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

Cosmological perturbation theory 297it is heavy enough (²10 −4 eV) to be far outside the sensitivity range of resonantgravitational detectors.The rest of this lecture will be devoted to discussing various theoreticaland phenomenological aspects of graviton production, in a general cosmologicalcontext and, in particular, in the context of the pre-big bang scenario. Let us startby recalling some basic notions of cosmological perturbation theory, which arerequired for the computation of the graviton spectrum.16.5 Cosmological perturbation theoryThe standard approach to cosmological perturbation theory is to start with a setof non-perturbed equations, for instance the Einstein or the string cosmologyequations,G µν = T µν , (16.39)to expand the metric and the matter fields around a given background solution,g µν → g (0)µν + δ(1) g µν , T µν → T (0)µν + δ(1) T µν , G (0)µν = T (0)µν , (16.40)and to obtain, to first order, a linearized set of equations describing the classicalevolution of perturbations,δ (1) G µν = δ (1) T µν . (16.41)In principle, the procedure is simple and straightforward. In practice, however,we have to go through a series of formal steps, that we list here in ‘chronological’order:• choice of the ‘frame’;• choice of the ‘gauge’;• normalization of the amplitude;• computation of the spectrum.16.5.1 Choice of the frameThe choice of the frame is that of the basic set of fields (metric included) usedto parametrize the action. The action, in general, can be expressed in terms ofdifferent fields. In string cosmology, for instance, there is a preferred frame, theS-frame, in which the lowest order gravidilaton action takes the form of (16.20).It is preferred because the metric appearing in the action is the same as the σ -model metric to which test strings are minimally coupled (see appendix A): withrespect to this metric, the motion of free strings is then geodesics. It is alwayspossible, however, through the field redefinition√2˜g µν = g µν e −2φ/(d−1) , ˜φ = φ, (16.42)d − 1