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

Cosmological perturbation theory 301Table 16.2. The four classes of accelerated backgrounds.α 0, ä > 0, Ḣ < 0α =−1 de Sitter ȧ > 0, ä > 0, Ḣ = 0−1 0, Ḣ > 0α>0 Accelerated contraction ȧ < 0, ä < 0, Ḣ < 0the growing mode is ‘gauged down’, i.e. it is suppressed enough to restore thevalidity of the linear approximation [51] (the off-diagonal part of the metricfluctuations remains growing, but the growth is suppressed in such a way thatthe amplitude, normalized to the vacuum fluctuations, keeps smaller than one forall scales k, provided the curvature is smaller than one in string units). This resultis also confirmed by a covariant and gauge invariant computation of the spectrum,according to the formalism developed by Bruni and Ellis [52].It should be stressed, however, that the presence of a growing mode, and theneed for choosing an appropriate gauge, is a problem typical of the pre-big bangscenario. In fact, let us come back to tensor perturbations, in the E-frame: for ageneric accelerated background the scale factor can be parametrized in conformaltime with a power α, as follows:a = (−η) α , η → 0 − , (16.63)and the perturbation equation (16.50) gives, for each Fourier mode, the Besselequationh ′′k + 2α η h′ k + k2 h k = 0, (16.64)with asymptotic solution, outside the horizon (|kη| ≪1):∫ ηdη ′h k = A k + B ka 2 (η ′ ) = A k + B k |η| 1−2α . (16.65)The solution tends to be constant for α1/2.It is now an easy exercise to re-express the scale factor (16.63) in cosmic time,dt = adη, a(t) ∼|t| α/(1+α) , (16.66)and to check that, by varying α, we can parametrize all types of acceleratedbackgrounds introduced in section 16.3: accelerated expansion (with decreasing,constant and growing curvature), and accelerated contraction, with growingcurvature (see table 16.2).In the standard, inflationary scenario the metric is expanding, α