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

Appendix B. Duality symmetry 325By inverting all the scale factors we obtain the transformation{a i ,φ}→{a −1i,φ− 2d∑ln a i } (16.158)which, in the isotropic case, corresponds in particular to the duality transformation(16.5).As a simple example, we consider here the particular isotropic solutioni=1a = t 1/√d , φ =−ln t, (16.159)which satisfies identically the set of equations (16.150)–(16.152). By applyinga duality and a time-reversal transformation we obtain the four different exactsolutions{a ± (t) = t ±1/√d , φ(t) =−ln t},{a ± (−t) = (−t) ±1/√d , φ(−t) =−ln(−t)}, (16.160)corresponding to the four branches illustrated in figure 16.2, and describingdecelerated expansion, a + (t), decelerated contraction, a − (t), acceleratedcontraction, a + (−t), accelerated expansion, a − (−t). The solution describesexpansion or contraction if the sign of ȧ is positive or negative, repectively, andthe solution is accelerated or decelerated if ȧ and ä have the same or the oppositesign, respectively.It is important to consider also the dilaton behaviour. According toequation (16.146):φ ± (±t) = φ(±t) + d ln a ± (±t) = (± √ d − 1) ln(±t). (16.161)It follows that, in a phase of growing curvature (t < 0, t → 0 − ), the dilaton isgrowing only for an expanding metric, a − (−t). This means that, in the isotropiccase, there are only expanding pre-big bang solutions, i.e. solutions evolving fromthe string perturbative vacuum (H → 0,φ →−∞), and then characterized by agrowing string coupling, ġ s = (exp φ/2)˙ > 0.In the more general, anisotropic case, and in the presence of contractingdimensions, a growing curvature solution is associated to a growing dilaton onlyfor a large enough number of contracting dimensions. To make this point moreprecisely, consider the particular, exact solution of equations (16.150)–(16.152)with d expanding and n contracting dimensions, and scale factors a(t) and b(t),respectively:a = (−t) −1/√ d+n , b = (−t) 1/√ d+n , φ =−ln(−t), t → 0 − . (16.162)This gives, for the dilaton,φ = φ + d ln a + n ln b = n − d − √ d + n√d + nln(−t), (16.163)