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

286 Elementary introduction to pre-big bang cosmologyand is manifestly invariant under the set of global transformations [11]:( )φ → φ, M → T M , T 0 Iη = η, η = , (16.10)I 0where I is the d-dimensional unit matrix, and η is the O(d, d) metric in offdiagonalform. The transformation (16.5), representing scale factor duality, isnow reproduced as a particular case of (16.10) with the trivial O(d, d) matrix = η, and for an isotropic background with B µν = 0.This O(d, d) symmetry holds even in the presence of matter sources,provided they transform according to the string equations of motion in the givenbackground [12]. In the perfect fluid approximation, for instance, the inversionof the scale factor corresponds to a reflection of the equation of state, whichpreserves however the ‘shifted’ energy ρ = ρ| det g ij | 1/2 :a →ã = a −1 , φ → φ, p/ρ →−p/ρ, ρ → ρ. (16.11)A detailed discussion of the duality symmetry is outside the purpose ofthese lectures. What is important, in our context, is the simultaneous presenceof duality and time-reversal symmetry: by combining these two symmetries, infact, it is possible in principle to obtain cosmological solutions of the ‘self-dual’type, characterized by the conditionsa(t) = a −1 (−t), φ(t) = φ(−t). (16.12)They are important, as they connect in a smooth way the phase of growingand decreasing curvature, and also describe a smooth evolution from the stringperturbative vacuum (i.e. the asymptotic no-interaction state in which φ →−∞and the string coupling is vanishing, g s = exp(φ/2) → 0), to the presentcosmological phase in which the dilaton is frozen, with an expectation value [13]〈g s 〉=M s /M p ∼ 0.3–0.03 (see figure 16.3).H g S = exp (ttFigure 16.3. Time evolution of the curvature scale H and of the string couplingg s = exp(φ/2) ≃ M s /M p , for a typical self-dual solution of the string cosmologyequations.