7.8 Bimetric theories

In the previous theories, the acceleration-dependence of MOND enters the equations through a free “MOND function” f (X ) acting either on the contracted gradient of an added scalar field, with dimensions of acceleration (Eq. 85View Equation), or on a scalar formed with the first derivatives of a vector field (Eq. 94View Equation) with a unit-norm constraint relating it to the gradient of the potential in the physical metric. The “MOND function” could not act directly on the Christoffel symbol because this is not a tensor, and such a theory would thus violate general covariance. However, if there is more than one metric entering gravitation, the difference between the associated Christoffel symbols is a tensor, and one can construct from it a scalar with dimensions of acceleration, on which the “MOND function” can act. Such theories in which there are two dynamical rank-2 symmetric tensor fields are called bimetric theories [204, 205, 369]. Milgrom [312Jump To The Next Citation Point, 317Jump To The Next Citation Point] proposed to construct a whole parametrized class of bimetric MOND theories (dubbed BIMOND), involving an auxiliary metric, with various phenomenological behaviors in the weak-field limit, ranging from Bekenstein–Milgrom MOND to QUMOND as well as a mix of both (see [318Jump To The Next Citation Point]). As one example (parameters α = − β = − 1 in the general class of BIMOND theories, for which we refer the reader to the review [318Jump To The Next Citation Point]), the auxiliary metric ˆgμν can, e.g., be introduced precisely in the same way as the auxiliary potential Φph in the QUMOND classical action of Eq. 34View Equation :
c4 ∫ √ --- S ≡ Sm [matter,gμν] + Sm[twin matter, ˆgμν] +------ d4x − g[R − Rˆ − 2l−2f(XBIMOND )], (96 ) 16 πG
where l ≡ c2∕a0, and
2 μν α β α β XBIMOND = lg (C μβCνα − C μνCβα ), (97 )
where α α ˆα Cμν = Γ μν − Γ μν. The MONDian modification of gravity is thus introduced through the interaction between the spacetime on which matter lives and the auxiliary spacetime (on which some “twin matter” might live). This modification is acceleration-based since the interaction involves the difference of Christoffel symbols, playing the role of acceleration. By varying the action w.r.t. both metrics, we obtain two sets of Einstein-like equations, which boil down in the static weak-field limit to ˆΦ = − Ψˆ and Φ = − Ψ in Eq. 73View Equation (so this yields the correct amount of gravitational lensing for normal photons w.r.t. the “matter metric” gμν), as well as the following generalized Poisson equations:
∇2 Φ = 4πG ρ + ∇. [f′(|∇ (Φ − ˆΦ )|2∕a2)∇ (Φ − Φˆ)] and ∇2Φˆ = 4πG ρˆ+ ∇. [f ′(|∇ (Φ − ˆΦ )|2∕a2)& 0 0
or, equivalently,
∇2 (Φ − ˆΦ ) = 4πG (ρ − ˆρ) and ∇2 Φ = 4πG ρ + ∇. [f ′(|∇ (Φ − ˆΦ )|2∕a20)∇ (Φ − Φˆ)]. (99 )
This is equivalent to QUMOND (Eq. 30View Equation) if the matter and twin matter are well separated (which is natural if they repel each other), the function f playing the role of H in Eq. 34View Equation, with f′(XBIMOND ) → 0 for XBIMOND ≫ 1 and f′(XBIMOND ) → X −1∕4 BIMOND for XBIMOND ≪ 1. Note that the existence of this putative twin matter is far from being necessary (putting ˆρ = 0 everywhere yields exactly QUMOND), but it might be suggested by the existence of the auxiliary metric within the theory. Again, it is mandatory to stress that the formulation of BIMOND sketched above is actually far from unique and can be suitably parametrized to yield a whole class of BIMOND theories with various phenomenological behaviors [312, 317, 318Jump To The Next Citation Point]. For instance, in matter-twin matter symmetric versions of BIMOND (α = β = 1, see [318]), and within a fully symmetric matter-twin matter system, a cosmological constant is given by the zero-point of the MOND function, naturally on the order of one, thereby naturally leading to 2 Λ ∼ a0 for the large-scale universe. Matter and twin matter would not interact at all in the high-acceleration regime, and would repel each other in the MOND regime (i.e., when the acceleration difference of the two sectors is small compared to a0), thereby possibly playing a crucial role in the universe expansion and structure formation [316Jump To The Next Citation Point].

This promising broad class of theories should be carefully theoretically investigated in the future, notably against the existence of ghost modes [69]. At a more speculative level, this class of theories can be interpreted as a modification of gravity arising from the interaction between a pair of membranes: matter lives on one membrane, twin matter on the other, each membrane having its own standard elasticity but coupled to the other one. The way the shape of the membrane is affected by matter then depends on the combined elasticity properties of the double membrane, but matter response depends only on the shape of its home membrane. Interestingly, bimetric theories have also been advocated [256] to be a useful ingredient for the renormalizability of quantum gravity (although they currently considered theories with only metric interactions, not derivatives like in BIMOND).

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