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

214 Sources of SGWB13.1.1 StringsTo illustrate the (cosmic) strings let us come back to the simple model consideredat the beginning of the preceding section (see equations (13.1) and (13.2)). Since〈φ〉 is single valued, the total change of the phase θ around any closed path inspace must be an integer multiple of 2π. Let us now consider a closed path withθ = 2π. If no singularity is encountered, as the path shrunk to a point, θcannot change continuously from 2π to zero, and, thus, we must encounter atleast one point where θ is undefined, i.e. 〈φ〉 =0. This means that at least onetube of false vacuum should be caught inside any path with θ ̸= 0. Such tubesof false vacuum, called strings, must either be close or infinite in length, otherwiseit would be possible to contract the path to a point without crossing the string.The simplest strings are produced in the phase transition associated to thespontaneous breaking of a local U(1) symmetry. In this case the Lagrangiancontains a gauge field A µ and a complex Higgs field φ which carries U(1) chargeg and with self-interaction of the form (13.1):Ä = (D µ φ) † (D µ φ) − 1 4 F µν F µν − V (φ) (13.4)whereD µ = ∂ µ − igA µ , F µν = ∂ µ A ν − ∂ ν A µ .In this case the string solution has a well-defined core outside of which φ containsno energy density in spite of non-vanishing gradients ∇φ: the gauge field A µ canabsorb the gradient, i.e. D µ φ = 0 when ∂ µ φ ̸= 0. The radius δ of the string coreis determined by the Compton wavelengths of the Higgs and vector bosons. Form φ ≪ m A , which is usually the case, one has:δ ∼ m −1φ= λ−1/2 η −1and the energy of the string per unit length within this width is finite and givenby:µ ∼ λη 4 δ 2 = η 2(independent of the coupling λ). The value of µ (or equivalently η) is the onlyfree parameter of the string. For a phase transition at the grand unification (GUT)energy scale, η = 10 16 GeV andδ ∼ 10 −30 cm, µ ∼ 10 22 gcm −1 . (13.5)Strings of cosmological interest have sizes much greater than their width. Inthis case the internal structure of the string is unimportant and physical quantitiesof interest, such as the energy-momentum tensor, can be averaged over the crosssection. For a long, thin, straight string lying along the z-axis we define˜T ν µ = δ(x)δ(y) ∫dx dyT ν µ .