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

Motivations: duality symmetry 285Expandingpre-big bangHExpandingpost-big bang~H (-t)H (t)tH (-t)~H (t)Contractingpre-big bangContractingpost-big bangFigure 16.2. The four branches of a low-energy string cosmology background.(16.5), is only a special case of a more general, O(d, d) symmetry of the stringeffective action, which is manifest already at the lowest order. In fact, the treelevelaction in general contains, besides the metric and the dilaton, also a secondrank antisymmetric tensor B µν , the so-called Kalb–Ramond ‘universal’ axion:S =− 1 ∫2λ d−1 d d+1 x √ [|g|e −φ R + (∂ µ φ) 2 − 1 ]s12 (∂ [µB να] ) 2 . (16.6)Given a background, even anisotropic, but with d Abelian isometries, this actionis invariant under a global, pseudo-orthogonal group of O(d, d) transformationswhich mix in a non-trivial way the components of the metric and of theantisymmetric tensor, leaving invariant the so-called ‘shifted’ dilaton φ:φ = φ − ln √ | det g ij |. (16.7)In the particular, ‘cosmological’ case in which we are interested, the disometries correspond to spatial translations (namely, we are in the case of ahomogeneous, Bianchi I type metric background). For this background, the action(16.6) can be rewritten in terms of the (2d × 2d) symmetric matrix M,defined bythe spatial components of the metric, g ij , and of the axion, B ij , as:(G−1−GB ≡ B ij , G ≡ g ij , M =−1 )BBG −1 G − BG −1 . (16.8)BIn the cosmic time gauge, the action takes the form (see appendix B)S =− λ s2∫dt e[( −φ ˙φ) 2 + 1 ]8 Tr Ṁ(M−1 )˙ , (16.9)