[Note] Geodesic equation for scalar, vector and tensor perturbations

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1 [Note] Geodesic equation for scalar, vector and tensor perturbations Toshiya Namikawa 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

2 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

3 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

4 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

5 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

6 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 ( ) D l, 2l 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