Physics Reports Chern–Simons modified general relativity

30 S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55where α = κ, commas in index lists stand for partial differentiation, η is conformal time and ˜ɛ ijk = ˜ɛ ηijk . Variation of thelinearized action with respect to the metric perturbation yields the linearized field equations, namely [29]¯□h j i := 1 a 2 ˜ɛpk(j [( ϑ ,ηη − 2Hϑ ,η)hi)k,pη + ϑ ,η ¯□h i)k,p], (177)where ¯□ is the D’Alembertian operator associated with the background, namely¯□f = f ,ηη + 2Hf ,η − δ ij f ,ij , (178)with f some function of all coordinates and the conformal Hubble parameter H := a ,η /a. One could have, of course, obtainedthe same linearized field equations by perturbatively expanding the C-tensor.One can see from Eq. (177) that the evolution of GW perturbations is governed by second and third derivatives of theGW tensor. Jackiw and Pi [22] were the first to point out that for the canonical choice of ϑ the GW evolution is governedby the D’Alembertian of flat space only, if we neglect corrections due to the expansion history of the Universe (i.e. if thisvanishes, then the linearized modified field equations for the GW perturbation are satisfied to linear order). Such a resultimplies there are two linearly independent polarizations that propagate at the speed of light.Let us now concentrate on gravitational wave perturbations, for which one can make the ansatzh ij = A ija(η) exp [ −i ( φ(η) − κn k χ k)] , (179)where the amplitude A ij , the unit vector in the direction of wave propagation n k and the conformal wavenumber κ > 0 areall constant. It is convenient to decompose the amplitude into definite parity states, such asA ij = A R e R ij + A Le L ij(180)where the circular polarization tensors e R,Lijare given in terms of the linear ones e +,×ijby [21]e R = 1 (kl√ e+kl+ ie × )kl2(181)e L kl = 1 √2(e+kl− ie × kl). (182)These polarization tensors satisfy the conditionn i ɛ ijk e R,Lkl= iλ R,L(ejl) R,L, (183)where λ R = +1 and λ L = −1.With this decomposition, the linearized modified field equations [Eq. (177)] reduce to[iφ,ηη R,L + ( )φ,ηR,L 2] (+ H ,η + H 2 − κ 2 1 − λ )R,Lκϑ ,η= iλ R,Lκ ( ) (ϑ,ηη − 2Hϑ ,η φR,La 2 a 2,η − iH) . (184)Before attempting to solve this equation for arbitrary ϑ(η) and a(η), it is instructive to take the flat-space limit, that isa → 1, and thus, ȧ → 0. Assuming further that time derivatives of the CS scalar do not scale with H, one finds that theabove equation reduces to [29,33](i ¨φ R,L + ˙φ 2 R,L − k2) ( 1 − λ R,L k ˙ϑ ) = iλ R,L k ¨ϑ ˙φ R,L , (185)where k is the physical wavenumber 3-vector, t stands for cosmic time and overhead dots stand for partial differentiationwith respect to time. Let us further assume that the GW phase satisfies φ ,tt /φ 2 ,t≪ 1, which is the standard short-wavelengthapproximation, as well as ¨ϑ = ϑ 0 = const. Then the above equation can be solved to first order in ϑ to findφ(t) = φ 0 + kt + iλ R,Lϑ 0kt + O(ϑ) 2 , (186)2where φ 0 is a constant phase offset and the uncontrolled remainder O(ϑ) 2 stands for terms of the form k 2 ˙ϑ 2 or k ˙ϑ ¨ϑ. Theimaginary correction to the phase then implies an exponential enhancement/suppression effect of the GW amplitude, as thispropagates in CS modified gravity. Recall that here we are interested in the propagation of GWs, which is why the right-handside of Eq. (185) implicitly omits the stress–energy tensor. We shall see in the next section that if there is a stress–energytensor, then the CS correction depends both on the first and second derivatives of the CS scalar. Lastly, if we had not assumedthat ¨ϑ = const. then the solution would have becomeφ(t) = φ 0 + kt + iλ R,L2 k ˙ϑ(t) + O(ϑ) 2 , (187)which still exhibits the exponential suppression/enhancement effect.