Physics Reports Chern–Simons modified general relativity

S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55 49The APS leptogenesis model can be naturally realized if the inflaton field is associated with a complex modulus field.In such a case, one must guarantee that the inflaton potential is sufficiently flat, such that the slow-roll conditions aresatisfied. The simplest model of this kind is that of single-field inflation and a pseudo-scalar φ as the inflaton, known asnatural inflation. Such a model can be expanded to include multiple axions, as in N-inflation models [184]. Such models fitinto extensions of the SM and in string-inspired inflationary models. 13In the remainder of this chapter, we shall use particle physics notation and conventions, where h = c = 1 andM Pl = (8πG) −1/2 ∼ 2.44 × 10 18 GeV is the reduced Planck mass. In particular, we shall follow [186] and choose α = 8κ,β = 0, and θ = f (φ), although we shall often work with the dimensionless functional F(φ) := f (φ)/M 2 Pl .8.3.2. Gravitational wave evolution during inflationAlthough GW solutions have already been discussed in Section 5, it is instructive to present the expansion of theLagrangian to second order in the metric perturbation. In the TT gauge, assuming a GW perturbed FRW metric and using aleft/right basis for the GW polarization, (see Eq. (175)):[( ∂2L = −(h L □h R + h R □h L ) + 16iF(φ)∂z h ∂ 22 R∂t∂z h L − ∂ )2∂z h ∂ 22 L∂t∂z h R( ∂2+ a 2 ∂t h ∂ 22 R∂t∂z h L − ∂ ) ( 2∂t h ∂ 2∂ 2 L∂t∂z h R + Ha 2 ∂t h ∂ 2R∂t∂z h L − ∂ )]∂t h ∂ 2L∂t∂z h R + O(h 4 ) (309)where t stands for cosmic time dt = a(η)dη and □ = ∂ 2 t+ 3H∂ t − ∂ 2 z /a2 . As is clear from Eq. (309), if h L and h R have thesame dispersion relation, ∗ R R vanishes, while otherwise ‘‘cosmological birefringence’’ is induced.From this Lagrangian, the equations of motion for h L and h R become□ h L = −2i Θ ḣ ′ La,□ h R = +2i Θ ḣ ′ Ra, (310)where dots denote time derivatives, and primes denote differentiation of F with respect to φ. The quantity Θ ∼ 2F ′ H ˙φ/M 2 Pl ,which with the functional form of F(φ) becomes Θ = √ 2ɛ/(2π 2 )H 2 /M 2 Pl N , where ɛ = ˙φ 2 /2 (HM Pl ) −2 is the slow-rollparameter of inflation [186]. We have here used the fact that the inflaton is purely time-dependent and we have neglectedterms proportional to ¨φ by the slow-roll conditions.Gravitational birefringence is present in the solution to the equations of motion. Let us focus on the positive frequencycomponent of the evolution of h L and adopt a basis in which h L depends on (t, z) only. In terms of conformal time, then, theevolution equation for h L becomesd 2dη h 2 L − 2 1 dη dη h L − d2dz h 2 L = −2iΘd2dηdz h L, (311)which is a special case of Eq. (177). If we ignore Θ for the moment and let h L ∼ e ikz , this becomes the equation of a sphericalBessel function, for which the positive frequency solution ish + L (k, η) = e+ik(η+z) (1 − ikη). (312)Let us now peel-off the asymptotic behavior of the solution with non-zero Θ. Let thenh L = e ikz · (−ikη)e kΘη g(η) (313)where g(η) is a Coulomb wave function that satisfies[d 2dη g + k 2 (1 − Θ 2 ) − 2 2 η − 2kΘ ]g = 0, (314)2 ηwhich in turn is the equation of a Schrödinger particle with l = 1 in a weak Coulomb potential. For h L , the Coulomb term isrepulsive, while for h R this potential is attractive. Such a fact leads to attenuation of h L and amplification of h R in the earlyuniverse, which is equivalent to the exponential enhancement/suppression effect discussed in Section 5.As we shall see, generation of baryon asymmetry is dominated by modes at short distances (sub-horizon modes) and atearly times. This corresponds to the limit kη ≫ 1. In this region, we can ignore the potential terms in Eq. (314) and take thesolution to be approximately a plane wave. More explicitly,g(η) = exp[ik(1 − Θ 2 ) 1/2 η(1 + α(η))], (315)where α(η) ∼ log η/η.13 For a concise review of string-inspired inflation see [185].