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

Topological defects 217(i)A loop radiates with powerP = ƔGµ 2 (13.9)where Ɣ is a dimensionless radiation efficiency that does not depend on theloop size, but only on its shape. Recent studies of realistic loop configurationindicate that the distribution of the values of this parameter has mean value〈Ɣ〉 ≈60. From equations (13.8) and (13.9), it follows that a loop formed attime t b at a time t > t b has lengthl(t, t b ) = f r αd H (t) − ƔGµ(t − t b ).The loop disappears when this length reaches zero, at a time(t d = 1 + f )r αd H (t b )t b = βt bƔGµt b(β is not function of t b because d H (t b ) ∼ t b ).(ii) The frequency of GWs emitted at time t by a loop formed at time t b is givenbyf n (t, t b ) =2n (n = 1, 2, 3,...). (13.10)l(t, t b )(iii) The fraction of the total power emitted in the nth oscillation mode atfrequency f n is given by the coefficient P n , whereƔ =∞∑P n . (13.11)n=1Analytic and numerical studies suggest that the radiation efficiencycoefficients behave as P n ∝ n −q .This model has several shortcomings. First, the spectral index q has not beenwell determined. Numerical simulations suggest q = 4/3 while from analyticcalculations q = 2 is obtained. The simulations have limited resolution of theimportant small-scale features of the long strings and loops and, therefore, theanalytic prediction may be more realistic. Second, the effect of the back reactionon the motion of the strings has been ignored. The authors of [35] have arguedthat this effect result in an effective high-frequency cut off in the oscillation modenumber n. Thus, the loop radiates only in a finite range of frequencies andequation (13.11) is replaced byƔ =n ∗ ∑n=1P n . (13.12)By comparing the back-reaction length scale to the loop size these authorsestimate that the cut off should be no larger than ∼(ƔGµ) −1 .