[Note] Geodesic equation for scalar, vector and tensor perturbations Toshiya Namikawa 212 1 Curl mode induced by vector and tensor perturbation 1.1 Metric Perturbation and Affine Connection The line element up to first order is given by ds 2 = a 2 (η)[ (1 + 2A)dη 2B i dηdx i + (γ ij + 2C ij )dx i dx j ]. (1) Since the path of null geodesic is not affected by conformal transformation, we set the scale factor a = 1. The first order perturbation of the affine connection, δγ µ νρ, is given by δγ µ νρ = 1 2ḡµν [ 2h ρσ Γσ νλ + h ρν,λ + h ρλ,ν h νλ,ρ ]. (2) Then, the affine connection Γ µ νρ(= Γ µ νρ + δγ µ νρ) up to first order becomes Γ = A, Γ i = Γ i = A,i, Γ ij = 1 2 (B i j + B j i ) + (C ij ), Γ i = γ ij A,j (γ ij B j ), Γ χ ij = (δ iθ δ jθ + δ iφ δ jφ sin 2 θ)χ, Γ i j = 1 2 (B i j Bi j) + (C i j), Γ θ ij = 1 χ (δ iθδ jχ + δ iχ δ jθ δ iφ δ jφ χ sin θ cos θ), Γ φ ij = 1 χ sin θ [(δ iφδ jχ + δ iχ δ jφ ) sin θ + (δ iφ δ jθ + δ iθ δ jφ )χ cos θ], δγ i jk = 1 2ḡii [ 2C il Γl jk + C ij,k + C ik,j C jk,i ], (3) where the prime denotes the derivative with respect to η. Note that components of the affine connection except for Γ i jk are first order quantity. 1
1.2 Geodesic Equation 1.2 Geodesic Equation The null geodesics parametrized by the affine parameter ˆλ are solutions of the geodesic equation, d 2 x µ (ˆλ) + Γ µ νρ(x(ˆλ)) dxν (ˆλ) dx ρ (ˆλ) =, (4) dˆλ 2 dˆλ dˆλ with ĝ µν (dx µ /dˆλ)(dx ν /dˆλ) = and dŝ 2 =. The -component of the geodesic equation (4) gives ( ) = d2 η dˆλ + dx ν dx ρ 2 2 Γ νρ dˆλ dˆλ = d2 η dη dˆλ + 2 Γ + 2Γ dη dx i i dˆλ dˆλ dˆλ + dx i dx j Γ ij dˆλ dˆλ ( ) 2 [ ] = d2 η dη dˆλ + Γ 2 + 2Γ dx i i dˆλ dη + dx i dx j Γ ij dη dη The i-component of the geodesic equation gives (5) ( ) = d2 x i dˆλ + dx ν dx ρ 2 Γi νρ dˆλ dˆλ = d2 x i 2 dη dˆλ + 2 Γi + 2Γ i dη dx j j dˆλ dˆλ dˆλ + dx j dx k Γi jk [ dˆλ dˆλ ( ) ] 2 [ ] ( ) dη d 2 x i = dˆλ dη + dxi d 2 η + Γ i 2 dη dˆλ 2 + 2Γ i dx j j dη + dx j dx k 2 dη Γi jk dη dη dˆλ [ ] ( ) = dxi d 2 η d 2 dη dˆλ + x i 2 dη + 2 Γi + 2Γ i dx j j dη + dx j dx k 2 dη Γi jk. (6) dη dη dˆλ Using Eq.(5), we obtain = F ẋ i + ẍ i + Γ i + 2Γ i jẋ j + Γ i jkẋ j ẋ k, (7) where the overdot represents the derivative with respect to η and we define 1.3 Solution of Geodesic Equations F Γ + 2Γ iẋ i + Γ ijẋ i ẋ j. (8) In the absence of the metric perturbation, the photons emitted at the last scattering surface would go straight. This means that, if we observe the photons in a direction (θ, ϕ), the unperturbed photons path is given by x i = (η η, θ, ϕ), where the quantity η is the conformal time at the observer. To solve Eq. (7), we introduce the perturbed components of the null geodesic, ξ i, such that x i = x i + ξ i. Then, at first order, Eq. (7) becomes = F δ i χ + ξ i + Γ i 2Γ i χ 2 Γ i χk ξ k + 2δΓ i χχ = ξ i 2 Γ i χk ξ k + F n i + Γ i 2Γ i χ + 2δΓ i χχ = ξ i 2 Γ i χk ξ k + G i, (9) where we define G i F δ i χ + Γ i 2Γ i χ + 2δΓ i χχ. (1) 2
