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

222 Sources of SGWBbang nucleosynthesis. It can be shown [38] that: gw ( f ) ∼ 6 × 10 −3 Gµh −20with Gµ º 8 × 10−4ln(t ∗ /t s ) . (13.17)Although t s and t ∗ are model-dependent (t s through the assumed cosmologicalscenario; t ∗ through µ and σ ), they do not have to be separated by many orders ofmagnitude. Following [38] if we assume ln(t ∗ /t s ) º 1 equation (13.17) gives:h 2 0 gw º 5 × 10 −6 . (13.18)At t > t ∗ these string loops spanned by domain walls will collapse intogravitational radiation (and other decay products) in about a Hubble time. Thisprocess takes place over a relatively short frequency range and may lead to a sharppeak in the spectrum. However, since the exact nature of this contribution displaya strong dependence on model phenomenology, it is not possible at present topredict whether or where such a peak should occur [37].Strings connected by monopolesMonopoles are point defects which form when the manifold Å of equivalentvacua contains unshrinkable surfaces. A simple model that illustrates themonopole solution is that of an SO(3) gauge theory spontaneously broken downto U(1) by a Higgs triplet φ a with a potentialV (φ) = 1 2 λ(φa φ a − η 2 m )2 , (a = 1, 2, 3).The magnitude of 〈φ a 〉 is fixed by the minimization of the potential to |φ| =η m ,but its direction in group space is not and the manifold Å is a sphere in thisspace. If we choose φ(⃗r) = (0, 0,η m ) the SO(3) symmetry is broken down toU(1) because φ is invariant under rotations about the ê 3 -axis. Another, less trivial,choice is represented byφ i (⃗r) = η m ˆr iwhere ˆr i is a unit vector in coordinate space. For this configuration of the fieldthere must be a point in space where φ = 0 and the energy density is nonvanishing.This point of false vacuum is a monopole.Therefore, the prototypical sequence of symmetry breakings resulting inmonopoles connected by strings isG → H ⊗ U(1) → Hwhere monopoles formed at first transition (T ∼ η m ) and strings get connectedby monopole/antimonopole (M ¯M) pairs at the second (T ∼ η s ). If the monopoleformingtransition occurs after any period of inflation which can have takenplace, then the average separation of the monopoles is always smaller than the