 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 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  (see also ).
Here, we first review the basics of the dark matter problem (Section 2) as well as the basic ingredients of the present-day 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 ). Up to that point, the challenges presented are purely based on observations, and are fully independent of any alternative theoretical framework3. 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 weak-acceleration regime , 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 generally-covariant 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.
Living Rev. Relativity 15, (2012), 10
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