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

236 Sources of SGWBa consequence of the high velocities and energy densities involved, when, withina Hubble expansion time, these bubbles collide a large fraction of the energy thatwas in the bubble walls is converted in gravitational radiation.Unlike the case of a network of cosmic strings, where gw ( f ) is flat as aconsequence of the existence of a ‘scaling’ solution, the spectrum of the GWsproduced by this bubble collision process is strongly peaked at a frequencycharacteristic of the particular cosmological time at which the phase transitionand bubble collisions took place. Under the assumption that the expansion of theuniverse has been adiabatic since the phase transition, for the present value of thischaracteristic frequency one has [61]:( )( ) βf max ≈ 5.2 × 10 −8 T∗ ( g∗) 1/6(13.37)H ∗ 1 GeV 100where, assuming an exponential bubble nucleation rate, β −1 is, roughly, theduration of the phase transition (see section 12.1 for the meaning of the otherquantities). In general, for the temperatures of interest (say, 1–10 16 GeV), it turnsout to be [62]:( )β MPl∼ 4ln ∼ 100, (13.38)H ∗ T ∗and, because g ∗ is also of order 100 in typical GUT models, from equation (13.37)we find that the most ‘promising’ cosmological phase transition, from thepoint of view of the VIRGO sensitivity band, would be one that occurred ata temperature comprised between 10 7 and 10 8 GeV (for which we have nocompelling candidate).The amplitude of the spectrum depends mostly upon the difference in the freeenergy between the inside and the outside of the bubble, driving the expansion ofthe bubble. By indicating with v the propagation velocity of the bubble walls, andwith α the ratio of vacuum energy to the thermal energy in the symmetric phase(the high-temperature phase before the transition), the amplitude at the frequencyf max is approximately [61]:h 2 0 gw( f max ) ≈ 1.1 × 10 −6 κ 2 (H∗β) 2 ( α) ( 21 + α) (100v 3 ) 1/30.24 + v 3 ,g ∗(13.39)where k is an increasing function of α quantifying the fraction of the availablevacuum energy that goes into kinetic (rather than thermal) energy of the fluid.The parameter α characterizes the strength of the phase transition, and the limitsα → 0, α → ∞ correspond to very weak and very strong first-order phasetransitions, respectively. For α ranging between these two extremes, typically,0.01 < κ < 1. For strongly first-order phase transitions it turns out to bealso v → 1, and, thus, for a transition at T ∼ 2 × 10 7 GeV, according toequations (13.37) and (13.39), one finds:h 2 0 gw( f ∼ 100 Hz) ≈ 10 −10 (13.40)