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

322 Elementary introduction to pre-big bang cosmologyUnfortunately, each term in the action, at each loop order, is multiplied bya dilaton ‘form factor’ which is different in general for different fields andfor different orders. This difference can lead to an effective violation of theuniversality of the gravitational interactions [85] in the low-energy, macroscopicregime, and this violation can be reconciled with the present tests of theequivalence principle only if the dilaton is massive enough, to make short enoughthe range of the non-universal dilatonic interactions.The tree-level relation (16.136) is valid also for a higher-dimensionaleffective action, provided e φ represents the shifted four-dimensional dilatonwhich includes the volume of the extra-dimensional, compact internal space, andwhich controls the grand-unification gauge coupling, α GUT , as [13]α GUT = exp〈φ〉 =(M s /M p ) 2 ∼ 0.1–0.001. (16.138)However, the relation (16.136) is no longer valid, in general, if the gaugeinteractions are confined in four dimensions and only gravity propagates in theextra dimensions. In that case the relation depends on the volume of the extradimensions, whose size may be allowed to be large in Planck units [86]. Forinternal dimensions of volume V n the relation becomes, in particular,M 2 p = M2+n s V n e − , (16.139)where is the dilaton in d = 3 + n dimensions. In this case, the string massparameter could be much smaller than the value expected from equation (16.138),provided the internal volume is correspondingly larger.Appendix B. Duality symmetryNotations and conventionsIn this paper we use the metric signature (+−−−), and we define the Riemannand Ricci tensor as follows:R β µνα = ∂ µ Ɣ β να + Ɣ β µρ Ɣ ρ να − (µ ↔ ν),R να = R µ µνα .Consider the gravidilaton effective action, in the S-frame, to lowest order inα ′ and in the quantum loop expansion:S =− 1 ∫2λ d−1 d d+1 x √ |g|e −φ [R + (∇φ) 2 ]. (16.140)sFor a homogeneous, but anisotropic, Bianchi I type metric background:φ = φ(t), g 00 = N 2 (t), g ij =−a 2 i (t)δ ij, (16.141)