5.1 Milgrom’s law and the dielectric analogy
Before such precise data were available, Milgrom [293
] 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
. 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
other.
This observational evidence for the universal appearance of
in galactic scaling relations was
not at all observationally evident back in 1983. What Milgrom [293
] then hypothesized was a modification of
Newtonian dynamics below this acceleration constant
, appropriate to the tiny accelerations encountered in
galaxies.
This new constant
would then play a similar role as the Planck constant
in quantum physics
or the speed of light
in special relativity. For large acceleration (or force per unit mass),
, everything would be normal and Newtonian, i.e.,
. Or, put differently,
formally taking
should make the theory tend to standard physics, just like recovering
classical mechanics for
. On the other hand, formally taking
(and
), or
equivalently, in the limit of small accelerations
, the modification would apply in the form:
where
is the true gravitational acceleration, and
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
[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 (4). For a test particle in circular motion around a point mass
, equilibrium between the
radial component of the force and the centripetal acceleration yields
. In the
weak-acceleration limit this becomes
The terms involving the radius
cancel, simplifying to
The circular velocity no longer depends on radius, asymptoting to a constant
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
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.