In order to ensure a smooth transition between the two regimes and , Milgrom’s law is written in the following way:

where the interpolating function Written like this, the analogy between Milgrom’s law and Coulomb’s law in a dielectric medium is clear, as noted in [56]. Indeed, inside a dielectric medium, the amplitude of the electric field generated by an external point charge located at a distance obeys the following equation: where is the relative permittivity of the medium, and can depend on . In the case of a gravitational field generated by a point mass , it is then clear that Milgrom’s interpolating function plays the role of “ gravitational permittivity”. Since it is smaller than 1, it makes the gravitational field stronger than Newtonian (rather than smaller in the case of the electric field in a dielectric medium, where ). In other words, the gravitational susceptibility coefficient (such that ) is negative, which is correct for a force law where like masses attract rather than repel [56]. This dielectric analogy has been explicitly used in devising a theory[60] where Milgrom’s law arises from the existence of a “gravitationally polarizable” medium (see Section 7).Of course, inverting the above relation, Milgrom’s law can also be written as

where However, as we shall see in Section 6, in order for to remain a conservative force field, these expressions (Eqs. 7 and 10) cannot be rigorous outside of highly symmetrical situations. Nevertheless, it allows one to make numerous very general predictions for galactic systems, or, in other words, to derive “Kepler-like laws” of galactic dynamics, unified under the banner of Milgrom’s law. As we shall see, many of the observations unpredicted by CDM on galaxy scales naturally ensue from this very simple law. However, even though Milgrom originally devised this as a modification of dynamics, this law is a priori nothing more than an algorithm, which allows one to calculate the distribution of force in an astronomical object from the observed distribution of baryonic matter. Its success would simply mean that the observed gravitational field in galaxies is mimicking a universal force law generated by the baryons alone, meaning that (i) either the force law itself is modified, or that (ii) there exists an intimate connection between the distribution of baryons and dark matter in galaxies.It was suggested, for instance, [218] that such a relation might arise naturally in the CDM context, if halos possess a one-parameter 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 and thus of (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 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 CDM is rather to explain all the different roles played by 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 Kepler-like 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.

Living Rev. Relativity 15, (2012), 10
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