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

Definitions 18712.2.2 The characteristic amplitudeIn a transverse-traceless (TT) gauge, the perturbation to the Minkowski metric ofspacetime introduced by the GW background can be written in terms of a planewave expansion as [4]h ij (t, ⃗r) =∑ ∫ ∞ ∫ ∫ 2πd f d ˆ dψ h A ( f, ˆ, ψ)e i2π f (t− ˆ·⃗r) εA=+,×−∞ S 2 ij A ( ˆ, ψ),0(12.13)where, since h ij is real, the Fourier amplitudes are complex arbitrary functionsthat satisfy the condition h A (− f, ˆ, ψ) = h ∗ A ( f, ˆ, ψ). The unit vector ˆ isalong the propagation direction of the wave, and, in terms of the standard polar(θ) and azimuthal (φ) angles on the 2-sphere, is:ˆ ≡ (cos φ sin θ,sin φ sin θ,cos θ), (12.14)with d ˆ = d cos θ dφ. The angle ψ describes the polarization of the wave.By introducing the following pair of orthogonal unit vectors lying in the planeperpendicular to ˆˆm( ˆ) ≡ (cos φ cos θ,sin φ cos θ,− sin θ), ˆn( ˆ) ≡ (sin φ,− cos φ,0),(12.15)ψ is the angle of which is rotated the intrinsic frame of the wave (where, in aTT gauge, h x ′ y ′ =−h y ′ x ′) respect to the frame ( ˆm, ˆn) (see [11], p 367). Thepolarization tensors can be written aswhereε + ( ˆ, ψ) = e + ( ˆ) cos 2ψ − e × ( ˆ) sin 2ψε × ( ˆ, ψ) = e + ( ˆ) sin 2ψ + e × ( ˆ) cos 2ψ, (12.16)e + ( ˆ) = ˆm( ˆ)⊗ ˆm( ˆ)−ˆn( ˆ)⊗ˆn( ˆ),with the normalizatione × ( ˆ) = ˆm( ˆ)⊗ˆn( ˆ)+ˆn( ˆ)⊗ ˆm( ˆ)(12.17)Tr{e A ( ˆ)e A′ ( ˆ)} =2δ AA′ .In the case of a stochastic background, we treat the complex Fourieramplitude h A as a random variable with zero mean value. If this backgroundis isotropic, unpolarized and stationary, the ensemble average of the product oftwo Fourier amplitudes satisfies:〈h ∗ A ( f, ˆ, ψ)h A ′( f ′ , ˆ ′ ,ψ ′ )〉=δ AA ′δ( f − f ′ ) δ2 ( ˆ, ˆ ′ )4πδ(ψ − ψ ′ )2π12 S h( f ),(12.18)