5.1 Milgrom’s law and the dielectric analogy

Before such precise data were available, Milgrom [293Jump To The Next Citation Point] already noted that other scales were also possible, and that one that is as unique to galaxies as size is acceleration. The typical centripetal acceleration of a star in a galaxy is of order ∼ 10−10 m s−2. This is eleven orders of magnitude less than the surface gravity of the Earth. As we have seen in Section 4, this acceleration constant appears “miraculously” in very different scaling relations that should not, in principle, be related to each other17. This observational evidence for the universal appearance of −10 −2 a0 ≃ 10 m s in galactic scaling relations was not at all observationally evident back in 1983. What Milgrom [293Jump To The Next Citation Point] then hypothesized was a modification of Newtonian dynamics below this acceleration constant a0, appropriate to the tiny accelerations encountered in galaxies18. This new constant a0 would then play a similar role as the Planck constant h in quantum physics or the speed of light c in special relativity. For large acceleration (or force per unit mass), F ∕m = g ≫ a0, everything would be normal and Newtonian, i.e., g = gN. Or, put differently, formally taking a0 → 0 should make the theory tend to standard physics, just like recovering classical mechanics for h → 0. On the other hand, formally taking a0 → ∞ (and G → 0), or equivalently, in the limit of small accelerations g ≪ a 0, the modification would apply in the form:
√ ----- g = gN a0, (4 )
where g = |g | is the true gravitational acceleration, and gN = |gN | the Newtonian one as calculated from the observed distribution of visible matter. Note that this limit follows naturally from the scale-invariance symmetry of the equations of motion under transformations (t,r) → (λt, λr) [315]. This particular modification was only suggested in 1983 by the asymptotic flatness of rotation curves and the slope of the Tully–Fisher relation. It is indeed trivial to see that the desired behavior follows from equation (4View Equation). For a test particle in circular motion around a point mass M, equilibrium between the radial component of the force and the centripetal acceleration yields Vc2∕r = gN = GM ∕r2. In the weak-acceleration limit this becomes
2 ∘ ------- V-c = GM--a0. (5 ) r r2
The terms involving the radius r cancel, simplifying to
V4c (r) = Vf4= a0GM. (6 )
The circular velocity no longer depends on radius, asymptoting to a constant Vf that depends only on the mass of the central object and fundamental constants. The equation above is the equivalent of the observed baryonic Tully–Fisher relation. It is often wrongly stated that Milgrom’s formula was constructed in an ad hoc way in order to reproduce galaxy rotation curves, while this statement is only true of these two observations: (i) the asymptotic flatness of the rotation curves, and (ii) the slope of the baryonic Tully–Fisher relation (but note that, at the time, it was not clear at all that this slope would hold, nor that the Tully–Fisher relation would correlate with baryonic mass rather than luminosity, and even less clear that it would hold over orders of magnitude in mass). All the other successes of Milgrom’s formula related to the phenomenology of galaxy rotation curves were pure predictions of the formula made before the observational evidence. The predictions that are encapsulated in this simple formula can be thought of as sort of “Kepler-like laws” of galactic dynamics. These various laws only make sense once they are unified within their parent formula, exactly as Kepler’s laws only make sense once they are unified under Newton’s law.

In order to ensure a smooth transition between the two regimes g ≫ a0 and g ≪ a0, Milgrom’s law is written in the following way:

( g ) μ --- g = gN, (7 ) a0
where the interpolating function
μ(x) → 1 for x ≫ 1 and μ (x ) → x for x ≪ 1. (8 )
Written like this, the analogy between Milgrom’s law and Coulomb’s law in a dielectric medium is clear, as noted in [56Jump To The Next Citation Point]. Indeed, inside a dielectric medium, the amplitude of the electric field E generated by an external point charge Q located at a distance r obeys the following equation:
Q μ (E)E = ------2, (9 ) 4π 𝜖0r
where μ is the relative permittivity of the medium, and can depend on E. In the case of a gravitational field generated by a point mass M, 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 μ > 1). In other words, the gravitational susceptibility coefficient χ (such that μ = 1 + χ) is negative, which is correct for a force law where like masses attract rather than repel [56Jump To The Next Citation Point]. This dielectric analogy has been explicitly used in devising a theory[60Jump To The Next Citation Point] 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

( ) g = ν gN- gN, (10 ) a0
where
ν (y) → 1 for y ≫ 1 and ν (y ) → y−1∕2 for y ≪ 1. (11 )
However, as we shall see in Section 6, in order for g to remain a conservative force field, these expressions (Eqs. 7View Equation and 10View Equation) 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 ΛCDM 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, [218Jump To The Next Citation Point] 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 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 a0 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 a0 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.


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