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

224 Sources of SGWBalso a stochastic background of gravitational waves. The basic concept is quitesimple: zero point quantum fluctuations of every sufficiently light field can beamplified when the size of the perturbation becomes larger than the event horizon.This is the classical interpretation of the well-known fact that a time-dependentgravitational field produces particles even if the initial state contains no quanta(see [45] for details).13.2.1 Classical pictureSince the final number of quanta created is typically very large, it comes out thata classical analysis of the phenomenon is appropriate and gives the right answer.So, let us show how this amplification works by studying the classical equation ofmotion for gravitational perturbations; we will return to the quantum descriptionlater.The starting point is to separate the metric into a ‘background’ 1 and a‘propagating’ part:g µν ∼ R 2 (η)(−dη 2 + d⃗x 2 ) + h µν ,where R is the scale factor of the universe, depending only on the conformal timeη (in terms of the usual cosmic time dη = dt/R(η)). The equation of motion forh µν in the FRW background is obtained by taking the first-order variation of theEinstein equations. One can defineh µ ν ∼ 1R(η) εµ ν ( ⃗k)ψ(η)e i ⃗k·⃗x , (13.20)where ε ν µ (⃗k) is the polarization tensor. It comes out that, with a suitable gaugechoice, ψ obeys the following equation:( )ψ ′′ (η) + k 2 − R′′ (η)ψ ∼ 0, (13.21)R(η)where ‘ ′ ’ denotes the derivative with respect to the conformal time. This is aSchrödinger-like equation with the potential given by V = R ′′ /R. Fork ≫ V (η)we have for ψ the obvious plane wave solution ψ ∼ e −ikη , so that |h µ ν |∼1/R;if, instead, k ≪ V (η), we get the two following solutions:ψ 1 ∼ R(η),∫ψ 2 ∼ R(η)dηR 2 (η) .As will be seen later, in the case of our interest ψ 1 is the dominant solution, so, inthis case, |h µ ν |∼1. This means that a solution characterized by a long wavelength(k ≪ V (η)) is amplified with respect to a short wavelength one by a factor R(η);1 For the sake of simplicity we consider a spatially flat universe.