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

Topological defects 215It can be shown [33] that, as a consequence of the invariance of the string underLorentz boost along z and the conservation law Tµ,ν ν one has:˜T ν µ= µδ(x)δ(y) diag(1, 0, 0, 1). (13.6)This expression shows a remarkable property of the strings: the pressure isnegative, i.e. is a string tension, and this tension is equal to the mass per unitlength µ. One recalls from classical mechanics that small transverse waves in astring with tension T move at speed (T/µ) 1/2 , so it is apparent that waves movealong a string at the velocity of light.Applied to strings, the Kibble mechanism implies that at the time of phasetransition a network of strings with typical length ξ(t c ) will form. According tonumerical simulations at formation about one fifth of the initial energy is in smallclosed loops and the remaining in ‘infinite’ long strings. The evolution of thisnetwork for t > t c is complicated. The key processes are:(i)The intercommutation of intersecting string segments, in which the twosegments swap partners, rather than passing through one another (see figurebelow). Based upon numerical simulations it appears that the probability forthis to occur is nearly unity. This process leads to the continual chopping oflong strings into smaller segments and/or loops.(ii) The decay of small loops through the emission of gravitational radiation.The strings oscillate relativistically under their own tension and, thus, a loopof characteristic radius R will radiate gravitational waves at a characteristicfrequency ω ∼ R −1 due to its time-varying quadrupole moment, Q ∼ µR 3 .For an order of magnitude estimate the power radiated in gravitational wavescan be calculated using the quadrupole formula (G is Newton’s constant)P ∼ G〈 ˙¨Q ˙¨Q〉 ∼Gµ 2 . (13.7)Because this power is constant, independent of the loop size, the mass-energyof the loop decreases linearly with time. In a characteristic timeτ ∼RGµthe loop shrinks to a point and vanishes. From equation (13.5), in the case ofa GUT phase transition, one has Gµ ∼ 10 −6 .