1.3 Solution of Geodesic Equations Using Eq. (3), we obtain = ξ χ + G χ, (11) = ξ a 2 χ ξ a + G a. (12) where a = 2 and 3. Under the condition ξ i (η ) = x i, the solutions of Eq. (12) are η η1 ξ χ (η) = dη 1 G χ dη 2, (13) η η η [ η1 ] [ ξ a (η) = x a 2dη η1 ( η2 )] 2 + dη 1 exp C a G a 2dη 3 dη 2 exp, (14) η η χ η χ with the quantities C θ and C φ being constant. If we use the Born approximation, the expression for ξ a becomes η [ ξ a (η) x a ɛ 2 dη η1 ( )] 1 G = lim C a a (η 2, χ(η 2 )ˆn)dη 2 (η η 2 ) 2 (15) ɛ η ɛ (η η 1 ) 2 η ɛ ɛ 2 η η1 (η η 2 ) 2 = dη 1 dη 2 η η (η η 1 ) 2 Ga (η 2, χ(η 2 )ˆn) (16) = = = η η dη 2 η η η η 2 dη 1 (η η 2 ) 2 (η η 1 ) 2 Ga (η 2, χ(η 2 )ˆn) (17) (η η 2 )(η η 2 ) dη 2 G a (η 2, χ(η 2 )ˆn) η η η (18) χ (χ χ ) G a (η χ, χ ˆn). χ (19) χ Using the expression for the affine connection, we can rewrite Eq. (1) as G a = Γ a 2Γ a χ + 2δΓ a χχ (2) = γ aa A,a (γ aa B a ) γ aa (B χ a B a χ ) 2(γ aa C aχ ) + ḡ aa (2C aχ,χ C χχ,a ) (21) = γ aa {A,a (B a + 2C aχ ) B χ,a + B a,χ + 2C aχ,χ C χχ,a } (22) { } = γ aa a (A B χ C χχ ) (B a + 2C aχ ) + B a,χ + 2C aχ,χ (23) { = γ aa a (A B χ C χχ ) d } (B a + 2C aχ ), (24) where we use Hereafter, we use d = η + dxi dη x. (25) i Ψ = A B χ C χχ, (26) Ω a = B a 2C aχ. (27) 3
1.4 Deflection Angle 1.4 Deflection Angle Using the basis in the polar coordinate the deflection angle, d, is defined by [1] e r = e x sin θ cos ϕ + e y sin θ sin ϕ + e z cos θ (28) = e x 1 µ2 cos ϕ + e y 1 µ2 sin ϕ + e z µ, (29) e θ = e x cos θ cos ϕ + e y cos θ sin ϕ e z sin θ (3) = e x µ cos ϕ + e y µ sin ϕ e z 1 µ2, (31) e ϕ = e x sin ϕ + e y cos ϕ, (32) d(θ, ϕ) e r (θ + ξ θ, ϕ + ξ ϕ ) e r (θ, ϕ) e θ ξ θ + e ϕ ξ ϕ sin θ + O(ξ 2 ). (33) The convergence and rotation (D and D [2]) are given by 2D d = (χ [ ( s χ) Ψ χ s χ e θ θ dω ) θ + e ( ϕ Ψ sin θ ϕ dω )] ϕ (34) = (χ [ ( s χ) χ s χ dω θ Ψ e θ + e )] (35) sin θ = (χ [ ( s χ) 2 dω θ Ψ e θ χ s χ + e )], (36) sin θ 2D ( ) d = (χ [ ( s χ) χ s χ ( ) dω θ Ψ e θ + e )] (37) sin θ = (χ ( s χ) e ϕ χ s χ θ e ) ( θ dω θ e θ sin θ ϕ + e ) (38) sin θ = (χ ( s χ) 1 dω θ χ s χ sin θ ϕ ) dω ϕ. (39) θ Note that, if we only consider the scalar perturbations in the metric, the rotation vanishes and the expression for convergence is consistent with the results obtained in Ref. [3]. 