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

Motivations: duality symmetry 287The explicit occurrence of self-dual solutions and, more generally, ofsolutions describing a complete and smooth transition between the phase of preandpost-big bang evolution, seems to require in general the presence of higherorder (higher loop and/or higher derivative) corrections to the string effectiveaction [14] (see, however, [15, 16]). So, in order to give only a simple exampleof combined {duality ⊕ time-reversal} transformation, let us consider here thelow-energy, asymptotic regimes, which are well described by the lowest ordereffective action.By adding matter sources, in the perfect fluid form, to the action (16.4), thestring cosmology equations for a d = 3, homogeneous, isotropic and conformallyflat background can be written as (see appendix C, equations (16.200), (16.202),(16.199), respectively):˙φ 2 − 6H ˙φ + 6H 2 = e φ ρ,Ḣ − H ˙φ + 3H 2 = 1 2 eφ p,2 ¨φ + 6H ˙φ − ˙φ 2 − 6Ḣ − 12H 2 = 0. (16.13)For p = ρ/3, in particular, they are exactly solved by the standard solution withconstant dilaton (see equations (16.216) and (16.217)),a ∼ t 1/2 , ρ = 3p ∼ a −4 , φ = constant, t →+∞, (16.14)describing decelerated expansion and decreasing curvature scale:ȧ > 0, ä < 0, Ḣ < 0. (16.15)This is exactly the radiation-dominated solution of the standard cosmologicalscenario, based on the Einstein equations. In string cosmology, however, to thissolution is associated a ‘dual complement’, i.e. an additional solution which canbe obtained by applying on the background (16.14) a time-reversal transformationt →−t, and the duality transformation (16.11):a ∼ (−t) −1/2 , φ ∼−3ln(−t), ρ =−3p ∼ a −2 , t →−∞. (16.16)This is still an exact solution of equations (16.13) (see appendix C), describinghowever accelerated (i.e. inflationary) expansion, with growing dilaton andgrowing curvature scale:ȧ > 0, ä > 0, Ḣ > 0. (16.17)We note, for future reference, that accelerated expansion with growing curvatureis usually called ‘superinflation’ [17], or ‘pole-inflation’, to distinguish it from themore conventional power-inflation, with decreasing curvature.The two solutions (16.14) and (16.16) provide a particular, explicitrepresentation of the scenario illustrated in figure 16.3, in the two asympotic