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

226 Sources of SGWBR 2MDR(η) (arbitrary units)R 110 0de SitterRD1 η2 1 3 η4 2 η50η (arbitrary units)Figure 13.3. The scale factor as a function of the conformal time. The universe undergoesa transition from a de Sitter- to a radiation-dominated phase at the time η 1 , and from aradiation to a matter dominated phase at the time η 2 . The present epoch corresponds to η 0 .At first sight, the first and the third modes written in equation (13.24), seemnot to be pure plane waves solutions. This comes from the fact that we are incurved spacetime: it can be easily verified that in the limit k →∞, i.e. when thewavelength is so short that the particle does not ‘feel’ the curvature of spacetime,all the modes in equation (13.24) reduce to the standard plane-wave form. Thefact that ψ r± is already in the standard form is due to the conformal invariance ofthe radiation-dominated spacetime.The coefficients α, β, γ and δ in equation (13.23) are calculated by requiringthe overall solution ψ(η) to be continuous with its first derivative at transitiontimes η 1,2 . Equation (13.21) says that this problem is similar to the wellknownquantum mechanics problem of tunnelling through a potential barrierV (η). Therefore, α, β, γ and δ can be considered as the transmission/reflectioncoefficients and one can show that the number of created gravitons with comovingfrequency k is( )3HN k ∼|δ k | 2 3= dsR(η 1 ) 4 218R(η 2 ) k 6 . (13.25)This expression should be taken with care, because the simple modeldescribed by equation (13.22) does not take into account two important physicaleffects that force equation (13.25) to hold only in a limited range of frequencies.First of all, the perturbations whose physical wavelength is greater than thepresent Hubble length (R(η 0 )/k > 1/H 0 ) do not contribute to the energy density;so k 0 = R ′ 0 /R 0, corresponding to a physical frequency f 0 ∼ 10 −19 Hz, provides