Cosmological Perturbation Theory, 26.4.2011 version

A GENERAL PERTURBATION 62A General PerturbationFrom Eq. (3.8) we have that the general perturbed metric (around the flat Friedmann model) isds 2 = a 2 (η) { −(1 + 2A)dη 2 − 2B i dηdx i + [(1 − 2D)δ ij + 2E ij ] dx i dx j} .(A.1)The Christoffel symbols areΓ 0 00 = H + A ′ (A.2)Γ 0 0i = −HB i + A ,iΓ 0 ij= H [(1 − 2A − 2D)δ ij + 2E ij ] + 1 2 (B i,j + B j,i ) − δ ij D ′ + E ′ ijΓ i 00 = −HB i − B ′ i + A ,iΓ i 0j= Hδ ij + 1 2 (B j,i − B i,j ) − D ′ δ ij + E ′ ijΓ i jk = Hδ jk B i − δ i jD ,k − δ i k D ,j + δ jk D ,i + E ij,k + E ik,j − E jk,i .and we have the Christoffel sumsΓ µ 0µ = 4H + A ′ − 3D ′ (A.3)Γ µ iµ= A ,i − 3D ,i .Note that the Christoffel sums contain only scalar perturbations. Thus for vector and tensorperturbations, these sums contain only the background value Γ µ 0µ = 4H.The Einstein tensor isG 0 0 = −3a −2 H 2 + a −2 [ −2∇ 2 D + 6HD ′ + 6H 2 ]A − 2HB i,i − E ik,ik (A.4)G 0 i = a −2 [ −2D ,i ′ − 2HA ,i − 1 2 (B i,kk − B k,ik ) − E ik,k′ ]G i 0 = a −2 [ 2D ,i ′ + 2HA ,i + 1 2 (B i,kk − B k,ik ) + 2H ′ B i − 2H 2 B i + E ik,k′ ]G i j= a −2 ( −2H ′ − H 2) δ ij+a −2 [ 2D ′′ − ∇ 2 (D − A) + H(2A ′ + 4D ′ ) + (4H ′ + 2H 2 )A − B k,k ′ − 2HB ]k,k − E kl,kl δij+a −2 [ (D − A) ,ij + 1 2 (B′ i,j + B′ j,i ) + H(B i,j + B j,i ) + E ij ′′ − ∇2 E ij + E ik,jk + E jk,ik + 2HE ij′ ]