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

288 Elementary introduction to pre-big bang cosmologyTable 16.1. Analogy between supersymmetry and duality.SupersymmetryDuality + Time-reversalPair of partners {bosons, fermions} {growing curvature, decreasing curvature}Known states photons, gravitons, . . . decelerated, standard post-big bangPredicted photinos, gravitinos, . . . accelerated, inflationary pre-big bangregimes of t large and positive, and t large and negative, respectively. The dualitysymmetry seems thus to provide an important motivation for the pre-big bangscenario, as it leads naturally to introduce a phase of growing curvature, and is acrucial ingredient for the ‘bell-like’ scenario of figure 16.3.It should be noted that pure scale factor duality, by itself, is not enough toconvert a phase of decreasing into growing curvature (see for instance figure 16.2,where it is clearly shown that H and ˜H, in the same temporal range, lead tothe same evolution of the curvature scale, H 2 ∼ ˜H 2 ). Time reflection is thusnecessarily required, if we want to invert the curvature behaviour. From this pointof view, time-reversal symmetry is more important than duality.In a thermodynamic context, however, duality by itself is able to suggestthe existence of a primordial cosmological phase with ‘specular’ properties withrespect to the present, standard cosmological phase [18]. It must be stressed, inaddition, that it is typically in the cosmology of extended objects that the phaseof growing curvature may describe accelerated expansion instead of contraction,and that the growth of the curvature may be regularized, instead of blowingup to a singularity. For instance, it is with the string dilaton [8], or with anetwork of strings self-consistently coupled to the background [19], that we arenaturally lead to superinflation. Also, in quantum theories of extended objects,it is the minimal, fundamental length scale of the theory that is expected tobound the curvature, and to drive superinflation to a phase of constant, limitingcurvature [20] asymptotically approaching de Sitter, as explicitly checked in astring theory context [21]. Duality symmetries, on the other hand, are typical ofextended objects (and of strings, in particular), so that it is certainly justified tothink of duality as of a fundamental motivation and ingredient of the pre-big bangscenario.Duality is an important symmetry of modern theoretical physics, and toconclude this section we would like to present an analogy with another veryimportant symmetry, namely supersymmetry (see table 16.1).According to supersymmetry, to any bosonic state is associated a fermionicpartner, and vice versa. From the existence of bosons that we know to bepresent in nature, if we believe in supersymmetry, we can predict the existenceof fermions not yet observed, like the photino, the gravitino, and so on.In the same way, according to duality and time-reversal, to any geometrical