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

182 Generalities on the stochastic GW backgroundHence, the gravitons are decoupled below the Planck scale. (At the Planckscale the above estimate of the interaction rate is not valid and nothing canbe said without a quantum theory of gravity). Just like the cosmic microwavebackground (CMB), the gravity-wave background is a randomly polarized relicof the Early universe. The important difference is that the electromagnetic (em)waves decoupled about 4 × 10 5 years after the big bang, while the gravitationalbackground could come from times as early as the Planck epoch at t Pl ≃5 × 10 −44 s. This means that the relic gravitational waves give information onthe state of the very early universe and, therefore, on physics at correspondinglyhigh energies, which cannot be accessed experimentally in any other way.Let us consider the standard Friedmann–Robertson–Walker (FRW)cosmological model, consisting of a radiation-dominated (RD) phase followedby the present matter-dominated (MD) phase, and let us call a(t) the FRW scalefactor. The RD phase goes backward in time until some new regime sets in. Thiscould be an inflationary epoch, for example, at the grand unification scale, or theRD phase could go back in time until Planckian energies are reached and quantumgravity sets in (t ∼ t Pl ). If the correct theory of quantum gravity is provided bystring theory, the characteristic mass scale is the string mass which is somewhatsmaller than the Planck mass and is presumably in the 10 17 –10 18 GeV region, andthe corresponding characteristic time is therefore one or two orders of magnitudelarger than t Pl . The transition between the RD and MD phases takes place att = t eq , when the temperature of the universe is of the order of only a few eV,so we are interested in graviton production which takes place well within the RDphase, or possibly at Planckian energies.A graviton produced with a frequency f ∗ at a time t = t 1 ∗ , within theRD phase has today (t = t 0 ) a red-shifted frequency f 0 = f ∗ a ∗ /a 0 . Tocompute the ratio a ∗ /a 0 one uses the fact that during the standard RD and MDphases the universe expands adiabatically: the entropy per unit comoving volumeS = g S (T )a 3 (t)T 3 is constant, where g S (T ) counts the effective number ofspecies [1]. From this one has [2]( )gS (T 0 ) 1/3 ( )T 0100 1/3 ( ) 1 GeVf 0 = f ∗ ≃ 8.0 × 10 −14 f ∗ , (12.1)g S (T ∗ ) T ∗ g S (T ∗ ) T ∗where we used the fact that T 0 = 2.728 ± 0.002 K [3] and according to thestandard model g S (T 0 ) ≃ 3.91 [1].The frequency f ∗ of the graviton produced when the temperature was T ∗ isdetermined by the Hubble constant H ∗ , i.e. the size of the horizon, at the time t ∗ ofproduction. The horizon size, physically, is the length scale beyond which causalmicrophysics cannot operate (see chapter 8.4 of [1]), and therefore, for causalityreasons, we expect that the production of gravitons or any other particles, attime t ∗ , with a wavelength longer than H∗−1 , will be exponentially suppressed.Therefore, we let λ ∗ = ɛ H∗−1,with ɛ ≤ 1. During RD, H ∗ 2 = (8π/3)Gρ rad. The1 Hereafter, the subscript α denotes the value assumed by a generic quantity at t = t α .