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

308 Elementary introduction to pre-big bang cosmologyfirst case, k ≫ 1/|η 1 |≡k 1 , we can approximate the Hankel functions with theirlarge argument limit, and we find that there is no particle production,|c + |≃1, |c − |≃0. (16.93)In practice, c − is not exactly zero, but is exponentially suppressed as a functionof the frequency, just like the quantum reflection probability for a wave with afrequency well above the top of a potential step. We will neglect such an effecthere, as we are mainly interested in a qualitative estimate of the perturbationspectrum.In the second case, k ≪ 1/|η 1 |≡k 1 , we can use the small argument limit ofthe Hankel functions,H (2)ν ∼ a(kη 1 ) ν − ib(kη 1 ) −ν , H (1)ν ∼ a(kη 1 ) ν + ib(kη 1 ) −ν , (16.94)and we find|c + |≃|c − |≃|kη 1 | −ν−1/2 , (16.95)corresponding to a power-law spectrum:dρ kdlnk = k4π 2 |c −(k)| 2 ≃ k4 1π 2 ( kk 1) 3−2ν, k < k 1 , (16.96)with a cut-off frequency k 1 = η1 −1 controlled by the height of the effectivepotential.For a comparison with present observations, it is finally convenient toexpress the spectrum in terms of the proper frequency, ω(t) = k/a(t), and inunits of critical energy density, ρ c (t) = 3Mp 2 H 2 (t)/8π. We then obtain thedimensionless spectral distribution,(ω,t) =ω dρ(ω)ρ c (t) dω ≃ 83πω14 ( ) ω 3−2νMp 2 H 2 ω 1≃ g 2 1 γ (t)( ωω 1) 3−2ν, ω < ω 1 , (16.97)whereω 1 = k 1a = 1 ≃ H 1a 1aη 1 ais the maximal amplified proper frequency, g 1 = H 1 /M p , and(16.98) γ (t) = ρ γρ c=(H1H) 2 (a1 ) 4(16.99)is the energy density (in critical units) of the radiation that becomes dominant att = t 1 , rescaled down at a generic time t (today, γ (t 0 ) ∼ 10 −4 ).a