(ed.). Gravitational waves (IOP, 2001)(422s).

160 Detection of scalar gravitational waves11.3.1 Scalar and Tensor GWs in the JBD TheoryIn the Jordan–Fierz frame, in which the scalar field mixes with the metric butdecouples from matter, the action reads [17]S = S grav [φ,g µν ] + S m [ψ m , g µν ]∫= c3d 4 x √ [−g16πφ R − ω BDφ]gµν ∂ µ φ∂ ν φ + 1 ∫cd 4 xL m [ψ m , g µν ](11.29)where ω BD is a dimensionless constant, whose lower bound is fixed to be ω BD ≈600 by experimental data [18], g µν is the metric tensor, φ is a scalar field and ψ mcollectively denotes the matter fields of the theory.As a preliminary analysis, we perform a weak-field approximation aroundthe background given by a Minkowskian metric and a constant expectation valuefor the scalar fieldg µν = η µν + h µνϕ = ϕ 0 + ξ. (11.30)The standard parametrization ϕ 0 = 2(ω BD +2)/G(2ω BD +3), with G the Newtonconstant, reproduces GR in the limit ω BD → ∞, which implies ϕ 0 → 1/G.Defining the new fieldθ µν = h µν − 1 2 η µνh − η µνξϕ 0(11.31)where h is the trace of the fluctuation h µν , and choosing the gaugeone can write the field equations in the following formwhereTS =−2(2ω BD + 3)∂ µ θ µν = 0 (11.32)∂ α ∂ α θ µν = − 16π τ µνϕ 0(11.33)∂ α ∂ α 8πξ =2ω BD + 3 S (11.34)τ µν = 1 (T µν + t µν ) (11.35)ϕ 0[ 12 ∂ α(θ∂ α ξ)+ 2 ∂ a (ξ∂ ξ)]α .ϕ 0(1 − 1 2 θ − 2 ξ ϕ 0)− 116π(11.36)