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

332 Elementary introduction to pre-big bang cosmologyIn terms of these variables, the time and space equations (16.200) and (16.202),and the dilaton equation (16.199), become, respectively:˙φ 2 − ∑ iH 2i = e φ ρ, (16.205)Ḣ i − H i ˙φ =12e φ p i , (16.206)2 ¨φ − ˙φ 2 − ∑ iH 2i = 0. (16.207)They are explicitly invariant under the scale-factor duality transformation:a i → ai −1 , φ → φ, ρ → ρ, p →−p (16.208)which implies, for a perfect fluid source, a ‘reflection’ of the equation of state,γ = p/ρ = p/ρ → −p/ρ = −γ (see the first paper in [8]). A generalO(d, d) transformation changes, however, the equation of state in a more drasticway (see [12]), introducing also shear and bulk viscosity.The above (d + 2) equations are a system of independent equations forthe (d + 2) variables {a i ,φ,ρ}. Their combination implies the usual covariantconservation of the energy density. By differentiating equation (16.205), andusing (16.206) and (16.207) to eliminate Ḣ i , ˙φ, respectively, we get in fact˙ρ + ∑ iH i p i = 0, (16.209)which, using the definitions (16.204), is equivalent to˙ρ + ∑ iH i (ρ + p i ) = 0. (16.210)In order to obtain exact solutions, it is convenient to include this energyconservation equation in the full system of independent equations.In these lectures we will present only a particular example of the matterdominatedsolution by considering a d-dimensional, isotropic backgroundcharacterized by a power-law evolution,a ∼ t α , φ ∼−β ln t, p = γρ. (16.211)We use (16.205), (16.207) and (16.209) as independent equations. The integrationof equation (16.209) gives immediatelyequation (16.205) is then satisfied providedρ = ρ 0 a −dγ ; (16.212)dγα+ β = 2. (16.213)