Physics Reports Chern–Simons modified general relativity

46 S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55I ≡ 〈 (a 1 ) 2 + (a 2 ) 2〉 , Q ≡ 〈 (a 1 ) 2 − (a 2 ) 2〉 , (290)U ≡ 〈2a 1 a 2 cos δ〉 , V ≡ 〈2a 1 a 2 sin δ〉 , (291)where δ ≡ ɛ 2 − ɛ 1 and the brackets signify a time average over a time long compared to ω −1 . The I parameter measures theintensity of the radiation, while the parameters Q , U, and V each carry information about the polarization of the radiation.Unpolarized radiation (so-called natural light) is described by Q = U = V = 0. The linear polarization of the radiation isencoded in Q and U, while the parameter V is a measure of elliptical polarization with the special case of circular polarizationocurring when a 1 = a 2 and δ = ±π/2. From here on we will simply refer to V as the measure of circular polarization, whichis technically correct if Q = 0.While I and V are coordinate independent, Q and U depend on the orientation of the coordinate system used on the planeorthogonal to the light’s direction of propagation. Under a rotation of the coordinate system by an angle φ, the parametersQ and U transform according toQ ′ = Q cos(2φ) + U sin(2φ), and U ′ = −Q sin(2φ) + U cos(2φ),while the angle defined byΦ = 1 ( ) U2 arctan ,Qgoes to Φ − φ following a rotation by the angle φ. Therefore, Q and U only define an orientation of the coordinate systemand not a particular direction in the plane: after a rotation by π they are left unchanged.Physically, such transformations are simply a manifestation of the oscillatory behavior of the electric field, which indicatethat Q and U are part of a second-rank symmetric trace-free tensor P ij , i.e. a spin-2 field in the plane orthogonal to thedirection of propagation. Such a tensor can be represented as( )P 0P ij = , (292)0 −Pin an orthonormal eigenbasis, where P = (Q 2 + U 2 ) 1/2 is ususally called the magnitude of linear polarization. For example,in two-dimensional, spherical polar coordinates (θ, φ), the metric is Ω ij = diag(1, sin 2 θ) and the polarization tensor is()Q (ˆn) −U(ˆn) sin θP ij (ˆn) =−U(ˆn) sin θ −Q (ˆn) sin 2 , (293)θwhere we recall that (i, j) run over the angular sector only.The temperature pattern on the CMB can be expanded in a complete orthonormal set of spherical harmonics:whereT(ˆn)T 0= 1 +a T (lm) = 1 T 0∫∞∑l=1 m=−ll∑a T (lm) Y (lm)(ˆn) (294)dˆn T(ˆn)Y(lm) ∗ (ˆn) (295)are the coefficients of the spherical harmonic decomposition of the temperature/polarization map and T 0 is the mean CMBtemperature. Likewise, we can also expand the polarization tensor in terms of a complete set of orthonormal basis functionsfor symmetric trace-free 2 × 2 tensors on the 2-sphere,P ij (ˆn)T 0=∞∑l=2 m=−ll∑ [aG(lm)Y G (ˆn) + (lm)ij aC (lm) Y C (ˆn)] (lm)ij, (296)where the expansion coefficients are given bya G (lm) = 1 ∫dˆnP ij (ˆn)Y G ij ∗(lm)T (ˆn), aC (lm) = 1 ∫0 T 0anddˆn P ij (ˆn)Y C ij ∗(lm)(ˆn). (297)The basis functions Y G (ˆn) (lm)ij and Y C (lm)ij(ˆn) are given in terms of covariant derivatives of spherical harmonics by(Y G = (lm)ij N l Y (lm):ij − 1 )2 Ω kijY (lm):k , (298)Y C = N l( )(lm)ij Y(lm):ik ɛ k j + Y (lm):jk ɛ k i , (299)2