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

Achievable sensitivities to the SGWB 201can take, for our estimate, ˜h f ∼ 10 −22 Hz −1/2 over a bandwidth f ∼ 1 kHz.Moreover, as clearly shown in figure 12.2, the overlap reduction function isapproximately equal to one up to f ∼ 1 kHz. Therefore, from equation (12.31)and (12.27) one has( ) 1 year 1/2 (h 2 0 min gw ( f ) ∼ 1.3 × 10−7 SNR 2 fT 100 Hz) 3(˜h f10 −22 Hz −1/2 ) 2.(12.51)which shows that correlating two VIRGO interferometers for 1 year we can detecta relic spectrum with h 2 0 gw(100 Hz) ∼ 3 × 10 −7 at SNR = 1.65, or 10 −7at SNR = 1. Compared to the case of a single interferometer with SNR = 1,equation (12.48), we gain five orders of magnitude. As already discussed, toobtain a precise numerical value one must however consider equation (12.12).This involves an integral over all frequencies, (that replaces the somewhatarbitrary choice of f made above) and depends on the functional form ofh 2 0 gw( f ). For h 2 0 gw( f ) independent of the frequency, using the analyticalapproximation of equation (12.49) for S n(i) ( f )(i = 1, 2) and equation (12.44) forγ(f ), we get 5h 2 0 min gw( ) 1 year 1/2 ≃ 7 × 10−8 SNR 2 , (h 2 0T gw( f ) = constant). (12.52)It is interesting to note that the main contribution to the integral comes fromthe region f < 100 Hz. In fact, neglecting the contribution of the regionf > 100 Hz, the result for h 2 0 min gw changes only by approximately 4%. Also,the lower part of the accessible frequency range is not crucial. For instance,restricting the numerical integration to the regions 20 Hz ≤ f ≤ 200 Hzand 30 Hz ≤ f ≤ 100 Hz the sensitivity on h 2 0 gw degrades by 1% and10%, respectively. This means that the most important source of noise for themeasurement of a flat stochastic background is the thermal noise 6 . In particular,the sensitivity is limited by the mirror thermal noise, which dominates in theregion 40 Hz º f º 200 Hz, while the pendulum thermal noise dominates below5 The integral has been evaluated numerically in the frequency interval 2 Hz–10 kHz. The analyticalfit of equation (12.49) underestimates the noise power spectrum in the region f < 2 Hz and, in anycase, this frequency region gives no appreciable contribution to the integral. Above 10 kHz the overlapreduction function is negligible (see figure 12.2).6 Note also that it is not very meaningful to give more decimal figures in the minimum detectablevalue of h 2 0 min gw . Apart from the various uncertainties which enter the computation of the sensitivitycurve, a trivial source of uncertainty is the fact that the computation of the thermal noises areperformed using a temperature of 300 K. A 5% variation, corresponding to an equally plausible valueof the temperature, gives a 5% difference in h 2 0 min gw . Quoting more figures is especially meaninglesswhen the minimum detectable h 2 0 gw is estimated using the approximate quantities h c ( f ), h n ( f ),i.e. approximating the integrand of equation (12.12) with a constant over a bandwidth f . Fora broadband detector these estimates typically give results which agree with the exact numericalintegration of equation (12.12) at best within a factor of two.