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

Definitions 189or, dividing by the critical density ρ c , gw ( f ) = 2π 23H 2 0f 2 h 2 c ( f ). (12.26)Inserting the numerical value of H 0 ,wefind ( [11], equation (65))h c ( f ) ≃ 1.26 × 10 −18 ( 1HzfFrom equations (12.21) and (12.26) one has) √h 2 0 gw( f ). (12.27) gw ( f ) = 4π 23H 2 0f 3 S h ( f ), (12.28)and defining S n ( f ) = (S n (1) ( f )S n(2) ( f )) 1/2 , equation (12.12) can be written in thefollowing more transparent form:SNR =[2F 2 T∫ ∞0d f γ 2 ( f ) S2 h ( f )S 2 n ( f ) ] 1/4. (12.29)The factor of two in front of the integral can be understood from ∫ ∞−∞ d f =2 ∫ ∞0d f .Finally, we mention another useful formula which expresses h 2 0 gw( f ) interms of the number of gravitons per cell of the phase space, n(⃗x, ⃗k). For anisotropic stochastic background n(⃗x, ⃗k) = n f depends only on the frequency f =|⃗k|/(2π) and ρ gw = ∫ n f 2π f d 3 k/(2π) 3 = 8π 2 ∫ ∞0d(ln f )n f f 4 . Thereforedρ gw /dln f = 8π 2 n f f 4 , and( n ) ( )h 2 0 f f 4gw( f ) ≃ 1.810 37 . (12.30)1 kHzAs we will discuss below, to be observable at the VIRGO/LIGO interferometers,we should have h 2 0 gw ∼ 10 −6 between 1 Hz and 1 kHz, corresponding to n fof order 10 31 at 1 kHz and n f ∼ 10 43 at 1 Hz. A detectable stochastic GWbackground is therefore exceedingly classical, n k ≫ 1.12.2.3 The characteristic noise levelWe have seen in the previous section that there is a very natural definition ofthe characteristic amplitude of the signal, given by h c ( f ), which contains all theinformation on the physical effects, and is independent of the apparatus. Wecan, therefore, associate with h c ( f ) a corresponding noise amplitude h n ( f ), thatembodies all the informations on the apparatus, defining h c ( f )/h n ( f ) in termsof the optimal SNR.