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

188 Generalities on the stochastic GW backgroundwhere δ 2 ( ˆ, ˆ ′ ) = δ(φ−φ ′ )δ(cos θ −cos θ ′ ). The function S h ( f ) defined by theabove equation has dimensions Hz −1 and satisfies S h ( f ) = S h (− f ). With thisnormalization, S h ( f ) is the quantity to be compared with the noise level S n ( f )defined in equation (12.10). Using equations (12.13) and (12.18) we get〈h ij (t, ⃗r)h ij (t, ⃗r)〉 =2= 4∫ ∞−∞∫ f =∞f =0d fS h ( f ) = 4We now define the characteristic amplitude h c ( f ) from〈h ij (t, ⃗r)h ij (t, ⃗r)〉 =2∫ ∞0d fS h ( f )d(ln f ) fS h ( f ). (12.19)∫ f =∞f =0d(ln f ) h 2 c ( f ). (12.20)Note that h c ( f ) is dimensionless, and represents a characteristic value of theamplitude, per unit logarithmic interval of frequency. The origin of the factorof two on the right-hand side of equation (12.20) is carefully explained in [5].Comparing equations (12.19) and (12.20), we geth 2 c ( f ) = 2 fS h( f ). (12.21)We now wish to relate h c ( f ) and h 2 0 gw( f ). The starting point is the expressionfor the energy density of gravitational waves, given by the 00-component of theenergy-momentum tensor. The energy-momentum tensor of a GW cannot belocalized inside a single wavelength (see, e.g., sections 20.4 and 35.7 in [12] fora careful discussion) but it can be defined with a spatial averaging over severalwavelengths:ρ gw = 132πG 〈ḣ ij ḣ ij 〉. (12.22)For a stochastic background, the spatial average over a few wavelengths is thesame as a time average at a given point, which, in Fourier space, is the ensembleaverage performed using equation (12.18). By inserting equation (12.13) intoequation (12.22) and using equation (12.18) one hasand, thusρ gw = 432πGdρ gwdln f∫ f =∞f =0d(ln f ) f (2π f ) 2 S h ( f ), (12.23)= π2G f 3 S h ( f ). (12.24)Comparing equations (12.24) and (12.21) we get the important relationdρ gwdln f= π4G f 2 h 2 c ( f ), (12.25)