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

Cosmological perturbation theory 307To give an explicit example we shall consider here a very simple modelconsisting of two cosmological phases, an initial accelerated evolution up to thetime η 1 , and a subsequent radiation-dominated evolution for η>η 1 :a ∼ (−η) α , η < η 1 ,a ∼ η, η > η 1 . (16.87)The effective potential for tensor perturbations in the E-frame, |a ′′ /a|, starts fromzero at −∞, grows like η −2 , reaches a maximum ∼ η1 −2 , and vanishes in theradiation phase. We must solve the canonical perturbation equation for ηη 1 . In the first phase the equation reduces to a Bessel equation:[]ψ k ′′ + k 2 α(α − 1)−η 2 ψ k = 0, (16.88)with general solution [60]ψ k =|η| 1/2 [AH (2)ν (|kη|) + BH (1)ν (|kη|)], ν =|α − 1/2|, (16.89)where H ν(1,2) are the first- and second-kind Hankel functions, of argument kη andindex ν =|α − 1/2| determined by the kinematics of the background. By usingthe large argument limit of the Hankel functions for η →−∞,H (2)ν (kη) ∼ 1 √ kηe −ikη , H (1)ν (kη) ∼ 1 √ kηe +ikη , (16.90)we choose initially a positive frequency mode, normalizing the solution to avacuum fluctuation spectrum,A = 1/2, B = 0. (16.91)In the second phase V = 0, and we have the free oscillating solution:ψ k = 1 √k(c + e −ikη + c − e +ikη ). (16.92)The matching of ψ and ψ ′ at η = η 1 gives now the coefficients c ± (moreprecisely, the matching would require the continuity of the perturbed metricprojected on a spacelike hypersurface containing η 1 , and the continuity of theextrinsic curvature of that hypersurface [61]; but in many cases these conditionsare equivalent to the continuity of the canonical variable ψ, and of its first timederivative).For an approximate determination of the spectrum, which is often sufficientfor practical purposes, it is convenient to distinguish two regimes, in whichthe comoving frequency k is much higher or much lower than the frequencyassociated to the top of the effective potential barrier, |V (η 1 )| 1/2 ≃ η −11. In the