### 7.8 Bimetric theories

In the previous theories, the acceleration-dependence of MOND enters the equations through a free
“MOND function” acting either on the contracted gradient of an added scalar field, with
dimensions of acceleration (Eq. 85), or on a scalar formed with the first derivatives of a vector field
(Eq. 94) 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 [312, 317] 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 [318]). As one example
(parameters in the general class of BIMOND theories, for which we refer the
reader to the review [318]), the auxiliary metric can, e.g., be introduced precisely in
the same way as the auxiliary potential in the QUMOND classical action of Eq. 34 :
where , and
where . 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 and
in Eq. 73 (so this yields the correct amount of gravitational lensing for normal
photons w.r.t. the “matter metric” ), as well as the following generalized Poisson equations:
or, equivalently,
This is equivalent to QUMOND (Eq. 30) if the matter and twin matter are well separated (which is natural
if they repel each other), the function playing the role of in Eq. 34, with for
and for . Note that the existence of this
putative twin matter is far from being necessary (putting 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, 318]. For instance, in matter-twin matter symmetric versions of
BIMOND (, 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 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 ), thereby possibly playing a crucial role in the universe expansion and structure
formation [316].
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).