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

292 Elementary introduction to pre-big bang cosmologyAfter integrationand the transformed solution takes the form:t ∼ ˜t d−1d+ √ d , (16.32)ã = (−˜t) 1/d ,e ˜φ = (−˜t) − √ 2(d−1)d, ˜t < 0, ˜t → 0 − . (16.33)One can easily check that this solution describes accelerated contraction withgrowing dilaton and growing curvature scale:dãd˜t< 0,d 2 ãd˜t 2 < 0,d ˜Hd˜t< 0,d ˜φd˜t> 0. (16.34)The same result applies if we transform other isotropic solutions from the stringto the Einstein frame, for instance the perfect fluid solution of appendix C,equation (16.216). We leave this simple exercise to the interested reader.Having discussed the ‘dynamical’ equivalence (in spite of the kinematicaldifferences) of the two classes of string cosmology metrics, IIa and IIb, it seemsappropriate at this point to stress the main dynamical difference between standardinflation, class I metrics, and pre-big bang inflation, class II metrics. Such adifference can be conveniently illustrated in terms of the proper size of the eventhorizon, relative to a given comoving observer.Consider in fact the proper distance, d e (t), between the surface of the eventhorizon and a comoving observer, at rest at the origin of an isotropic, conformallyflat background [22]:∫ tMd e (t) = a(t) dt ′ a −1 (t ′ ). (16.35)tHere t M is the maximal allowed extension, towards the future, of the cosmic timecoordinate for the given background manifold. The above integral converges forall the above classes of accelerated (expanding or contracting) scale factors. Inthe case of class I metrics we have, in particular,∫ ∞d e (t) = t β dt ′ t ′−β =tβ − 1 ∼ H −1 (t) (16.36)tfor power-law inflation (β >1, t > 0), and∫ ∞d e (t) = e Ht dt ′ e −Ht′ = H −1 (16.37)for de Sitter inflation. For class II metrics (β