Physics Reports Chern–Simons modified general relativity

S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55 43For convenience, we choose specify the CS coupling functional f [φ] via [28,31]N φf =. (263)16π 2 M 2 MPl Plwhere M Pl ≡ m Pl / √ 8π is the reduced Planck mass and N a number that can be related to the string scale. In terms of theslow-roll parameters ɛ ≡ −Ḣ/H 2 , δ ≡ − ¨φ/(H ˙φ) and ξ ≡ (˙ɛ − ˙δ)/H (an overhead dot means a derivative with respect tocosmic time), we have at leading order in these parameters (see also Ref. [167])a(η) ∼ (−η) −1−ɛ , φ ′ ≃ −M Pl H √ 2ɛ. (264)From these expressions, one deduces that to leading ordera = − N2 16π 2 M 2 Plf ′H √2ɛ ≃a 2N16π 2 (HinfM Pl) 2√2ɛη (265)because H ≃ −(1 + ɛ)/η.The equation of motion is still in a rather involved form, but it can be simplified further by introducing the characteristicscale k C and Θ parameter [28,31]k C ≡ kN ( ) 2 Hinf √2ɛ = k Θ 32π 2 M Pl 16 ,Θ ≡ N2π 2 ( HM Pl) 2√2ɛ. (266)Let us further introduce the variable x ≡ Θkη/8 < 0, such that Eq. (261) takes the form[ ]d 2 µ 64dx + 2 Θ − f s(x) µ = 0, (267)2withf s (x) = 2 + 3ɛx 2 +λ sx(1 − λ s x) − 1 14 (1 − λ s x) , (268)2Using slow-roll perturbation theory and the functional relations for f (φ), the equation of motion can be maximally simplifiedinto[d 2 µ L kdτ + − 1 2 4 + iΘ16τ + 1 ]µ L4τ 2 k= 0. (269)where we have defined τ ≡ 16i(1+x)/Θ. We recognize Eq. (269) as the Whittaker equation [see eg. Eq. 9.220.1 of Ref. [168]],whose corresponding, normalized solution isµ L = − 4√ [ ]πl Plk√ e ikη i16i(1 + x)e −πΘ/32 W iΘ/16,0 , (270)2k Θwhere W κ,µ (z) is the Whittaker function.8.1.2. CS corrected inflationary gravitational waves on large scalesLet us now concentrate on super-horizon scales, i.e. x ∼ 0, such that we can make contact with CMB observables. Theeffective potentials can then be approximated viaf s (x) ≃ 2 + 3ɛx 2+ λ sx − 1 4 , (271)where the first term gives the standard slow-roll term behavior, while the second term represents the birefringent CScorrection. With this potential, the equation of motion becomesd 2 µ s kdx 2 +[ 64Θ + 1 2 4 − λ sx − 2 + 3ɛ ]µ sx 2 k= 0. (272)Using results from Ref. [169], we define√y ≡ i 1 + 256Θ x, κ ≡ iλ s√1 2 + 256/Θ , ξ ≡ 3 + ɛ. (273)2 2