Physics Reports Chern–Simons modified general relativity

S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55 47where ɛ ij is the completely antisymmetric tensor on the 2-sphere, a colon in an index list stands for covariant differentiationon the 2-sphere, and N 2 l≡ 2(l − 2)!/(l + 2)! is a normalization factor. Since the Y (lm) ’s provide a complete basis for scalarfunctions on the sphere, the Y G (lm)ij and Y C (lm)ijtensors provide a complete basis for gradient-type (G) and curl-type (C) STFtensors, respectively. This G/C decomposition is also known as the scalar/pseudo-scalar decomposition, which is similar tothe tensor spherical harmonic decomposition of Section 5.Integration by parts transforms Eqs. (297) into integrals over scalar spherical harmonics and derivatives of thepolarization tensor:a G (lm) = N lT 0∫a C (lm) = N lT 0∫dˆn Y ∗ (lm) (ˆn) P ij :ij (ˆn), (300)dˆn Y ∗ (lm) (ˆn) P ij :ik (ˆn)ɛ k j , (301)where the second equation uses the fact that ɛ ij :k = 0. Given that T and P ij are real, all of the multipoles must obey thereality condition a X (lm) ∗ = (−1)m a X (l,−m) , where X = {T, G, C}. The spherical harmonics Y (lm) and Y G (lm)ij have parity (−1)l , butthe tensor harmonics Y C (lm)ij have parity (−1)l+1 .The two-point statistics of the temperature/polarization map is then given viaC XX′l≡ 〈a X (lm) (aX′ (lm) )∗ 〉, (302)where the averaging is over all 2l + 1 values of m and over many realizations of the sky. This two-point statistic is thuscompletely specified by the six (TT , GG, CC, TG, TC, and GC) sets of multipole moments. If the temperature/polarizationdistribution is parity invariant, then C TCland C GClmust vanish due to the symmetry properties of the G/C tensor sphericalharmonics under parity transformations.Parity conservation, however, is a theoretical bias. Lue, Wang and Kamionkowski [134] provided the first time physicalscenario where C TCl= C GCl≠ 0 due to parity violation in the GW power spectrum of the CMB 12 . This physical scenarioconsisted of GWs sourced by the CS interaction term in Eq. (3) with (α, β) = (1, 0) and ϑ = f (φ) some polynomial functionof the inflaton field φ. As we have discussed, CS modified gravity leads to amplitude birefringence in GW propagation, whichin turn leads to an excess of left- over right-cicularly polarized GWs that lead to a non-vanishing C TCl[134].In order to understand this, consider GWs during the inflationary epoch. These waves stretch and become classical atwavelengths on the order of λ ∼ 1/µ, where µ ∼ 1/f ′ is some CS energy scale, until they eventually freeze as they becomecomparable to the Hubble radius. When the waves exit this radius, the fraction of the accumulated discrepancy betweenleft- and right-polarized GWs can be estimated through the indexɛ ∼ (M p /µ)(H/M P ) 3 ( ˙φ/H 2 ) 2 , (303)where H is the Hubble scale and f ′′ ∼ 1/µ 2 . The factor H 2 / ˙φ ∼ 10 −5 is associated with the amplitude of scalar densityperturbations, while H/M P < 10 −6 is related to the amplitude of tensor perturbations [134].Since long wavelength GWs produce temperature anisotropies of curl type, an excess of left- over right-polarized GWsproduces a nonzero C TCl. This is because the multipole coefficients a T,C(lm)will be non-vanishing (right) for circularly polarizedGW{a T (lm) = (δm,2 + δ m,−2 )A T l(k) even l (+),−i(δ m,2 − δ m,−2 )A T (k) l odd l (×), (304){a C (lm) = (δm,2 + δ m,−2 )A C l(k) even l (×),−i(δ m,2 − δ m,−2 )A C (k) l odd l ( + ), (305)where A T,Clare temperature brightness functions and (+, ×) stand for plus or cross, linear GW polarizations (see eg. [173]).Likewise the multipole coefficients for the gradient component of the CMB polarization are similar, with the replacementof A T,Clfor polarization brightness functions. For a left, circularly polarized GW, the sign of the even-l moments is reversed.The above equations allow one to understand why parity is not violated with linearly polarized GWs. For example, let usassume that only + modes are present, then C TClby construction. However, a right or left, circularly polarized GW possessesboth + and x modes, and thus the cross-correction is non-vanishing:C TCl= 2(2l + 1) −1 A T l (k)AC l(k). (306)Lue, Wang and Kamionkowski [134] conclude that the parameter ɛ in Eq. (303) could in principle be measured by apost-Planck experiment with a sensitivity of 35 µK, a result that was later confirmed by the more detailed study of Saito,et. al. [165]. Such results have aroused interest in the polarization detection community, pushing them to improve their12 Lue et al. noticed that the CS term violates both parity and time reflections. Thus, since gravity is insensitive to charge, CPT is conserved.