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

328 Elementary introduction to pre-big bang cosmologyWe may note, at this point, that M is a symmetric matrix of the pseudoorthogonalO(d, d) group. In fact,for any B and G. Therefore,M T ηM = η, M = M T (16.179)Mη = ηM −1 , (Ṁη) 2 = η(M −1 )˙Mη, (16.180)and the action can be finally rewritten asS =− λ ∫ [ ]sdt e −φ 1 ˙φ2 +28 Tr Ṁ(M−1 )˙ . (16.181)This form is explicitly invariant under the global O(d, d) transformations (16.10),preserving the shifted dilaton:φ → φ, M → T M , T η = η. (16.182)In factTr ˙˜M( ˜M −1 )˙=Tr[ T Ṁ −1 (M −1 )˙( T ) −1 ] = Tr Ṁ(M −1 )˙. (16.183)In the absence of the antisymmetric tensor M is diagonal, and the special O(d, d)transformation with = η corresponds to an inversion of the metric tensor:M = diag(G −1 , G),˜M = T M = ηMη = diag(G, G −1 ) ⇒ ˜G = G −1 . (16.184)For a diagonal metric G = a 2 I, and the invariance under the scale factor dualitytransformation (16.5) is recovered as a particular case of the global O(d, d)symmetry of the low-energy effective action.Appendix C. The string cosmology equationsIn order to derive the cosmological equations let us include in the action, forcompleteness, the antisymmetric tensor B µν , a dilaton potential V (φ), and alsothe possible contribution of other matter sources represented by a Lagrangiandensity L m :S = − 1 ∫d d+1 x √ [|g|e −φ R + (∇φ) 2 − 1 ]12 H µνα 2 + V (φ)2λ d−1 s∫+ d d+1 x √ |g|L m . (16.185)In a scalar–tensor model of gravity, especially in the presence of higherderivative interactions, it is often convenient to write the action in the language of