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

Cosmological perturbation theory 305|h(k,t)|43with α’without α’21-40 -20 20 40 60 tFigure 16.6. Amplification of tensor fluctuations, with and without the higher-curvaturecorrections included in the canonical perturbation equation.the pump field is simply the scale factor. The typical cosmological background inwhich we are interested in should describe a transition fom an initial accelerated,inflationary evolution,η →−∞, a ∼|η| α , V ∼ η −2 , (16.79)to a final standard, radiation-dominated phase,η →+∞, a ∼ η, V = 0. (16.80)In this context, the evolution of fluctuations, initially normalized as inequation (16.77), can be described as a scattering of the canonical variableby an effective potential, according to the Schrödinger-like perturbationequation (16.75) (see figure 16.7).However, the differential variable in equation (16.75) is (conformal) time,not space. As a consequence, the eigenfrequencies represent (comoving) energies,not momenta. Thus, even normalizing the initial state to a positive frequencymode, as in equation (16.77), the final state is, in general, a mixture of positiveand negative frequency modes, i.e. of positive and negative energy states,η →+∞, ψ out ∼ c + e −ikη + c − e +ikη . (16.81)In a quantum field theory context, such a mixing represents a process of pairproduction from the vacuum. The coefficients c ± are the so-called Bogoliubovcoefficients, parametrizing a unitary transformation between |in〉 and |out〉 states.In matrix form, they connect the set of |in〉 annihilation and creation operators,{ψ in , b k , b † k } to the out ones {ψ out, a k , a † k}, as follows:a k = c + b k + c ∗ − b† −k , a† −k = c −b k + c ∗ + b† −k . (16.82)