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

Cosmological perturbation theory 299In terms of the conformal time coordinate, dη = dt/a, this wave equation can befinally rewritten, for each polarization mode, as˜h ′′ + 2 ã′ã ˜h ′ −∇ 2 ˜h = 0, (16.50)(where we have explicitly reinserted the tilde, and where a prime denotesdifferentiation with respect to the conformal time, which is the same in theEinstein and in the string frame, according to equations (16.27) and (16.31)).Let us now repeat the computation in the S-frame, where the backgroundequations for the metric (equation (16.196) with no contribution fromH µνα , T µν , V and σ ) can be written explicitly asPerturbing to first order,R µ ν + g να (∂ µ ∂ α φ − Ɣ µα ρ ∂ ρ φ) = 0. (16.51)δ (1) R µ ν − (δ (1) g να Ɣ µα 0 + g να δ (1) Ɣ µα 0 ) ˙φ = 0. (16.52)The (0, 0) component, as well as the perturbation of the dilaton equation, aretrivially satisfied. The (i, j) components, using again the identities (16.48), leadto [31]:£h j i − ˙φḣ j i = 0. (16.53)In conformal time, and for each polarization component,)h ′′ +(2 a′a − φ′ h ′ −∇ 2 h = 0. (16.54)This last equation seems to be different from the E-frame equation (16.50).Recalling, however, the relation (16.27) between a and ã, wehave2 ã′ã = 2 a′a − φ′ , (16.55)so that we have the same equation for h and ˜h, the same solution, and the samespectrum when the solution is expanded in Fourier modes. The perturbationanalysis is thus frame-independent, and we can safely choose the more convenientframe to compute the spectrum.16.5.2 Choice of the gaugeThe second step is the choice of the gauge, i.e. the choice of the coordinate systemwithin a given frame. The perturbation spectrum is, of course, gauge-independent,but the the perturbative analysis is not, in general. It is possible, in fact, that thevalidity of the linear approximation is broken in a given gauge, but still valid in adifferent, more appropriate gauge.