1.5 Angular Power Spectrum Let us consider the vector mode in the metric perturbation, Ω; Ω(η χ, χ, θ, ϕ) = e ik (χer) {Ω (2π) 3 k,+ e + + Ω k, e }. (4) The perturbation Ω is divergence-less ( Ω = ). Note that the quantities Ω k,+ and Ω k, are the function of χ. If we choose we obtain k = ke z, e ± = e x ± ie y, (41) k e r = kµ, e ± e θ = µe ±iϕ, e ± e ϕ = sin ϕ ± i cos ϕ = ±i e ±iϕ. (42) 4
1.5 Angular Power Spectrum Since Ω i dx i = Ω dx (43) = ( Ω r e r + Ω θ e θ + Ω ϕ e ϕ ) (e r + χe θ dθ + χe ϕ sin θdϕ) (44) = Ω r + Ω θ χdθ + Ω ϕ χ sin θdϕ (45) we obtain Ω θ = χ(ω e θ ) = χµ (2π) 3 e ikχµ {Ω k,+ e iϕ + Ω k, e iϕ }, (46) Ω ϕ = χ 1 µ 2 (Ω e ϕ ) = iχ 1 µ 2 (2π) 3 e ikχµ {Ω k,+ e iϕ Ω k, e iϕ }, (47) Ω θ,ϕ = iχµ (2π) 3 e ikχµ {Ω k,+ e iϕ Ω k, e iϕ }, (48) Ω ϕ,θ = (2π) [iχµ 3 kχ2 (1 µ 2 )]e ikχµ {Ω k,+ e iϕ Ω k, e iϕ } (49) Substituting this into Eq. (39), the rotation becomes D = 1 2 = 1 2 (χ s χ) d χ s χ (2π) 3 e ikχµ kχ 2 1 µ 2 {Ω k,+ e iϕ Ω k, e iϕ } (5) (2π) 3 e ikχµ k 1 µ 2 {Ω k,+ e iϕ Ω k, e iϕ }. (51) The multipole coefficients of the rotation then become D l,m = 1 2 (2π) 3 = 1 2l + 1 (l m)! 2 4π (l + m)! 1 1 dµe ikχµ k(1 µ 2 ) 1/2 P l,m (µ) = 1 2l + 1 (l m)! 2 4π (l + m)! (2π) 3 1 2l + 1 (l 1)! = π 4π (l + 1)! 2π dµ dϕy l,m (ˆn)e ikχµ k(1 µ 2 ) 1/2 {Ω k,+ e iϕ Ω k, e iϕ } (52) (2π) 3 2π dϕe imϕ {Ω k,+ e iϕ Ω k, e iϕ } (53) dµe ikχµ k(1 µ 2 ) 1/2 2π{Ω k,+ P l, (µ)δ m, Ω k, P l,1 (µ)δ m,1 } (54) d 3 k (2π) 3 k {Ω k,+ () m δ m, Ω k, δ m,1 } 1 dµe ikχµ (1 µ 2 ) 1/2 P l,1 (µ). (55) 5
REFERENCES REFERENCES Using the following relation 1 dµe ±ikχµ (1 µ 2 ) 1/2 P l,1 (µ) = (±i) l+1 2l(l + 1) j l(x) x, (56) x=kχ Eq. (55) is rewritten as D l,m = 2l + 1 (l 1)! j l (x) 2π( i)l+1 l(l + 1) 4π (l + 1)! (2π) 3 x k {Ω k,+ () m δ m, Ω k, δ m,1 }. x=kχ (57) Finally, the angular power spectrum of rotation becomes References C 1 l l = D l,m 2l + 1 2 = 1 ( ) D l, 2l + 1 2 + D l,1 2 (58) m= l = 4π2 l 2 (l + 1) 2 2l + 1 (l 1)! d 3 k d 3 k (59) 2l + 1 4π (l + 1)! (2π) 3 (2π) 3 k j l(x) x k j l(x ) ( x=kχ x Ω k,+ Ω k,+ + Ω k, Ω k, ) (6) x =k χ l(l + 1) = dk k 2 j 2π χ χ l (kχ)j l (kχ )P ω (k, χ, χ ). (61) [1] A. Lewis, Lensed CMB simulation and parameter estimation, Phys. Rev. D71 (25) 838, [astro-ph/52469]. [2] C. Li and A. Cooray, Weak lensing of the cosmic microwave background by foreground gravitational waves, Phys. Rev. D74 (26) 23521, [astro-ph/64179]. [3] A. Lewis and A. Challinor, Weak Gravitational Lensing of the CMB, Phys. Rept. 429 (26) 1 65, [astro-ph/61594]. 6