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

304 Elementary introduction to pre-big bang cosmologyIt is important to stress that there is no need to introduce the canonicalvariable to study the classical evolution of perturbations, but that such variableis needed for the initial normalization to a vacuum fluctuation spectrum. Wecan also normalize perturbation in a different way, of course but in that case weare studying the amplification not of the vacuum fluctuations, but of a differentspectrum [55].At this point, two remarks are in order. The first concerns the frameindependenceof the spectrum. The above procedure can also be applied in theE-frame, to define a canonical variable ˜ψ: one then obatins for ˜ψ k the canonicalequation (16.75), with a pump field that depends only on the metric, ˜z = ã.However, by using the conformal transformation connecting the two frames, itturns out that the two pump fields are the same, ˜z = ã = ae −φ/2 = z, so thatfor ψ and ˜ψ we have the same potential, the same evolution equation, the samesolution, and thus the same spectrum.The second remark is that the canonical procedure can be applied to anyaction, and in particular to the string effective action including higher curvaturecorrections of order α ′ , which can be written as [21]:∫S = d 4 x √ {}−ge −φ −R − ∂ µ φ∂ µ φ + α′4 [R2 GB − (∂ µφ∂ µ φ) 2 ] (16.78)where RGB2 ≡ R2 µναβ − 4R2 µν + R2 is the Gauss–Bonnet invariant (we havechosen a convenient field redefinition that removes terms with higher-than-secondderivatives from the equations of motion, see appendix A). From the quadraticperturbed action we obtain α ′ corrections to the pump fields. The canonicalequation turns out to be the same as before, but with a k-dependent effectivepotential [54], and such an equation can be used to estimate the effects of thehigher curvature corrections on the amplification of tensor perturbations. Anumerical integration [54], in which the metric fluctuations are expanded aroundthe high-curvature background solution of [21], leads in particular to the resultsillustrated in figure 16.6.The qualitative behaviour is similar, both with and without α ′ correctionsin the perturbed equations: the fluctuations are oscillating inside the horizon andfrozen outside the horizon, as usual. However, the final amplitude is enhancedwhen α ′ corrections are included, and this suggests that the energy spectrum ofthe gravitational radiation, computed with the low-energy perturbation equation,may represent a sort of lower bound on the total amount of produced gravitons.16.5.4 Computation of the spectrumThe final, amplified perturbation spectrum is to be obtained from the solutionsof the canonical equation (16.75). In order to solve such an equation weneed explicitly the effective potential V [z(η)] which, in general, vanishesasymptotically at large positive and negative values of the conformal time.Consider, for instance, the tensor perturbation equation in the E-frame, so that