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 [81Jump To The Next Citation Point] 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 [81Jump To The Next Citation Point]). 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:
4 ∫ ∘ --- S = − --c--- d4x − &tidle;g[X + 2V (ϕ )], (90 ) ϕ 8 πGl2
with 2 μν X = l &tidle;g ϕ,μ ϕ,ν and 2 2 2 V (ϕ ) = lm ϕ ∕2. The physical metric gμν is then disformally related to the Einstein metric through (see Eq. 5.11 of [81Jump To The Next Citation Point]):
g ≡ A2 &tidle;g + B ϕ, ϕ, , (91 ) μν μν μ ν
with the functionals
A (ϕ,X ) = eηϕ − ϕh(Y )Y∕ η,B (ϕ, X ) = − 4 ϕη−1Y ∕X, (92 )
where 1∕2 −1 −1∕4 Y = (ηa0 ) c X. The free function h(Y ) is the “MOND function” playing the role of Milgrom’s μ. 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 [82Jump To The Next Citation Point]. 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 m 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 [82Jump To The Next Citation Point]). 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 [82Jump To The Next Citation Point], namely to make the matter coupling (or, equivalently, the baryon-scalar DM interaction) depend on the local density of matter51: 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.
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