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

184 Generalities on the stochastic GW backgroundcorrect, and one in which it can fail by several orders of magnitudes. Bothexamples will, in general, illustrate the fact that the argument does not involveonly kinematics, but also the dynamics of the production mechanism.From equation (12.4) we see that the temperatures of the early universeexplored detecting today relic GWs at a frequency f 0 are, for constant g ∗ , smallerby a factor approximately equal to ɛ, compared to those estimates with ɛ = 1.Equivalently, a signal produced at a given temperature T ∗ could in principle showup today in the VIRGO/LIGO frequency band when a naive estimate with ɛ = 1suggests that it falls at lower frequencies.There is, however, another effect, which instead gives hopes of exploringthe universe at much higher temperatures than naively expected, using GWexperiments. In fact, the characteristic frequency that we have discussed is thevalue of the cut-off frequency in the graviton spectrum. Above this frequency, thespectrum decreases exponentially [6], and no signal can be detected. Below thisfrequency, however, the form of the spectrum is not fixed by general arguments.Thermal spectra have a low-frequency behaviour dρ/d f ∼ f 2 , but, since thegravitons below the Planck scale interact too weakly to thermalize, there is noa priori reason for a ∼ f 2 dependence. The gravitons will retain the form ofthe spectrum that they had at the time of production, and this is a very modeldependentfeature. However, as shown in [5], from a number of explicit examplesand general arguments we learn that spectra flat or almost flat over a large rangeof frequencies seem to be not at all unconceivable.This fact has potentially important consequences. It means that, even if aspectrum of gravitons produced during the Planck era has a cut-off at frequenciesmuch larger than the VIRGO/LIGO frequency band, we can still hope to observein the 10 Hz–1 kHz region the long low-frequency tails of these spectra.12.2 Definitions12.2.1 gw ( f ) and the optimal SNRThe intensity of a stochastic background of gravitational waves (GWs) can becharacterized by the dimensionless quantity gw ( f ) = 1 ρ cdρ gwdln f , (12.6)where ρ gw is the energy density of the stochastic background of gravitationalwaves, f is the frequency and ρ c is the present value of the critical energy densityfor closing the universe. In terms of the present value of the Hubble constant H 0 ,the critical density is given byρ c = 3H 2 08πG . (12.7)