7.5 Bi-Scalar-Tensor-Vector theory

In TeVeS [33Jump To The Next Citation Point], the “MOND function” f (XTeVeS ) of Eq. 76View Equation, where XTeVeS ∼ (&tidle;gμν − U μU ν)ϕ,μ ϕ,ν, could also be expressed as a potential V of a non-dynamical scalar field q, i.e., a scalar action for TeVeS of the form:
∫ ∘ ---[ ] Sϕ ∝ − d4x − &tidle;g 1q2XTeVeS + V (q) . (88 ) 2
After variation of the action w.r.t. this non-dynamical field, one gets qX = − V′(q), and variation w.r.t. ϕ yields the usual BM Poisson equation for ϕ (Eq. 17View Equation), with 2 √ --- q ∝ &tidle;μ( X ). 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 [390Jump To The Next Citation Point] proposed to make this field dynamical by adding a kinetic term &tidle;gμνq, q, μ ν in the action, leading to the following very general action for the scalar fields ϕ and q:
∫ ∘ --- [ ] S (ϕq) ∝ − d4x − &tidle;g 1(&tidle;gμνq,μq,ν +H (q)(&tidle;gμν + U μUν)ϕ,μ ϕ,ν ) − F(q)U μU νϕ,μϕ,ν +V (q) .(89 ) 2
In this theory (dubbed BSTV for bi-scalar-tensor-vector theory), the physical metric has the same form as in TeVeS, meaning that ϕ is the matter-coupling scalar field, while q only influences the strength of that coupling. A remarkable achievement of the theory is that the quasi-static field equation for ϕ can be obtained only in a cosmological context, and thereby naturally explains the connection between a0 and H0 [390Jump To The Next Citation Point]. What is more, oscillations of the q 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 [81Jump To The Next Citation Point], thus most likely ruling out this theory, at least in its present form.
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