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

284 Elementary introduction to pre-big bang cosmologyTo any standard cosmological solution H (t), describing decelerated expansionand decreasing curvature (H > 0, Ḣ < 0), is thus associated with a ‘reflected’solution, H (−t), describing a contracting universe because H is negative.This is the situation in general relativity. In string theory the action, inaddition to the metric, contains at least another fundamental field, the scalardilaton φ. At the tree-level, namely to lowest order in the string couplingand in the higher-derivatives (α ′ ) string corrections, the effective action whichguarantees the absence of conformal anomalies for the motion of strings in curvedbackgrounds (see apppendix A) can be written as:S =− 1 ∫2λ d−1 d d+1 x √ |g|e −φ [R + (∂ µ φ) 2 ] (16.4)s(λ s = Ms−1 is the fundamental string length scale; see appendix B for notationsand sign conventions). In addition to the invariance under time-reversal, the aboveaction is also invariant under the ‘dual’ inversion of the scale factor, accompaniedby an appropriate transformation of the dilaton (see [9] and the first paper of [8]).More precisely, if a(t) is a solution for the cosmological background (16.2) , thena −1 (t) is also a solution, provided the dilaton transforms as:a →ã = a −1 , φ → ˜φ = φ − 2d ln a (16.5)(this transformation implements a particular case of T -duality symmetry, usuallycalled ‘scale factor duality’, see appendix B).When a goes into a −1 , the Hubble parameter H again goes into −H sothat, to each one of the two solutions related by time reversal, H (t) and H (−t),is associated a dual solution, ˜H(t) and ˜H(−t), respectively (see figure 16.2).The space of solutions is thus reached in a string cosmology context. Indeed,because of the combined invariance under the transformations (16.3) and (16.5),a cosmological solution has in general four branches: two branches describeexpansion (positive H ), two branches describe contraction (negative H ). Also,as illustrated in figure 16.2, for two branches the curvature scale (∼ H 2 ) growsin time, with a typical ‘pre-big bang’ behaviour, while for the other two branchesthe curvature scale decreases, with a typical ‘post-big bang’ behaviour.It follows, in this context, that to any given decelerated expandingsolution, H (t) > 0, with decreasing curvature, Ḣ(t) < 0 (typical of thestandard cosmological scenario), is always associated a ‘dual partner’ describingaccelerated expansion, ˜H(−t) >0, and growing curvature,˙˜H(−t) >0. Thisdoubling of solutions has no analogue in the context of Einstein cosmology, wherethere is no dilaton, and the duality symmetry cannot be implemented.It should be stressed, before proceeding further, that the duality symmetry isnot restricted to the case of homogeneous and isotropic backgrounds like (16.2),but is expected to be a general property of the solutions of the string effectiveaction (possibly valid at all orders [10], with the appropriate generalizations).The inversion of the scale factor, in particular, and the associated transformation