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

equation gives immediately(ρgwρ γ)NSObservational bounds 209= 7 8 (N ν − 3), (12.61)where the subscript NS reminds us that this equality holds at the time ofnucleosynthesis. If more extra species, not included in the standard model,contribute to g ∗ (N ν ), then the equals sign in the above equation is replaced byless than or equal. The same happens if there is a contribution from any otherform of energy present at the time of nucleosynthesis and not included in theenergy density of radiation, like, for example, primordial black holes.To obtain a bound on the energy density at the present time, we notethat from the time of nucleosynthesis to the present time ρ gw scaled as 1/a 4 ,while, as a consequence of the assumed adiabatic expansion of the universe (seesection 12.1), ρ γ ∼ T 4 ∼ 1/(a 4 g 4/3 ). Therefore, one has(ρgwS( ( )ρgw gS (T 0 ) 4/3 ( ( )ρgw 3.914/3==. (12.62)ρ γ)0ρ γ)NSg S (1 MeV) ρ γ)NS10.75Therefore we get the nucleosynthesis bound at the present time,(ρgwρ γ)0≤ 0.227(N ν − 3). (12.63)Of course this bound holds only for GWs that were already produced at thetime of nucleosynthesis (T ∼ 1 MeV, t ∼ 1 s). It does not apply toany stochastic background produced later, like backgrounds of astrophysicalorigin (see section 13.5). Note that this is a bound on the total energydensity∫in gravitational waves, integrated over all frequencies. Writing ρ gw =d(ln f ) dρgw /dln f , multiplying both ρ gw and ρ γ in equation (12.63) by h 2 0 /ρ c,and inserting the numerical value h 2 0 ρ γ /ρ c ≃ 2.474 × 10 −5 [29], we get∫ f =∞f =0d(ln f ) h 2 0 gw( f ) ≤ 5.6 × 10 −6 (N ν − 3). (12.64)The bound on N ν from nucleosynthesis is subject to various systematic errorsin the analysis, which have to do mainly with the issues of how much of theobserved 4 He abundance is of primordial origin, and of the nuclear processing of3 He in stars, and as a consequence over the last five years have been quoted limitson N ν ranging from 3.04 to around 5. The situation has been recently reviewedin [30]. The conclusion of [30] is that, until current astrophysical uncertaintiesare clarified, N ν < 4 is a conservative limit. Using extreme assumptions, ameaningful limit N ν < 5 still exists, showing the robustness of the argument.Correspondingly, the right-hand side of equation (12.64) is, conservatively, oforder 5 × 10 −6 and anyway cannot exceed 10 −5 .