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

Topological defects 219• A peak near f ∼ 10 −12 Hz. The shape of this portion depends criticallyon the model for the emission by a loop. The overall height of the spectrumdepends linearly on Gµ, while the frequency at which the peaked spectrummerges to the red noise portion depends inversely on α. The important resultis that for values n ∗ < 10 2 and q ≥ 4/3 the spectrum drops off as 1/f forany value.Because the frequency band is accessible to VIRGO, in the following weshall concentrate on the ‘red noise’ portion of the spectrum, referring to [34, 35]for details about the region of the peak.An analytic expression for the ‘red noise’ portion of the GW spectrum isgiven as follows: gw ( f ) = 8π 9AƔG 2 µ 2α(1 −〈v 2 〉) β3/2 − 11 + z eq. (13.13)Let us remark that the above expression for the spectrum does not accountfor the reduction in the number of relativistic degrees of freedom that occurs everytime the temperature falls through a particle mass threshold. This has the effectof modifying equation (13.13) by a factorN =(g∗ag ∗b) 1/3where g ∗a (g ∗b ) is the number of relativistic degrees of freedom at a temperatureabove (below) the particle mass threshold. Within the standard model SU(3) C ⊗SU(2) L ⊗ U(1) Y , one has:⎧1, f ∈ (10 −8 , 10 −10 α −1 ) Hz( )⎪⎨ 3.36 1/3= 0.68, f ∈ (10 −10 α −1 , 10 −4 α −1 ) HzN = 10.75(13.14)( ) 3.36 1/3⎪⎩= 0.32, f ∈ (10 −4 α −1 , 10 8 ) Hz106.75where we take α ∼ ƔGµ in evaluating the above frequency range. Hence, thethermal history of the cosmological fluid reflects on the red noise spectrum bya series of steps down in amplitude with increasing frequency. The detection ofsuch a shift would provide unique insight into the particle content of the earlyuniverse (at times much earlier than the electroweak phase transition).There is a substantial body of astronomical evidence which suggests thatthe cosmological density parameter 0 is less than one, i.e. the universe is open.The evolution of strings in an open universe will differ from that in the flat caseonly after the time at which the expansion of the universe becomes curvaturedominated:after this time the linear regime no longer exists. This is importantfor consideration of GWs created with low frequencies during the matter era, but,