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

Chapter 13Sources of SGWBHere we review the present knowledge about the potential processes, both ofcosmological and astrophysical origin, from which a stochastic GW backgroundmight arise. We examine in some detail the mechanisms at work in each case andthe features of the corresponding spectrum of GW radiation.13.1 Topological defectsThe concept of the spontaneous symmetry breaking, the idea that there areunderlying symmetries of Nature that are not manifest in the structure of thevacuum, play a crucial role in the modern description of the particle interactions.Of particular interest for cosmology is the theoretical expectation that at hightemperatures, symmetries that are spontaneously broken today were restored[31]. In the context of the hot big bang cosmology this implies a sequence ofphase transitions in the early universe, with critical temperatures related to thecorresponding symmetry breaking scales.To illustrate the phenomenon of the high-temperature symmetry restorationwe consider a complex scalar field with a ‘Mexican-hat’ potentialV (φ) = 1 2 λ(φ† φ − η 2 ) 2 (λ > 0). (13.1)The Lagrangian is invariant under the group U(1) of the global phasetransformations, φ → e iα φ. The minima of the potential are at nonzero valuesφ, and so the symmetry is spontaneously broken and φ acquires a vacuumexpectation value (VEV)〈φ〉 =ηe iθ (13.2)where the phase θ is arbitrary. We thus have a manifold Å of degenerate vacuumstates corresponding to different choices of θ, that, in this case, is the circle|φ| =η in the complex φ plane.211