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

192 Generalities on the stochastic GW backgroundwhere the symmetric, traceless tensor D(ˆr) describes the orientation andgeometry of the detector located at ⃗r. In terms of this tensor the gravitationalwavestrain sensed by this detector (see equation (12.9)) is given by [10]h(t, ⃗r) = D ij (ˆr)h ij (t, ⃗r). (12.39)For an interferometer, indicating with û and ˆv the unit vectors in the directions ofits arms, one has:D(ˆr) = 1 2{û(ˆr) ⊗û(ˆr) −ˆv(ˆr) ⊗ˆv(ˆr)}. (12.40)For the lowest longitudinal mode of a cylindrical GW antenna with axis in thedirection individued by the unit vector ˆl, one has [13]D(ˆr) = ˆl(ˆr) ⊗ ˆl(ˆr) − 1 3I, (12.41)where I is the unit matrix. Finally, for the lowest five degenerate quadrupolemodes (m =−2,...,+2) of a spherical detector, the corresponding tensors areD (0) (ˆr) = 12 √ 3 {e+ (ˆr) + 2g + (ˆr)} ∼ 12 √ 3 {2 f + (ˆr) − e + (ˆr)}D (+1) (ˆr) =− 1 2 g× (ˆr), D (−1) (ˆr) =− 1 2 f × (ˆr) (12.42)D (+2) (ˆr) = 1 2 e+ (ˆr), D (−2) (ˆr) =− 1 2 e× (ˆr)wheref + (ˆr) = ˆm(ˆr) ⊗ˆm(ˆr) −ˆr ⊗ˆr,g + (ˆr) =ˆn(ˆr) ⊗ˆn(ˆr) −ˆr ⊗ˆr,f × (ˆr) = ˆm(ˆr) ⊗ˆr +ˆr ⊗ˆm(ˆr)g × (ˆr) =ˆn(ˆr) ⊗ˆr +ˆr ⊗ˆn(ˆr),and e +,× (ˆr) are the tensors of equation (12.17) written in terms of the unit vectorsˆm(ˆr) and ˆn(ˆr) lying on the plane perpendicular to ˆr. From these expressions forthe tensors D ij and interpreting each of the five modes of a sphere as a singledetector, it is possible to show that in the case of coincident detectors one has:〈F1 A (ˆr, ˆ, ψ)F2 B (ˆr, ˆ, ψ)〉 ˆ,ψ ∼ c 12δ AB , (A, B =+, ×) (12.43)where c 12 depends only on the geometry and the relative orientations of thetwo detectors. The corresponding values of F (see equation (12.37)) for thethree different geometries considered (interferometer, cylindrical bar, sphere) aresummarized in table 12.1.By introducing the following notation⃗r = dŝ,η = 2π fd,where ŝ is the unit vector along the direction connecting the two detectors andd is the distance between them, it can be shown [10] that the overlap reductionfunction assumes the following form (D k ≡ D(ˆr k )):γ(f ) = ρ 0 (η)D ij1 D 2ij + ρ 1 (η)D ij1 Dk 2i s j s k + ρ 2 (η)D ij1 Dkl 2 s is j s k s l (12.44)