### 7.5 Bi-Scalar-Tensor-Vector theory

In TeVeS [33], the “MOND function” of Eq. 76, where ,
could also be expressed as a potential of a non-dynamical scalar field , i.e., a scalar action for TeVeS
of the form:
After variation of the action w.r.t. this non-dynamical field, one gets , and variation
w.r.t. yields the usual BM Poisson equation for (Eq. 17), with . Inspired by an older
theory (Phase Coupling Gravity [32, 381]) devised in a partially successful attempt to eliminate
superluminal propagation from RAQUAL (but plagued with the same gravitational lensing problem as
RAQUAL, and with additional instabilities), Sanders [390] proposed to make this field dynamical by
adding a kinetic term in the action, leading to the following very general action for the scalar
fields and :
In this theory (dubbed BSTV for bi-scalar-tensor-vector theory), the physical metric has the
same form as in TeVeS, meaning that is the matter-coupling scalar field, while only
influences the strength of that coupling. A remarkable achievement of the theory is that the
quasi-static field equation for can be obtained only in a cosmological context, and thereby
naturally explains the connection between and [390]. What is more, oscillations of
the field around its expectation value can be considered as massive dark matter, and is
allowing an explanation of the peaks of the angular power spectrum of the Cosmic Microwave
Background [390]. Unfortunately, various instabilities and a Hamiltonian unbounded by below have been
evidenced in Section IV.A of [81], thus most likely ruling out this theory, at least in its present
form.