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

Motivations: duality symmetry 283coupling constants), the so-called ‘string perturbative vacuum’. In this scenariothe phase of high, but finite (nearly Planckian) curvature is what replaces the bigbang singularity of the standard scenario. It thus comes naturally, in a stringcosmology context, to call ‘pre-big bang’ [8] the initial phase with growingcurvature, in contrast to the subsequent, standard, ‘post-big bang’ phase withdecreasing curvature.At this point, a number of questions may arise naturally. In particular:• Motivations: why such a cosmological scenario, characterized by a ‘belllike’shape of the curvature, seems to emerge in a string cosmology contextand not, for instance, in the context of standard cosmology based on theEinstein equations?• Kinematics: in spite of the differences, is the kinematic of the pre-big bangphase still appropriate to solve the well-known problems (horizon, flatness...) of the standard scenario? After all, we do not want to lose the mainachievements of the conventional inflationary models.• Phenomenology: are there phenomenological consequences that candiscriminate between string cosmology models and other inflationarymodels? Are such effects observable, at least in principle?In the following sections we will present a quick discussion of the threepoints listed above.16.2 Motivations: duality symmetryThere are various motivations, in the context of string theory, suggesting acosmological scenario like that illustrated in figure 16.1. All the motivations arehowever related, more or less directly, to an important property of string theory,the duality symmetry of the effective action.To illustrate this point, let us start by recalling that in general relativity thesolutions of the standard Einstein action,S =− 1 ∫2λ d−1 d d+1 x √ |g|R (16.1)p(d is the number of spatial dimensions, and λ p = Mp−1 is the Planck lengthscale), are invariant under ‘time-reversal’ transformations. Consider, for instance,a homogeneous and isotropic solution of the cosmological equations, representedby a scale factor a(t):ds 2 = dt 2 − a 2 (t) dxi 2 . (16.2)If a(t) is a solution, then also a(−t) is a solution. On the other hand, when t goesinto −t, the Hubble parameter H =ȧ/a changes sign,a(t) → a(−t), H =ȧ/a →−H. (16.3)