Physics Reports Chern–Simons modified general relativity

44 S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55and simplify the equation of motion in the limit 256/Θ 2 ≫ 1, which corresponds to large scale behavior, such that Eq. (272)becomesd 2 µ s k+[− 1dy 2 4 + κ ( ) ]1 1y + 4 − ξ 2 µ sy 2 k= 0. (274)The exact general solution to this equation is given in terms of Whittaker functionsµ s k (η) = C s 1 (k)W κ,ξ (y) + C s 2 (k)W −κ,ξ (−y) , (275)where C s(k) 1 and C s 2(k) are two constants of integration.Initial conditions are determined in the divergence-free region −1 < x ≪ 0, where we assume µ s kbehavior [169]has a plane waveC s (k) = − 4√ ( )πl Pl1√ e iqη iexp − λs πΘ, C s 2(k) = 0, (276)2k32where l Pl is the Planck length and we have used that W κ,ξ (y) ∼ e −y/2 y κ as y → +∞. This assumption is equivalent torequiring that non-linear phenomena occurring near the divergence of the effective potential does not affect the initialconditions in the region x > −1. Indeed, such an assumption is also made in inflationary cosmology, where the vacuumis assumed to be the correct initial state, in spite of the fact that modes of astrophysical interest today originate from thetrans-Planckian region [170]. A possible weakness of the above comparison is that, in the case of the trans-Planckian problemof inflation [170,171], one can show that (under certain conditions) the final result can be robust to changes in the shortdistance physics [170,171]. In the present context, however, it is more difficult to imagine that the non-linearities will notaffect the initial conditions. On the other hand, in the absence of a second-order calculations and as a first approach to theproblem, this seems to be quite reasonable.8.1.3. Power spectrum and tensor-to-scalar ratioThe power spectrum can be calculated either as the two-point correlation function in the vacuum state or as a classicalspatial average. We take here the latter view, since a fully consistent quantum formulation of the present theory is not yetavailable. The power spectrum is then defined as∫〈h ij (η, x) h ij (η, x)〉 = 1 Vdx h ij (η, x) h ij (η, x) , (277)with V = ∫ dx is the total volume. Using the properties of the polarization tensor, straightforward calculations show that〈h ij (η, x) h ij (η, x)〉 = 1π 2∑s=L,R∫ +∞dkfrom which we deduce the power spectrum∣k 3 P s k3 ∣∣∣∣h(k) =π 2∣µ s ∣∣∣∣2ka(η) √ 1 − λ s kf ′ /a 20∣ k k3 hs ∣ 2k, (278). (279)The power spectrum is usually proportional to 2k 3 /π 2 , where here the factor of 2 is missing because we have not summedover polarizations.The CS corrected power spectrum could be calculated exactly in terms of the Whittaker function, but only the large scalebehavior is needed so, in this regime, one hask 3 P s = 16 l 2 Plk −2ɛ Ɣ 2 (2ξ)hπ a 2 2 2ξ |Ɣ (1/2 + ξ − iλ0s Θ/16) | 2 e−λs πΘ/16 . (280)where Ɣ[·] is the Gamma function and α 0 is the value of the scale factor today. Eq. (280) can be expanded to first order inthe slow-roll parameter to obtainwith,k 3 P s h (k) = 16H2 infπm 2 Pl12 As (Θ)[1 − 2 (C + 1) ɛ − 2ɛ ln k ]− ɛB(Θ) , (281)k ∗(π π2A s ≡ 1 − λ s16 Θ + 384 − 1 )Θ 2 + O ( Θ 3) , B ≡ 4Ψ − 2Ψ (2), (282)256where Ψ is related to the derivative of the Gamma function via Ψ = Ɣ ′ /Ɣ. The amplitude of the CS corrected rightpolarizationstate is reduced while the one of the left-polarization state is enhanced. Moreover, at leading order in theslow-roll parameter, the spectral index remains unmodified, since n s = ( T d ln k 3 Ph) s /d ln k = −2ɛ for each polarizationstate.