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

Practical applications of the Isaacson energy 55which is a very large energy flux even for this weak a wave. It is twice the energyflux of a full moon! Integrating over a sphere of radius r, assuming a total durationof the event τ, and solving for h, again with appropriate normalizations, gives[ ] 1 [ ]h = 10 −21 E gw 2 r −1 [ ]f −1 [ τ ] − 120.01M ⊙ c 2 . (5.17)20 Mpc 1 kHz 1msThis is the formula for the ‘burst energy’, normalized to numbers appropriate to agravitational collapse occurring in the Virgo cluster. It explains why physicistsand astronomers regard the 10 −21 threshold as so important. However, thisformula could also be applied to a binary system radiating away its orbitalgravitational binding energy over a long period of time τ, for example.5.3.1 Curvature produced by wavesWe have assumed that the background metric satisfies the vacuum Einsteinequations to linear order, but now it is possible to view the full action principleas a principle for the background with a wave field h µν on it, and to let the waveenergy affect the background curvature [24]. This means that the background willactually solve, in a self-consistent way, the equationG αβ [g µν ] = 8πTαβ GW [g µν + h µν ]. (5.18)This does not contradict the vanishing of the first variation of the action, which weneeded to use above, because now we have an Einstein tensor that is of quadraticorder in h µν , contributing a term of cubic order to the first-variation of the action,which is of the same order as other terms we have neglected.5.3.2 Cosmological background of radiationThis self-consistent picture allows us to talk about, for example, a cosmologicalgravitational wave background that contributes to the curvature of the universe.Since the energy density is the same as the flux (when c = 1), we haveϱ gw = π 4 f 2 h 2 , (5.19)but now we must interpret h in a statistical way. This will be treated in thecontribution by Babusci et al, but basically it is done by replacing h 2 by astatistical mean square amplitude per unit frequency (Fourier transform power),so that the energy density per unit frequency is proportional to f 2 | ˜h| 2 . It is thenconventional to talk about the energy density per unit logarithm of frequency,which means multiplying by f . The result, after being careful about averagingover all directions of the waves and all independent polarization components, isdϱ gwdln f= 4π 2 f 3 | ¯h( f )| 2 . (5.20)