6.2 The interpolating function
The basis of the MOND paradigm is to reproduce Milgrom’s law, Eq. 7, in highly symmetrical systems,
with an interpolating function asymptotically obeying the conditions of Eq. 8, i.e.,
for
and
for
. Obviously, in order for the relation between
and
to be univocally
determined, another constraint is that
must be a monotonically increasing function of
, or
equivalently
or equivalently
Even though this leaves some freedom for the exact shape of the interpolating function, leading to
the various families of functions hereafter, let us insist that it is already extremely surprising,
from the dark matter point of view, that the MOND prescriptions for the asymptotic behavior
of the interpolating function did predict all the aspects of the dynamics of galaxies listed in
Section 5.
As we have seen in Section 6.1, an alternative formulation of the MOND paradigm relies on Eq. 10,
based on an interpolating function
In that case, we also have that
must be a monotonically increasing function of
.
Finally, as we shall see in detail in Section 7, many MOND relativistic theories boil down to multifield
theories where the weak-field limit can be represented by a potential
, where each
obeys a
generalized Poisson equation, the most common case being
where
obeys the Newtonian Poisson equation and the scalar field
(with dimensions of a potential)
plays the role of the phantom dark matter potential and obeys an equation of either the type of Eq. 17 or
of Eq. 30. When it obeys a QUMOND type of equation (Eq. 30), the
-function must be replaced by the
-function of Eq. 32. When it obeys a BM-like equation (Eq. 17), the classical interpolating
function
acting on
must be replaced by another interpolating function
acting on
, in order for the total potential
to conform to Milgrom’s
law.
In the absence of a renormalization of the gravitational constant, the two functions are related
through [145
]
For
(the deep-MOND regime), one has
and
, yielding
, i.e., although it is generally different,
has the same low-gravity asymptotic behavior as
.
In spherical symmetry, all these different formulations can be made equivalent by choosing equivalent
interpolating functions, but the theories will typically differ slightly outside of spherical symmetry (i.e., the
curl field will be slightly different). As an example, let us consider a widely-used interpolating
function [141
, 166
, 402
, 508
] yielding excellent fits in the intermediate to weak gravity regime of galaxies
(but not in the strong gravity regime of the Solar system), known as the “simple”
-function (see
Figure 19):
This yields
, and thus
and
yields the “simple”
-function:
It also yields
, and hence
, yielding for the “simple”
-function:
A more general family of
-functions is known as the
-family [15
],
valid for
and including the simple function of the
case
:
corresponding to the following family of
-functions:
The
case is sometimes referred to as “Bekenstein’s
-function” (see Figure 19) as it was used
in [33
]. The problem here is that all these
-functions approach
quite slowly, with
in their
asymptotic expansion for
,
. Indeed, since
, its
asymptotic behavior is
. So, if
,
for
as well as for
,
which would imply that
would be a multivalued function, and that the
gravity would be ill-defined. This is problematic because even for the extreme case
,
the anomalous acceleration does not go to zero in the strong gravity regime: there is still a
constant anomalous “Pioneer-like” acceleration
, which is observationally
excluded
from very accurate planetary ephemerides [154
]. What is more, these
-functions, defined only in the
domain
, would need very-carefully–chosen boundary conditions to avoid covering
values of
outside of the allowed domain when solving for the Poisson equation for the scalar
field.
The way out to design
-functions corresponding to acceptable
-functions in the strong gravity
regime is to proceed to a renormalization of the gravitational constant[145
]: this means that the bare value
of
in the Poisson and generalized Poisson equations ruling the bare Newtonian potential
and the
scalar field
in Eq. 40 is different from the gravitational constant measured on Earth,
(related to
the true Newtonian potential
). One can assume that the bare gravitational constant
is related to
the measured one through
meaning that
where
,
, and
. We then
have for Milgrom’s law:
In order to recover
for
, it is straightforward to show [145
] that it suffices
that
for
, and that
. Then, if
in the asymptotic
expansion
, one has
. This second linear term
allows
to go to infinity for large
and thus
to be single-valued. On the other
hand, for the deep-MOND regime, the renormalization of
implies that
for
.
