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

306 Elementary introduction to pre-big bang cosmologyFigure 16.7. Scattering and amplification of the canonical variable.Thus, even starting from the vacuum,n in =〈0|b † b|0〉 =0, (16.83)we end up with a final number of produced pairs which is nonzero, in general,and is controlled by the Bogoliubov coefficient c − asn out =〈0|a † a|0〉 =|c − | 2 ̸= 0. (16.84)In a second quantization approach, the amplification of perturbations canthus be seen as a process of pair production from the vacuum (or from anyotherwise specified initial state), under the action of a time-dependent externalfield (the gravidilaton background, in the string cosmology case). Equivalently,the process can be described as a ‘squeezing’ of the initial state [56] (thisdescription is useful to evaluate the associated entropy production [57]), or, in asemiclassical language, as a ‘parametric amplification’ [58] of the wavefunctionψ k , which is scattered by an effective potential barrier through an ‘antitunnelling’process [59]. Quite independently of the adopted language, the differential energydensity of the produced radiation, for each mode k, depends on the number ofproduced pairs, and can be written asd 3 kdρ k = 2kn k(2π) 3 , n k =|c − (k)| 2 . (16.85)The computation of the so-called energy spectrum, defined as the spectral energydensity per logarithmic interval of frequency,dρ kdlnk ≡ k dρ kdk = k4π 2 |c −(k)| 2 , (16.86)thus requires the computation of c − (k), and then the knowledge of the asymptoticsolution of the canonical pertubation equation at large positive times.