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

300 Elementary introduction to pre-big bang cosmologySince this effect is particularly important, let me give, in short, an explicitexample for the scalar perturbations of the metric tensor in a d = 3, isotropicand conformally flat background, in the E-frame (we will omit the tilde, forsimplicity). The perturbed metric, in the so-called longitudinal gauge, dependson the two Bardeen potentials and as [49]:ds 2 = a 2 [(1 + 2) dη 2 − (1 − 2)dxi 2 ]. (16.56)By perturbing the Einstein equations (16.43), the dilaton equation, and combiningthe results for the various components, one obtains to first order that = , andthat the metric fluctuations satisfy the equation: ′′ + 6 a′a ′ −∇ 2 = 0. (16.57)We now consider the particular, exact solution of the vacuum string cosmologyequations in the E-frame,a(η) =|η| 1/2 , φ(η) =− √ 3ln|η|, η → 0 − , (16.58)corresponding to a phase of accelerated contraction and growing dilaton (i.e. thepre-big bang solution (16.33), written in conformal time, for d = 3). For thisbackground, the perturbation equation (16.57) becomes a Bessel equation for theFourier modes k , ′′k + 3 η ′ k + k2 k = 0, ∇ 2 k =−k 2 k , (16.59)and the asymptotic solution, for modes well outside the horizon (|kη| ≪1),ϕ k = A k ln |kη|+B k |kη| −2 (16.60)contains a growing part which blows up (∼ η −2 ) as the background approachesthe high curvature regime (η → 0 − ). In this limit the linear approximationbreaks down, so that the longitudinal gauge is not, in general, consistent withthe perturbative expansion around a homogeneous, inflationary pre-big bangbackground, as scalar inhomogeneities may become too large.In the same background (16.58) the problem is absent, however, for tensorperturbations, since their growth outside the horizon is only logarithmic. Fromequation (16.50) we have in fact the asymptotic solutionh k = A k + B k ln |kη|, |kη| ≪1. (16.61)This may suggest that the breakdown of the linear approximation, for scalarperturbations, is an artefact of the longitudinal gauge. This is indeed confirmed bythe fact that, in a more appropriate off-diagonal (also called ‘uniform curvature’[50]) gauge,ds 2 = a 2 [(1 + 2ϕ)dη 2 − dx 2 i − 2∂ i B dx i dη], (16.62)