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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 w0 and quality factor Q. From the equation of geodesic deviation, the infinitesimal displacement q of the mass along the line of separation from its equilibrium position satisfies the equation of motion
2w L ( ) ¨q + --0q + w20q = -- F+(h, f, y)¨h+(t) + F× (h,f,y)¨h ×(t) , (73) Q 2
where F+(h,f, y) and F ×(h,f, y) are “beam-pattern” factors that depend on the direction of the source (h,f) and on a polarization angle y, 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 x -y 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 dik - N^i ^N k djl- N^jN^l - -- dij- N^iN^j dkl- N^kN^l , (75) 2
where ^N 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 x -y plane makes an angle f with the x axis, the beam pattern factors are given by F+ = cos2f and F × = sin2f. For a resonant cylinder oriented along the laboratory z axis, and for source direction (h,f), they are given by F+ = sin2h cos 2y, F × = sin2h sin 2y (the angle y 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 q defined as the differential displacement along the two arms, the beam pattern functions are 1 2 F+ = 2(1 + cos h)cos 2f cos2y - cosh sin 2f sin2y and F × = 12(1 + cos2h) cos2f sin2y + cosh sin2f cos 2y.

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) = - ----(2pMfb)2/3(1 + cos2 i) cos2Pb(t), (76) R 2M-- 2/3 h×(t) = - R (2pMfb) 2cos i cos 2Pb(t), (77)
where integral t ' ' Pb(t) = 2p fb(t )dt is the orbital phase.


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