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4.5 Gravitational wave detection

A gravitational wave detector can be modelled as a body of mass M at a distance L from a fiducial laboratory point, connected to the point by a spring of resonant frequency ω0 and quality factor Q. From the equation of geodesic deviation, the infinitesimal displacement ξ of the mass along the line of separation from its equilibrium position satisfies the equation of motion
2ω L ( ) ¨ξ + --0ξ˙+ ω20ξ = -- F+ (θ,φ, ψ)¨h+ (t) + F× (θ,φ,ψ )¨h ×(t) , (73 ) Q 2
where F+(θ,φ, ψ) and F ×(θ,φ, ψ) are “beam-pattern” factors that depend on the direction of the source (θ,φ ) and on a polarization angle ψ, and h (t) + and h (t) × are gravitational waveforms corresponding to the two polarizations of the gravitational wave (for a review, see [255]). In a source coordinate system in which the xy plane is the plane of the sky and the z-direction points toward the detector, these two modes are given by
h (t) = 1-(hxx(t) − hyy (t)), h (t) = hxy (t), (74 ) + 2 TT TT × TT
where hij TT represent transverse-traceless (TT) projections of the calculated waveform of Equation (68View Equation), given by
ij [( )( ) 1 ( ) ( )] hTT = hkl δik − Nˆi ˆN k δjl − NˆjNˆl − -- δij − NˆiNˆj δkl − NˆkNˆl , (75 ) 2
where ˆN j is a unit vector pointing toward the detector. The beam pattern factors depend on the orientation and nature of the detector. For a wave approaching along the laboratory z-direction, and for a mass whose location on the xy plane makes an angle φ with the x-axis, the beam pattern factors are given by F+ = cos2 φ and F × = sin2φ. For a resonant cylinder oriented along the laboratory z-axis, and for source direction (θ,φ ), they are given by F+ = sin2 θ cos2ψ, F × = sin2θ sin 2ψ (the angle ψ measures the relative orientation of the laboratory and source x-axes). For a laser interferometer with one arm along the laboratory x-axis, the other along the y-axis, and with ξ defined as the differential displacement along the two arms, the beam pattern functions are 1 2 F+ = 2(1 + cos θ)cos 2φ cos2ψ − cosθ sin 2φ sin2ψ and F × = 12(1 + cos2θ) cos2φ sin2 ψ + cosθ sin2φ cos 2ψ.

The waveforms h+ (t) and h ×(t) depend on the nature and evolution of the source. For example, for a binary system in a circular orbit, with an inclination i relative to the plane of the sky, and the x-axis oriented along the major axis of the projected orbit, the quadrupole approximation of Equation (70View Equation) gives

2ℳ h+(t) = − ----(2πℳfb )2∕3(1 + cos2 i) cos2Φb (t), (76 ) R 2ℳ-- 2∕3 h×(t) = − R (2πℳfb ) 2cos i cos 2Φb(t), (77 )
where ∫t ′ ′ Φb(t) = 2π fb(t )dt is the orbital phase.


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