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

220 Sources of SGWBapart from a shift in the frequency corresponding to equal matter-radiation hasa negligible effect on the red noise spectrum, which is produced in the radiationera [35].The values of the dimensionless parameters appearing in equation (13.13)are not completely known. Numerical simulations provide a reasonable estimatefor A and 〈v 2 〉 in the radiation-dominated era: A = 52 ± 10, 〈v 2 〉=0.43 ±0.02. Comparing detailed calculations of large angular scale CMB temperatureanisotropies induced by strings [36] with observations, the string mass per unitlength has been normalized toGµ = 1.05 +0.35−0.20 × 10−6 .This value is below the upper bound obtained from the pulsar timingand nucleosynthesis constraints on the gravitational radiation spectrum [35].However, the value of α (the size of the loop at formation) is still unknowm. Thehigh-resolution numerical simulations show that α < 10 −2 . This surprisinglysmall relative size is a result of the small-scale structure on the long strings and,since this is cut off by gravitational back-reaction, we may reasonably expectthat α > ƔGµ ≈ 6 × 10 −5 [33]. This uncertanty in the value of α leads toa wide uncertainty in β that, because this parameter governs the lifetime of theloop, has a large effect on the spectrum. However, it is easy to verify that thespectral density gw ( f ) reach a minimum value when α → 0. Therefore, in theradiation-dominated era (d H = 2t): gw ( f ) ≥ 8π 3 Af r Gµ 1 −〈v2 〉1 + z eqN, f ∈ (10 −8 , 10 10 ) Hz. (13.15)Since 1 + z eq = 2.32 × 10 4 0 h 2 0 , in the case 0 = 1 one has:h 2 0 gw ≥ 5.0 × 10 −9 N.In the case of the standard model thermal scenario, the value of N to be inserted inthis expression is 0.32 (see equation (13.14)). But, as anticipated in section 12.1,the evolution of N with the temperature is known only up to T ∼ 10 3 , i.e. upto frequencies f ∼ 10 −3 α −1 ∼ 10 Hz. If the particle physics model has moredegrees of freedom beyond this temperature ( f ² 10 Hz) there could be othersteps in the function N(T ) associated with other phase transitions. However,the dependence of the spectral density on the number of degrees of freedom isreasonably weak and, thus, we can conservatively estimatein the frequency range explored by VIRGO.h 2 0 gw ≥ 1.6 × 10 −9 (13.16)