Modified Gravity theory (MOND) is a simple empirical relationship that has been predictive (most recently here) at explaining gravitational dynamics without dark matter at galactic scale, although it understates "dark matter" effects at galactic cluster scales. It predicts not just velocity dispersion of objects in galaxies but subtle effects like the impact of proximity to a host galaxy on dwarf galaxy behavior. There are good reasons to doubt that its mechanism is correct and that a more conventional dark matter theory is the right mechanism for causing these effects.
But, the great predictive success of a very simple, one parameter MOND theory over very large data sets and involving new kinds of data not used to generate the theory long in advance, implies that it must be possible to derive the MOND rule at the proper galaxy level scale from any correct dark matter theory. Likewise, if a simple one parameter formula can explain all of that data, any dark matter theory must itself be very simple. The simply theory is obviously flawed in some respects (e.g. in the original version it is not relativistic). But, it can be generalized without losing its essential features (e.g. in the TeVeS formulation that is fully relativistic).
It is also possible that MOND, "dim matter" and some kind of "cluster dark matter" that is abundant in galactic clusters, but almost absent everywhere else could be at work.
Another attractive feature of MOND is that the dark matter particles that particle physics was supposed to provide as dark matter candidates have not been detected. But, if MOND is correct, we don't need them.
There are a variety of ways to work MOND effects into modifications of general relativity. Some flow from the observation that the MOND constant has a strong coincidence with the size of the universe, suggesting that MOND may arise from the suppression of gravity waves with amplitudes larger than the size of the universe itself.
UPDATE August 30, 2013:
The observation that MOND works and predictive is more than an observation of a mere coincidence or even as I noted before an strong indication that any dark matter mechanism, if there is a dark matter mechanism is very likely very simple because the MOND theory itself is (although it is possible that the complex bits are simply small contributions to the overall result in the same way that the general relativity corrections to Newtonian gravity, while very deep and complex are usually negligible).
But, the fact that MOND works and predictive implies something else about the correct theory that produces this phenomenological relationship. While correlation does not imply causation, correlation does imply that some cause, direction unknown and possibly indirect, causes that correlation. Robust and predictive correlations happen for a reason even if that reason is not a direct causal relationship between the two data sets.
What is MOND?
The MOND hypothesis is that there is a functional relationship between the gravitational fields that would be generated by luminous matter in a galaxy and the "dark matter" effects in that galaxy that are observable only in the parts of the luminous matter gravitational field that are weak which is defined as having gravitational acceleration below the MOND acceleration constant a0. MOND argues that gravity gets weaker according to a conventional 1/r2 law (where r is the distance between the two objects which are attracted to each other by gravity) in fields stronger than a0 and according to a "new physics" 1/r relationship in fields weaker than a0. An ad hoc interpolation function is used to estimate the force of gravity around the transitional field strength and data don't allow meaningful ways to distinguish between the alternative transition formulas.
Because GMm/r2 << G'Mm/r where G' is the constant that produces the MOND gravity prediction in the limit as r >> r at a0, the simplest interpolation is simply to assume that MOND gravity equals Newtonian gravity + G'Mm/r gravity where the second term is too small to discern experimentally in gravitational fields that are strong relative relative to a0= approximately 1.2*10^-10 ms^-2, and the first term is too small to discern relative to the second experimentally in gravitational fields that are weak relative to a0.
What does this imply?
One of the most profound implications of the fact that MOND works and is predictive is that there is a direct and reasonable precise functional relationship between the input into MOND's black box formula, the amount and distribution of luminous matter in a galaxy, and the output, which is the "dark matter" effects that are observed empirically in a galaxy.
This means that in any dark matter theory that accurately replicates reality, the distribution of dark matter particles in the dark matter halo of a galaxy must be functional related to the amount and distribution of luminous matter in that galaxy.
There are several ways that this could be possible. To illustrate this point, here are three broad kinds of scenarios that could cause this to be true. I marginally favor the first, although I don't rule out the second. The third, I consider to be very unlikely, but include it for completeness.
First, it could be that galaxies differ from each other in a very simple, more or less one dimensional way as a result of the way that galaxies evolve. Galaxies of a particular mass may always have a particular or one of a couple of particular luminous matter distributions and any factor that impacts how a galaxy of a particular size evolves impacts the distribution of dark matter in that galaxy in a way that corresponds to the distribution of luminous matter in that galaxy. Thus, the MOND relationship between the galaxy's luminous matter distribution and its dark matter halo distribution arises because the evolution of both kinds of matter distributions is a process that in each case is almost entirely gravity dominated and is shared by all of the matter luminous and dark in a given galaxy. In this process, Newtonian gravitational effects predominate over additional general relativistic gravitational effects, and this very simple gravitational law produces very simple and characteristic distributions of matter than can be summed up in the empirical MOND relationship that is observed. Deriving the MOND relationship from this process may take some pretty clever analytical modeling of the evolution of galaxies that exhibits shrewd understanding of how this process can be drastically simplified without loss of significant accuracy.
