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

208 Generalities on the stochastic GW backgroundapart from occasional weak processes, is dominated by the decay of free neutrons,the ratio n n /n p remains frozen at the value exp(−Q/T f ), where T f is the value ofthe temperature at the time of freeze-out. This number, therefore, determinesthe density of neutrons available for nucleosynthesis, and since practically allneutrons available will eventually form 4 He, the final primordial abundance ofthis nucleus is very sensitive to the freeze-out temperature T f . If we assumefor simplicity Ɣ pe→nν ≃ G 2 F T 5 (which is really appropriate only in the limitT ≫ Q), where G F is the Fermi constant, from equation (12.3) turns out that T fis determined by the conditionG 2 F T 5f≃(4π 3 g ∗45) 1/2T2fM Pl. (12.58)This shows that T f ∼ g∗1/6 , at least with the approximation that we used forƔ pe→nν . A large energy density in relic gravitons gives a large contribution to thetotal density ρ and therefore to g ∗ . This results in a larger freeze-out temperature,more available neutrons and then in overproduction of 4 He. This is the idea behindthe nucleosynthesis bound [28]. More precisely, since the density of 4 He increasesalso with the baryon to photon ratio η, we could compensate an increase in g ∗ witha decrease in η, and therefore we also need a lower limit on η, which is providedby the comparison with the abundance of deuterium and 3 He.Rather than g ∗ , N ν is often used as an ‘effective number of neutrino species’and is defined as follows. In the standard model, at T ∼ a few MeV, the activedegrees of freedom are the photon, e ± , neutrinos and antineutrinos, and they havethe same temperature, T i = T . Then, for N ν families of light neutrinos, one hasg ∗ (N ν ) = 2 + 7 8 (4 + 2N ν), (12.59)where the factor of two comes from the two elicity states of the photon, fourfrom e ± in the two elicity states, and 2N ν counts the N ν neutrinos and the N νantineutrinos, each with their single elicity state. According to the standardmodel, N ν = 3 and therefore g ∗ = 43/4. Therefore, we can define an ‘effectivenumber of neutrino species’ N ν fromg ∗ (N ν ) ∼ 434 + ∑i=extra bosonsg i(TiT) 4+ 7 8∑i=extra fermions( ) 4 Tig i . (12.60)TOne extra species of light neutrino, at the same temperature as the photons, wouldcontribute one unit to N ν , but all species, weighted with their energy density,contribute to N ν , which, of course, in general is not an integer. For i = gravitons,we have g i = 2 and (T i /T ) 4 = ρ gw /ρ γ , where ρ γ = 2(π 2 /30)T 4 is the photonenergy density. If gravitational waves give the only extra contribution to N ν ,compared to the standard model with N ν = 3, using equation (12.59), the above