We can then use, even in multifield theories,
-functions quickly asymptoting to 1. For each of these
functions, there is a one-parameter family of corresponding
-functions (labelled by the parameter
), obtained by inserting
into
and making sure that the function is
increasing and thus invertible. A useful family of such
-functions asymptoting more quickly towards 1
than the
-family is the
-family:
The case
is again the simple
-function, while the case
has been extensively used in
rotation curve analysis from the very first analyses [28
, 223], to this day [401
], and is thus known as the
“standard”
-function (see Figure 19). The corresponding
-function for
has a very peculiar
shape of the type shown in Figure 3 of [81
] (which might be considered a fine-tuned shape, necessary to
account for solar system constraints). On the other hand, the corresponding
-function family is:
As the simple
-function (
or
) fits galaxy rotation curves well (see Section 6.5.1) but is
excluded in the solar system (see Section 6.4), it can be useful to define
-functions that have a gradual
transition similar to the simple function in the low to intermediate gravity regime of galaxies, but a more
rapid transition towards one than the simple function. Two such families are described in [325
] in terms of
their
-function:
and
Finally, yet another family was suggested in [274
], obtained by deleting the second term of the
-family,
and retaining the virtues of the
-family in galaxies, but approaching one more quickly in the solar
system:
To be complete, it should be noted that other
-functions considered in the literature include [304
, 505
]
(see also Section 7.10):
and
This simply shows the variety of shapes that the interpolating function of MOND can in principle
take.
Very precise data for rotation curves, including negligible errors on the distance and on the stellar
mass-to-light ratios (or, in that case, purely gaseous galaxies) should allow one to pin down its precise form,
at least in the intermediate gravity regime and for “modified inertia” theories (Section 6.1.1) where
Milgrom’s law is exact for circular orbits. Nowadays, galaxy data still allow some, but not much, wiggle
room: they tend to favor the
simple function [166
] or some interpolation between
and
[141
], while combined data of galaxies and the solar system (see Sections 6.4 and 6.5) rather
tend to favor something like the
function of Eq. 52 and Eq. 53 (which effectively
interpolates between
and
, see Figure 19), although slightly higher exponents (i.e.,
or
) might still be needed in the weak gravity regime in order to pass solar system
tests involving the external field from the galaxy [62
]. Again, it should be stressed that the
most salient aspect of MOND is not its precise interpolating function, but rather its successful
predictions on galactic scaling relations and Kepler-like laws of galactic dynamics (Section 5.2), as
well as its various beneficial effects on, e.g., disk stability (see Section 6.5), all predicted from
its asymptotic form. The very concept of a pre-defined interpolating function should even in
principle fully disappear once a more profound parent theory of MOND is discovered (see,
e.g., [22
]).
To end this section on the interpolating function, let us stress that if the
-function asymptotes as
for
, then the energy of the gravitational field surrounding a massive body is
infinite [38
]. What is more, if the
function of relativistic multifield theories asymptotes in the same way
to zero before going to negative values for time-evolution dominated systems (see Section 9.1), then a
singular surface exists around each galaxy, on which the scalar degree of freedom does not propagate, and
can therefore not provide a consistent picture of collapsed matter embedded into a cosmological
background. A simple solution [145
, 380] consists in assuming a modified asymptotic behavior of the
-function, namely of the form
In that case there is a return to a Newtonian behavior (but with a very strong renormalized gravitational
constant
) at a very low acceleration scale
, and rotation curves of galaxies are only
approximately flat until the galactocentric radius
Thus, one must have
to not affect the observed phenomenology in galaxies. Note that the
-function will never go to zero, even at the center of a system. Conversely, in QUMOND and the like,
one can modify the
-function in the same way: