7.7 Generalized Einstein-Aether theories

All theories reviewed so far are best expressed in the “Einstein frame”, and involve a form for the physical metric to which matter couples (an form expressed as a function of the Einstein metric and of the other additional fields). However, the work of [513] has shown that, for instance, TeVeS (Sections 7.3 and 7.4) is expressible as a pure Tensor-Vector theory in the matter frame, and that the physical metric then both satisfies the Einstein–Hilbert action and couples minimally to the matter fields, just like in GR. In fact, the modification of gravity in TeVeS thus only comes from the coupling of the physical metric to the vector field. The idea of Zlosnik et al. [514Jump To The Next Citation Point] was then that a similar, but simpler, modification of gravity could be obtained by devising a simple tensor-vector theory in the matter frame, with no a priori on the geometry of the physical metric. Starting from the extensively studied Einstein-Aether theories [206Jump To The Next Citation Point, 207], with a vector action of the type of Eq. 86View Equation, the idea is to make the k-essence free function f (X ) (the “MOND function” of Eq. 76View Equation) act directly on the vector field rather than on an additional scalar field. This leads to vector k-essence, or Generalized Einstein-Aether (GEA) theories (also called non-canonical Einstein-Aether theories), in which the Einstein–Hilbert and matter actions remain as in GR, but with an additional unit-norm vector field with the following action [431Jump To The Next Citation Point, 514]:
∫ --c4--- 4 √ ---[ 2 μν ] SU ≡ − 16πGl2 d x − g f (Xgea) − lλ(g U μU ν + 1) , (93 )
where (see Eq. 87View Equation and replacing μν &tidle;g by μν g)
X = l2K αβμνU U . (94 ) gea β,α ν,μ
The unit-norm constraint fixes the vector field in terms of the metric, and from there we have that, in the weak-field limit, Xgea ∝ − |∇ Φ |2, with Φ defined as in Eq. 73View Equation. The Einstein equation in the weak-field limit then yields a BM type of Poisson equation (Eq. 17View Equation) for the full gravitational potential Φ, with ′ ′ μ = f + (1 − f )∕(1 − C∕2 ) and C = c1 − c4 [431Jump To The Next Citation Point]. In the deep-MOND limit, the usual choice for f is of the type f(Xgea) ∝ (− Xgea)3∕2 + 2Xgea∕C, and the length-scale must be fixed as:
2 l ≡ (2 −-C)c--. (95 ) 3∕2C3 ∕2a0
Let us note that this weak-field limit of GEA theories is different from that of RAQUAL or TeVeS, where only the scalar field ϕ obeys a BM-like equation governed by an interpolating function μ&tidle;(s), and where the total potential is given by Eq. 40View Equation.

The remarkable feature of GEA theories allowing for the desired enhancing of gravitational lensing without any on the form of the physical metric is that, writing the metric as in Eq. 73View Equation, it can be shown [431] that in the limit Xgea → 0 the action of Eq. 93View Equation is only a function of ϒ = Φ + Ψ and is thus invariant under disformal transformations [Φ → Φ + β (r);Ψ → Ψ − β (r)], of the type of Eq. 83View Equation. These GEA theories are currently extensively studied, mostly in a cosmological context (see Section 9), but also for their parametrized post-Newtonian coefficients in the solar system [65] or for black hole solutions [451].

Interestingly, it has been shown that all these vector field theories (TeVeS, BSTV, GEA) are all part of a broad class of theories studied in [183]. Yet other phenomenologically-interesting theories exist among this class, such as, for instance, the V Λ models considered by Zhao & Li [502, 506, 510] with a dynamical norm vector field, whose norm obeys a potential (giving it a mass) and has a non-quadratic kinetic term à-la-RAQUAL, in order to try reproducing both the MOND phenomenology and the accelerated expansion of the universe, while interpreting the vector field as a fluid of neutrinos with varying mass [504, 505]. This has the advantage of giving a microphysics meaning to the vector field. Such vector fields have also been argued to arise naturally from dimensional reduction of higher-dimensional gravity theories [34, 261], or, more generally, to be necessary from the fact that quantum gravity could need a preferred rest frame [206] in order to protect the theory against instabilities when allowing for higher derivatives to make the theory renormalizable (e.g., in Hořava gravity [64, 195]). Inspired by this possible need of a preferred rest frame in quantum gravity, relativistic MOND theories boiling down to particular cases of GEA theories in which the vector field is hypersurface-orthogonal have, for instance, been proposed in [61, 396].

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