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

Gravitational wave radiation in the JBD theory 165Finally, substituting (11.65), (11.66) in (11.67) and integrating over time, oneobtains( ) 1/42 12ω BD + 19 G 3 m 1 m 2 md = 215 ω BD + 2 c 5 τ 4 (11.68)where we have defined τ = t c − t, t c being the time of the collapse between thetwo bodies.From (11.49), (11.53)–(11.56) one can deduce the form of the scalar field(see appendix B of the third reference in [6] for details) which, for equal masses,is2µ [ξ(t) =−v 2 + m r(2ω BD + 3) d − (ˆn ·⃗v)2 + m ]d 3 (ˆn · ⃗d) (11.69)where r is the distance of the source from the observer, and ˆn is the versor of theline of sight from the observer to the binary system centre of mass. Indicatingwith γ the inclination angle, that is the angle between the orbital plane and thereference plane (defined to be a plane perpendicular to the line of sight), and withψ the true anomaly, that is the angle between d and the x-axis in the orbital planex–y, yields ˆn · ⃗d = d sin γ sin ψ. Then, from (11.69) one obtains2Gµmξ(t) =(2ω BD + 3)c 4 dr sin2 γ cos(2ψ(t)) (11.70)which can also be written asξ(τ) = ξ 0 (τ) sin(χ(τ) +¯χ) (11.71)where ¯χ is an arbitrary phase and the amplitude ξ 0 (τ) is given by2Gµmξ 0 (τ) =(2ω BD + 3)c 4 dr sin2 γ( )1 ωBD + 2 1/4 ( ) 15G 1/4 5/4M c=2(2ω BD + 3)r 12ω BD + 19 2c 11 τ 1/4 sin2 γ. (11.72)In the last expression, we have introduced the definition of the chirp massM c = (m 1 m 2 ) 3/5 /m 1/5 .11.3.5 Detectability of the scalar GWsLet us now study the interaction of the scalar GWs, a spherical GW detector.As usual, we characterize the sensitivity of the detector by the spectraldensity of strain S h ( f ) [Hz] −1 . The optimum performance of a detector isobtained by filtering the output with a filter matched to the signal. The energysignal-to-noise ratio (SNR) of the filter output is given by the well known formula:SNR =∫ +∞−∞|H ( f )| 2S h ( f ) d f (11.73)