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

216 Sources of SGWBThese two processes combine to create a mechanism by which the infinitestring network loses energy (and length as measured in comoving coordinates),preventing the network from dominating the universe. Indeed, based uponcomputer simulations and analytical arguments, there is strong evidence that thecosmological evolution of the network becomes self-similar, approaching what iscalled a ‘scaling’ solution. In the simplest scale-invariant model, the correlationlength ξ of the network is proportional to its causality bound:ξ(t) ∼ t ∼ d H (t)and, thus, the statistical properties of the network are time independent if all thedistances are scaled to the causal horizon.This scaling property can be used to obtain qualitative relations. Forexample, the energy density of long strings is given by:ρ ∞ = A µdH 2 (t) ∼ A µ t 2where A is a dimensionless constant representing the number of long stringspresent per horizon sized volume. Numerical simulations suggest the valueA = 52 in the radiation-dominated era, and A = 31 in the matter-dominatedone. True scale invariance implies that the size of a newly formed loop producedby the network is a fixed fraction of the horizonl(t) = αd H (t) ∼ αt.Although the loops are observed to form with relativistic peculiar velocities v i (theloop centre of mass is moving with respect to the rest frame of the cosmologicalfluid), these are rapidly redshifted to zero by the expansion of the universe, leavinga generic loop with only a fraction f r = (1 − v 2 i )1/2 of its initial energy. Thisredshifting of peculiar velocities does not affect the loop production rate, but itdoes change the loop size immediately after its formation to:l(t) = f r αd H (t). (13.8)Hence, only a fraction f r of the total loop energy is converted into GWs. Bynumerical simulations this fraction turns out to be 0.71.This scale-invariant model is implemented by the assumption that theuniverse is described by a spatially flat ( = 1) FRW cosmology. A full treatmentof this model is developed in [34]. In particular, the effect of the string networkon the expansion of the universe and the rate of loop formation are calculated.The spectrum of GWs produced by a network of string loops can beobtained implementing the scale-invariant model described above with a modelof the emission of gravitational radiation by string loops. This model has beendeveloped in [34] and is composed of the following three elements: