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

The overlap reduction function 191Of course, the reduction factor in the noise level is larger if the integration timeis larger, and if we increase the bandwidth f over which a useful coincidenceis possible. Note that h 2 0 gw is quadratic in h c ( f ), so that an improvement insensitivity by two orders of magnitudes in h c means four orders of magnitude inh 2 0 gw.12.3 The overlap reduction functionWhen we consider the correlation between the signals of two GW detectors wehave a reduction in sensitivity due to the fact that, in general, these detectorswill not be either coincident or coaligned. This effect is quantified by theoverlap reduction function γ(f ) appearing in the expression of the SNR (seeequation (12.12)). This is a dimensionless function of frequency f which takesinto account the relative positions and orientations of the two detectors and for anisotropic and unpolarized background is defined as [10]γ(f ) = 1 ∑〈e i2π f ˆ·⃗r F1 AF(ˆr 1, ˆ, ψ)F2 A (ˆr 2, ˆ, ψ)〉 ˆ,ψ ∼ Ɣ( f )FA(12.35)where ⃗r =⃗r 1 −⃗r 2 is the separation vector between the two detector sites, F Aiis the pattern function characterizing the response of the ith detector (i = 1, 2) tothe A =+, × polarization, and the following notation〈···〉ˆ,ψ = ∫S 2 d ˆ4π∫ 2π0dψ(···) (12.36)2πhas been introduced to indicate the average over the propagation direction (θ, φ)and the polarization angle ψ of the GW. The normalization factor F is given by:F = ∑ A〈F A1 (ˆr 1, ˆ, ψ)F A2 (ˆr 2, ˆ, ψ)〉 ˆ,ψ | 1≡2, (12.37)where the notation 1 ≡ 2 is a compact way to indicate that the detectors arecoincident and coaligned and, if at least one of them is an interferometer, theangle between its arms is equal to π/2 (L-shaped geometry). In this situation, bydefinition, γ(f ) = 1. When the detectors are shifted apart (so there is a phaseshift between the signals in the two detectors), or rotated out of coalignment (sothe detectors have different sensitivity to the same polarization) it turns out that:|γ(f )| < 1.The pattern functions (or orientation factors) of a GW detector can be writtenin the following formF A (ˆr, ˆ, ψ) = Tr{D(ˆr)ε A ( ˆ, ψ)} (12.38)