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

302 Elementary introduction to pre-big bang cosmologycontraction is fast enough, i.e. α > 1/2 (in fact, the growing mode problemwas first pointed out in the context of Kaluza–Klein inflation and dynamicaldimensional reduction [53], where the internal dimensions are contracting). Forthe low-energy string cosmology background (16.58) we have α = 1/2, thegrowth is simply logarithmic (see (16.61)), and the linear approximation canbe applied consistently, provided the curvature remains bounded by the stringscale [51]. However, for α > 1/2 the growth of the amplitude may require adifferent gauge for a consistent linearized description.16.5.3 Normalization of the amplitudeThe linearized equations describing the classical evolution of perturbations canbe obtained in two ways:• by perturbing directly the background equations of motion;• by perturbing the metric and the matter fields to first order, by expanding theaction up to terms quadratic in the first order fluctuations,g → g + δ (1) g, δ (2) S ≡ S[(δ (1) g) 2 ], (16.67)and then by varying the action with respect to the fluctuations.The advantage of the second method is to define the so-called ‘normal modes’for the oscillation of the system {gravity + matter sources}, namely the variableswhich diagonalize the kinetic terms in the perturbed action, and satisfy canonicalcommutation relations when the fluctuations are quantized. Such canonicalvariables are required, in particular, to normalize perturbations to a spectrumof quantum, zero-point fluctuations, and to study their amplification from thevacuum state up to the present state of the universe.Let us apply such a procedure to tensor perturbations, in the S-frame, for ad = 3 isotropic background. In the syncronous gauge, the transverse-traceless,first-order metric perturbations h µν = δ (1) g µν satisfy equation (16.46). Weexpand all terms of the low-energy gravidilaton action (16.20) up to order h 2 :δ (1) g µν =−h µν , δ (2) g µν = h µα h ν α ,δ (1)√ −g = 0, δ (2)√ −g =− 1 √4 −ghµν h µν , (16.68)and so on for δ (1) R µν , δ (2) R µν (see, for instance, [54]). By using the backgroundequations, and integrating by part, we finally arrive at the quadratic actionδ (2) S = 1 ∫ ()d 4 xa 3 e −φ ḣ ji4ḣi j + h j ∇ia 2 hi j . (16.69)By separating the two physical polarization modes, i.e. the standard ‘cross’ and‘plus’ gravity wave components,h ji hi j = 2(h2 + + h2 × ), (16.70)