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

324 Elementary introduction to pre-big bang cosmologyso that the quadratic action (16.147) is clearly invariant under the inversion of anyscale factor preserving the shifted dilaton,a i →ã i = a −1i, φ → φ (16.149)(for ‘scale factor duality’, see [9] and the first paper of [8]).In order to derive the field equations, it is convenient to use the variablesβ i = ln a i , so that H i = ˙β i , Ḣ i = ¨β i , and the action (16.147) is cyclic in β i .By varying with respect to N,β i and φ, and subsequently fixing the cosmic timegauge N = 1, we obtain, respectively,˙φ 2 − ∑ iH 2i = 0, (16.150)Ḣ i − H i ˙φ = 0, (16.151)2 ¨φ − ˙φ 2 − ∑ iH 2i = 0. (16.152)This is a system of (d + 2) equations for the (d + 1) variables {a i ,φ}. However,only (d + 1) equations are independent (see, for instance, [21]: equation (16.150)represents a constraint on the set of initial data).The above equations are invariant under a time reversal transformationt →−t,H →−H,˙φ →−˙φ, (16.153)and also under the duality transformation (16.149). If we invert k ≤ d scalefactors, ã 1 = a1 −1 ,...,ã k = ak −1 , the shifted dilaton is preserved, φ = ˜φ,providedfrom which:φ = φ −d∑ln a i = ˜φ −i=1k∑ln ã i −i=1˜φ = φ − 2d∑i=k+1ln a i , (16.154)k∑ln a i . (16.155)i=1Given an exact solution, represented by the set of variables{a 1 ,...,a d ,φ}, (16.156)the inversion of k ≤ d scale factors defines then a new exact solution, representedby the set of variables{a −11 ,...,a−1 k, a k+1 ,...,a d ,φ− 2lna 1 ,...,−2lna k }. (16.157)