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

Inflation 225as a first approximation we can assume that this amplification takes place as longas k < V (η). The last inequality, apart from irrelevant numerical factors due toour approximation, turns out to have the same analytical form of the condition:kR < R′R 2 ≡ H (η),that relates the physical momentum of the perturbation to the Hubble parameterH , whose inverse corresponds to the horizon size: this is just the conditionanticipated above, i.e. that a perturbation is amplified when its wavelengthbecomes larger than the event horizon.13.2.2 Calculation of the spectrumTo calculate the spectrum of perturbations we need to find particular solutionsof the Schrödinger-like equation previously introduced. Let us, then, specify ourmodel: the form of the potential V is determined by the evolution of R(η); forour purposes it is sufficient to consider a three-stage model in which, as shownin figure 13.3, a de Sitter phase (characterized by a constant Hubble parameterH ds ) is followed by a radiation-dominated (RD) and, then, by a matter-dominated(MD) era:⎧− 1H ds η , η < η 1 < 0 de Sitter⎪⎨ η − 2η 1R(η) ∼H ds η 2 , η 1