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

212 Sources of SGWBV (Τ φ)High TLow T−η+ηφFigure 13.1. The effective potential.At finite temperature the effective potential for φ acquires additional,temperature-dependent terms. In the high-temperature limit, one has:V T (φ) = AT 2 φ † φ + V (φ) (13.3)where the dimensionless constant A depend on the self-coupling λ and is assumedto be positive. From equations (13.1) and (13.3) we see that the effective mass ofthe field φ at temperature T is:( ) λ 1/2m 2 (T ) = 2A(T 2 − Tc 2 ) where T c = η.AFrom this expression we see that for T > T c , m 2 (T ) is positive, the minimumof V T (φ) is at φ = 0 and so the symmetry is restored (see figure 13.1).The temperature T c is the critical temperature of the phase transition from thesymmetric (〈φ〉 =0) to the broken-symmetry (〈φ〉 ̸=0) phase. Unless λ is verysmall, T c ∼ η. In this example the transition is second order: the symmetric phasecorresponds to a maximum for V T at T < T c and the transition occurs smoothly.More complicated models can lead to first-order transitions, where the symmetricphase remains a local minimum at T < T c separated by a barrier from the minimaat φ ̸= 0. In this case the transition occurs through the bubble nucleation.In a cosmological context, as the universe cools through the criticaltemperature the Higgs field φ develop an expectation value 〈φ〉 correspondingto some point in the manifold Å. Since all points of this manifold are equivalent,the choice depends on random fluctuations and is different in different regionsof space. Therefore, associated to the phase transition there is a length scale ξrepresenting the maximum distance over which the Higgs field can be correlated.