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

Astrophysical sources 237which is three orders of magnitude too small to be observed by any of the groundbasedinterferometers presently under construction.13.5 Astrophysical sourcesTo the stochastic background of gravitational radiation that should pervadeour universe, the GWs associated with unresolved astrophysical sources alsocontribute. Since the generation of these waves dates back to more recent epochs,when galaxies and stars started to form and evolve, their contribution to gw isnot subject to the nucleosynthesis bound. Therefore, our first concern is whetherthis component of the stochastic background can give a contribution to h 2 0 gw( f )larger than the bound (12.64), or anyway larger than the expected relic signal,thereby masking the background of cosmological origin.A first observation is that there is a maximum frequency at whichastrophysical sources can radiate. This comes from the fact that a source of massM, even if very compact, will be at least as large as its gravitational radius 2GM,the bound being saturated by black holes. Even if its surface were rotating at thespeed of light, its rotation period would be at least 4πGM, and the source cannotemit waves with a period much shorter than that. Therefore, we have a maximumfrequency [63],1f º4πGM ∼ M 104 ⊙Hz. (13.41)MTo emit near this maximum frequency an object must presumably have a massof the order of the Chandrasekhar limit ∼1.2M ⊙ , which gives a maximumfrequency of the order of 10 kHz [63], and this limit can be saturated onlyby very compact objects (see [64] for a recent review of GWs emitted in thegravitational collapse to black holes, with typical frequencies f º 5 kHz). Thesame numbers, apart from factors of order one, can be obtained using the fact thatfor a self-gravitating Newtonian system with density ρ, radius R, there is a naturaldynamical frequency [65]f dyn = 1( ) 3GM 1/22π (πGρ)1/2 =16π 2 R 3 . (13.42)With R ≥ 2GM we recover the same order of magnitude estimate apart from afactor (3/8) 1/2 ≃ 0.6. This is already an encouraging result, because it showsthat the natural frequency domains of cosmological and astrophysical sources canbe very different. We have seen in section 12.1 that the natural frequency scale forPlanckian physics is the GHz, while no astrophysical objects can emit above, say,(6–10) kHz. Therefore, a stochastic background detected above these frequencieswould be unambiguously of cosmological origin.However, ground-based interferometers have their maximum sensitivityaround 100 Hz, where astrophysical sources hopefully produce interestingradiation (since these sources were the original motivation for the construction