### 7.6 Non-minimal scalar-tensor formalism

As a consequence of the inability of RAQUAL (the scalar-tensor k-essence of Section 7.1) to enhance
gravitational lensing, all other attempts reviewed so far (Sections 7.2 to 7.5) have been plagued with an
aesthetically unpleasant growth of additional fields and free parameters. This has led Bruneton &
Esposito-Farèse [81] to consider models with fewer additional fields. They first considered pure
metric theories in which matter is not only coupled to the metric but also non-minimally to its
curvature (Eqs. 5.1 and 5.2 of [81]). While they showed that such models can indeed reproduce the
MOND dynamics, they also concluded that they are generically unstable if locality is to be
preserved (but see Section 7.10). They then considered models in which at most one scalar field is
added, without any additional vector field, but where this field is coupled non-minimally to
matter, in the sense that the matter-coupling depends on the scalar field itself but also on its first
derivatives. In other words, the gradient of the scalar field is replacing the dynamical vector
field of TeVeS. The simple scalar field action is just the normal action of a massive scalar field:
with and . The physical metric is then disformally related to
the Einstein metric through (see Eq. 5.11 of [81]):
with the functionals
where . The free function is the “MOND function” playing the role of
Milgrom’s . An alternative formulation of the model is obtained by separating the matter action into a
normal matter action and an “interaction term” between the scalar field, the metric and the matter
fields [82]. Considering the massive scalar field as a dark matter fluid, this model can thus be interpreted as
a non-standard baryon–dark-matter interaction leading to the MOND behavior. If the scalar mass is
small enough, it is a pure MOND theory, but if it is higher, it can lead to a “DM+ MOND” behavior,
especially noteworthy in regions of high gravity such as the center of galaxy clusters (see Section 6.6.4 and
discussions in [82]). Let us note that, while this theory exhibits superluminal propagations outside of
matter, it is, in principle, not a problem for causality [80]. It has also been possible to study the
behavior of the theory within matter, e.g., within the dilute HI gas inside galaxy disks (an
analysis, which is mostly too difficult to perform in other models reviewed so far): this led to
a deadly problem, i.e., that the Cauchy problem becomes ill-posed and the solutions to field
equations ill-defined. A possible solution was proposed in [82], namely to make the matter
coupling (or, equivalently, the baryon-scalar DM interaction) depend on the local density of
matter:
this can also lead to an interesting phenomenology, where only gas-rich systems behave according to
Milgrom’s law, while others would behave in a CDM way [143]. A lot remains to be studied within this
framework.