In particular, there is a fair amount of evidence to suggest that inferred dark matter halo shapes are strongly related to the shape of a galaxy's inner bulge, but are fairly indifferent to the distribution of matter at the fringe of a galaxy. The shape of a galaxy's inner bulge, in turn is largely a function of the nature of a galaxy's central black hole. If the distribution of the luminous matter in a galaxy and the distribution of the dark matter in a galaxy are largely a function of the nature of the central black hole of the galaxy, then it would follow that luminous matter distributions in a galaxy and dark matter distributions in a galaxy should be functionally related to each other. Moreover, is a central black hole in a galaxy of a given mass is pretty much like every other central black hole in a galaxy that has the same mass, then the distribution of both luminous matter and dark matter in galaxies should both be a function of a single number - the mass of the central black hole of the galaxy.
One version of this kind of scenario is one in which apparent "dark matter" effects are actually driven by ordinary "dim matter" emitted by the central black hole mostly in the "upward" and "downward" directions of the axis of rotation of that central black hole and the galaxy that arises around it. A 1/r relationship between force and distance is precisely the relationship one would expect in a simple Newtonian gravitational scenario in which there is a long, narrow, axial distribution of dim matter in both directions from the central black hole of a galaxy. If the axial distribution of ordinary "dim matter" is long enough and coherent enough that is generates its own 1/r gravitational field to a distance at least as great as the most distant star for which the galaxy's gravitational influence can be observed by an astronomer, then this would generate apparent dark matter effects that approximately follow the phenomenological MOND law.
The combined distribution of luminous and non-luminous matter in a galaxy in the scenario discussed above would look something like the image above, but with thinner and longer extensions up and down out of its axis containing matter and energy that is in rapid motion away from the galaxy.
It should be fairly elementary, moreover, for anyone with a year or two of calculus based physics under their belt to use the MOND constant a0 to calculate the characteristic ratio of axial dim matter to galactic ordinary matter in such a scenario (I could do it this weekend if I could find the time in an hour or two). With a few additional data points about the most distant stars that have been observed to experience MOND-like effects in the vicinity of a galaxy one could also fairly easily establish a minimum length of this axial dim matter and the amount of mass per linear distance of axial dim matter that would be anticipated in a typical galaxy, although any bound on the width of this axial mass distribution would be fairly weak. Since there are at least two different processes observed by astronomers by which black holes are known to emit matter and energy in a more or less axial direction and much of that matter is "dim" and the speed of the emitted matter and emitted energy and the minimum age of a galaxy can be determined to within reasonable bounds, the extent to which known processes could account for axial dim matter giving rise to MOND-like effects wouldn't be too hard to estimate, and the amount of axial "dim matter" that would necessarily have a source in some other unknown form of black hole emissions could also be estimated fairly accurately. It wouldn't be surprising if the sum of the total amount of axial dim matter in the universe resolved much of the "missing baryon" problem - that the number of baryons in the universe according to widely held baryongenesis hypotheses is smaller than the number that are present in all observed luminous matter by a substantial amount without giving rise to any notable cosmological effects that have been attributed to this missing baryonic matter.
Of course, given what my weekend looks like - violin supply store, CostCo, bank deposits, working on my cases the weekend, writing course packs, buying groceries, BBQing, getting someone to a tennis lesson, weeding, mulching, fertilizing, laundry, etc., the efforts of anyone else interested in doing so and posting the results in the commons would be welcome. Scientific discoveries can't be patented and I would love to know the answer and have no deep need to be the one who finds it.
This black hole emitted matter unaccounted for by known processes could be created by the extreme conditions that exist only the large central black holes in the center of galaxies (which would explain why we can't produce this kind of matter in particle accelerators), or it could simply be ordinary matter that does coalesce into stars or other large objects when emitted from a black hole in this way because it is emitted in a diffuse spray of fast moving particles whose speed and common direction prevent them from gaining the critical mass necessary to combine into compact objects that astronomers can observe.
Perhaps astronomers looking for this very particular and well defined kind of dim matter signature could find a way to measure it in some natural experiment arrangement of stellar objects somewhere among the billions and billions of stars in the universe that we can observe on all manner of wavelengths.
Any such process would, by definition, produce neither non-baryonic dark matter, nor ordinary dim matter that ends up in the galaxy's disk of rotation. So, direct dark matter detection experiments conducted in the vicinity of Earth which is in the plane of the Milky Way's disk are doomed to fail if this hypothesis is correct, because in this model, there is no dark or dim matter in that part of the galaxy.
This would also explain why estimated cosmological amounts of dark matter are on the same order of magnitude as estimated cosmological amounts of ordinary matter, another great unsolved question in physics.
In any case, if the missing matter, whether in the form of "novel" dark matter particles of an unknown type, or merely "dim matter" has a distribution that is driven by the same central black hole gravitational effects that drive the distribution of luminous matter in galaxies, the key to reconciling MOND theories and dark matter theories would be at hand.
It is not clear, however, that such a theory would adequately fill the role that dark matter plays in the highly predictive six parameter lamdaCDM model of cosmology, or would be consistent with bottom up galaxy formation models that have been highly successful in 2 keV warm dark matter scenarios that help address problems with the cold dark matter model like "cuspy halos" and the "missing satellites" problem. This hypothesis could create as many new problems as it solves for the dark matter paradigm.
The warm dark matter literature is surprisingly devoid is simple diagrams like the one above illustrating the inferred shape of the Milky Way's dark matter halo in one recent study. But, this illustration is closer to the conventionally expected warm dark matter halo shape. The dark matter paradigm favors structures that are blob-like rather than cylindrical for dark matter halos because it is hard to make nearly collisionless particles with significant kinetic energy that interact primarily only through gravity form more compact structures. The differential effects of the central mass of the galaxy prevent the dark matter halos from behaving like an ideal gas, but only modestly.
The non-spherical shape of the halo, however, is critical to generating the apparent 1/r gravitational field strengths that are observed.
(It is also worth noting that the roughly 2 keV sterile neutrino-like warm dark matter particles that seem to be the best fit to the empirical data within the dark matter paradigm are virtually undetectable in existing direct dark matter detection experiments which are designed to observe weakly interacting dark matter particles with masses in the GeV or heavier mass range.)
A result like this that involves only ordinary "dim matter" however, would be a huge blow to physicists longing for "new physics." It would explain the biggest unsolved problem in physics when it comes to the fundamental laws of physics and their observable effects using only a deeper understanding of processes that occur entirely according to already well understood "old physics." The biggest empirical arrow pointing towards undiscovered types of stable fundamental particles would turn out to have been a mere mirage.
Without the "crutch" of some sort of theory to explain dynamically the evolution of dark or dim matter halo shapes in galaxies parallel to luminous matter distributions in those galaxies, no dark matter theory can be considered complete.
Second, the MOND law could, for whatever, reason actually constitute the true law of gravity when suitably formulated in a general relativistic form (something that has actually been done successfully in several different varieties of proof of concept efforts). As noted above, this would call for some kind of quantum gravity theory or perhaps something related to the impact of a bounded universe of finite size on the way that gravity waves behave.
This would be exciting news for quantum gravity researchers and bad news for particle physics theorists. A 1/r relationship would quite plausibly derive from some process that reduced the effective dimensionality of space-time from three spatial dimensions to two. Perhaps, for example, due to quantum entanglement of distant points between which a particle has traveled or because gravity models have underestimated the gravitational effect of the angular momentum of a spinning galaxy due to some subtle flaw in the normal formulation of general relativity or the way that this formulation is inaccurately applied in models of complex massively many bodied systems like galaxies.
Of course, in particular, this would also be bad news for direct dark matter detection experiments because in this scenario there is no dark matter to detect anywhere except possibly in galactic clusters - all of which are a very, very long way from planet Earth making direct detection of cluster dark matter virtually impossible. Making sense of anomalous gravitational effects that might be due to dark or dim matter in galactic clusters is hard. This is because the structure and non-luminous ordinary matter content of galactic clusters is far less well understood and is far more complex, than the structure and non-luminous ordinary content of ordinary individual spiral, elliptical and dwarf galaxies.
This mechanism for a MOND theory also directly and transparently explains why it doesn't work as well in galactic clusters, whether or not "cluster dark matter" exists. The MOND relationship, in any variation of this hypothesis flows from the parallel evolution processes that are more or less the same for any one given galactic central black hole of a given size, it makes sense that these relationships might not hold for a system with many galactic central black holes in close proximity to each other and different typical ages in the galaxy formation process. Galactic clusters may be profoundly more complex to such an extent that no simple model like MOND can explain them.
Third, there could be a non-gravitational interaction between luminous matter and dark matter that causes dark matter halos to be distributed in a particular way relative to luminous matter. For example, the flux of photons out of a galaxy is roughly proportional to the Newtonian component of the gravitational field of the luminous matter in that galaxy at any given distance from the galaxy. So, if dark matter had very weak electromagnetic interactions with the outgoing flux of photons, this could produce a dark matter distribution that tracks the distribution of luminous matter in a system, while still having a character as collisionless, non-self-interacting particles. Of course, since photon flux, like graviton flux has a 1/r2 relationship to distance from the luminous matter source, this doesn't easily explain a 1/r MOND effect. Also, the photon flux generated by a star is not all that perfectly related to the mass of the star generating the flux, so far as I know (more accurately, I have no idea one way or the other how tight the relationship is between photon flux and stellar mass). Perhaps, at long distances, the geometry of a galaxy impacts this flux somehow in a manner different than for graviton flux.
This kind of explanation would be a field day for particle physicists, because no known fundamental particle has this kind of interactions. I don't see it as a likely option, but one should consider all possibilities for unexplained phenomena for sake of completeness.
Dark matter of this variety ought to be highly amenable to detection by a model driven direct dark matter detection experiment, although existing direct dark matter detection experiments, which involve a very different paradigm and model, might be useless at detecting it.