Physics Reports Chern–Simons modified general relativity

S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55 31The exponentially growing modes could be associated with instabilities in the solution to gravitational wave propagation.One must be careful, however, to realize that the results above have been obtained within the approximation k 2 ˙ϑ 2 ≪ 1 ≫k ˙ϑ ¨ϑ. Thus, provided ˙ϑ is smaller than the age of the universe, then the instability time scale will not have enough time toset in. For larger values of ˙ϑ, the approximate solutions we presented above break down and one must account for higherorder corrections. A final caveat to keep in mind is that these results are derived within the non-dynamical formulation ofthe theory; GW solutions in the fully dynamical theory are only now being actively investigated.Let us now return to the field equation for the phase in an FRW background [Eq. (184)]. The solution to this equation isnow complicated by the fact that the scale factor also depends on conformal time, and thus, one cannot find a closed formsolution prior to specifying the evolution of a(η). Let us then choose a matter-dominated cosmological model, in whicha(η) = a 0 η 2 = a 0 /(1 + z), where a 0 is the value of the scale factor today and z is the redshift. It then follows that theconformal Hubble parameter is simply given by H = 2/η = 2(1 + z) 1/2 . With this choice, one can now compute the CScorrection to the accumulated phase as the plane-wave propagates from some initial conformal time to η, namely [33]∫ 1[ 1∆φ (R,L) = iλ R,L kH 04 ϑ ,ηη(η) − 1 ] dηη ϑ ,η(η)η + 4 O(ϑ)2 , (188)ηwhich one can check reduces to the flat space result of above in the right limit. Note that the exponentialenhancement/suppression effect now depends on an integrated measure of the evolution of the CS scalar and the scale factor.The CS correction to the GW amplitude derives from a modification to the evolution equations of the gravitationalperturbation, but it also leads to important observational consequences. One of these can be understood by considering aGW generated by a binary black hole system in the early inspiral phase. The GW produced by such a system can be describedas followsh R,L = √ 2 M ( )Mk0 (t) 2/3(1 + λR,L cos ι ) 2 [ ( )]exp −i Ψ (t) + ∆φR,L , (189)d L 2where d L = a 0 η (1 + z) is the luminosity distance to the binary’s center of mass, Ψ (t) is the GW phase described by GR,k 0 (t) is the instantaneous wave number of the gravitational wavefront passing the detector and M is the comoving chirpmass, which is a certain combination of the binary mass components. The inclination angle ι, the angle subtended by theorbital angular momentum and the observer’s line of sight, can be isolated ash R= 1 + cos ι [ ] 2k(t)1 − cos ι exp ζ = 1 + cos ῑ1 − cos ῑ , (190)h LH 0where we have defined∫ 1[ 1ζ := H 2 0η 4 ϑ ,ηη(η) − 1 ] dηη ϑ ,η(η)η . (191)4We see then that from an GW observational standpoint, the CS correction leads to an apparent inclination angle ῑ, whicheffectively modifies the actual inclination angle by a factor that depends on the integrated history of the CS correction:sinhcos ῑ =cosh(k 0 (t)ξ(z))H 0+ cosh( )k 0 (t)ξ(z)H 0+ sinh(k 0 (t)ξ(z))cosH 0ι( )k 0 (t)ξ(z)cosH 0ι∼ cos ι + k 0(t)ξ(z)H 0sin 2 ι + O ( ξ 2) . (192)We see then that the CS correction effectively introduces an apparent evolution of the inclination angle, which tracks thegravitational wave frequency.The interpretation of the CS correction as inducing an effective inclination angle should be interpreted with care. In GR, if agravitational wave propagates along the line of sight, such that the actual inclination angle is (eg. 0 or π), then the amplitudeis a maximum. In CS gravity, however, the amplitude can be either enhanced or suppressed, depending on whether the waveis right- or left-circularly polarized. When the CS effect suppresses the GW amplitude, one can think of this as an effectivemodification of the inclination angle away from the maximum. However, when the CS effect enhances the amplitude, thereis no real inclination angle that can mimic this effect (i.e. the effective angle would have to be imaginary).The evolution equation for the gravitational wave perturbation depends sensitively on the scale factor evolution[see, eg. Eq. (177)]. Alexander and Martin [29] have investigated gravitational wave solutions when the scale factorpresents an inflationary behavior. Suffice it to say in this section that Eq. (177) can be recast in the form of a parametricoscillator equation, with a non-trivial effective potential. In certain limits appropriate to inflation, one can solve thisdifferential equation in terms of Whittaker functions, which can be decomposed into products of trigonometric functionsand exponentials. In essence, the solutions present the same structure as that of a matter-dominated cosmology. Once thegravitational wave modes have been computed, one can proceed to calculate the power spectrum, but these results will bediscussed further in Section 8.1.