Gravitational wave detection

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

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 x–y plane is the plane of the sky and the z-direction points toward the detector, these two modes are given by

where $hij TT$ represent transverse-traceless (TT) projections of the calculated waveform of Equation (68

), given by

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 x–y 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 (70 ) 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.