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

344 Post-Newtonian computation of binary inspiral waveformsNotice that the direction p ≡ x is one of the ‘ascending node’ N of the binary,namely the point at which the bodies cross the plane of the sky moving towardthe detector. Thus, the polarization vectors p and q lie, respectively, along themajor and minor axis of the projection onto the plane of the sky of the (circular)orbit, with p pointing toward N using the standard practice of celestial mechanics.Finally, let us denote by ξ the polarization angle between p and the vertical planeβ = constant; that is, ξ is the angle between the vertical and the direction of thenode N. Wehaven =−X sin α cos β − Y sin α sin β − Z cos α, (17.23)p = X(cos ξ cos α cos β + sin ξ sin β)+ Y (cos ξ cos α sin β − sin ξ cos β) − Z cos ξ sin α, (17.24)q = X(− sin ξ cos α cos β + cos ξ sin β)+ Y (− sin ξ cos α sin β − cos ξ cos β) + Z sin ξ sin α. (17.25)Defining all these angles, the relative orientation of the binary with respect to theinterferometric detector is entirely determined. Indeed using (17.22) and (17.25)one relates the triad (x, y, z) associated with the source to the triad ( X, Y , Z)linked with the detector.The gravitational wave as it propagates through the detector in the wavezone of the source is described by the so-called transverse and traceless (TT)asymptotic waveform h TTij= (g ij −δ ij ) TT , where g ij denotes the spatial covariantmetric in a coordinate system adapted to the wave zone, and δ ij is the Kroneckermetric. Neglecting terms dying out like 1/R 2 in the distance to the source, thetwo polarization states of the wave, customarily denoted h + and h × , are given byh + = 1 2 (p i p j − q i q j )h TTij , (17.26)h × = 1 2 (p iq j + p j q i )h TTij , (17.27)where p i and q i are the components of the polarization vectors. The detector isdirectly sensitive to a linear combination of the polarization waveforms h + andh × given byh(t) = F + h + (t) + F × h × (t), (17.28)where F + and F × are the so-called beam-pattern functions of the detector, whichare some given functions (for a given type of detector) of the direction of thesource α, β and of the polarization angle ξ. This h(t) is the gravitational-wavesignal looked for in the data analysis of section 17.2, and used to construct theoptimal filter (17.10). In the case of the laser-interferometric detector we haveF + = 1 2 (1 + cos2 α) cos 2β cos 2ξ + cos α sin 2β sin 2ξ, (17.29)F × = − 1 2 (1 + cos2 α) cos 2β sin 2ξ + cos α sin 2β cos 2ξ. (17.30)The orbital plane and the direction of the node N are fixed so the polarizationangle ξ is constant (in the case of spinning particles, the orbital plane precesses