Cosmological Perturbation Theory, 26.4.2011 version

8 CONFORMAL–NEWTONIAN GAUGE 18andδΓ 0 00 = Φ′ δΓ 0 0k = Φ ,k δΓ 0 ij = − [2H(Φ + Ψ) + Ψ′ ] δ ijδΓ i 00 = Φ ,i δΓ i 0j = −Ψ′ δj i δΓ i kl = − ( Ψ ,l δk i + Ψ ,kδl) i + Ψ,i δ kl .The Ricci tensor is(8.10)R µν = Γ α νµ,α − Γ α αµ,ν + Γ α αβ Γβ νµ − Γ α νβ Γβ αµ= ¯R µν + δΓ α νµ,α − δΓ α αµ,ν + ¯Γ α αβ δΓβ νµ + ¯Γ β νµδΓ α αβ − ¯Γ α νβ δΓβ αµ − ¯Γ β αµδΓ α νβ . (8.11)Calculation givesR 00 = −3H ′ + 3Ψ ′′ + ∇ 2 Φ + 3H(Φ ′ + Ψ ′ )R 0i = 2(Ψ ′ + HΦ) ,iR ij= (H ′ + 2H 2 )δ ij+ [ −Ψ ′′ + ∇ 2 Ψ − H(Φ ′ + 5Ψ ′ ) − (2H ′ + 4H 2 )(Φ + Ψ) ] δ ij+ (Ψ − Φ) ,ij (8.12)Next we raise an index to get R µ ν. Note that we can not just raise the index of the backgroundand perturbation parts separately, sinceR µ ν = g µα R αν = (ḡ µα + δg µα )( ¯R αν + δR αν ) = ¯R µ ν + δg µα ¯Rαν + ḡ µα δR αν . (8.13)We getR 0 0 = 3a −2 H ′ + a −2 [ −3Ψ ′′ − ∇ 2 Φ − 3H(Φ ′ + Ψ ′ ) − 6H ′ Φ ]R 0 i = −2a −2 ( Ψ ′ + HΦ ) ,iR i 0 = −R 0 i = 2a −2 ( Ψ ′ + HΦ ) ,iR i j= a −2 (H ′ + 2H 2 )δ i jand summing for the curvature scalar+ a −2 [ −Ψ ′′ + ∇ 2 Ψ − H(Φ ′ + 5Ψ ′ ) − (2H ′ + 4H 2 )Φ ] δ ij+ a −2 (Ψ − Φ) ,ij . (8.14)R = R 0 0 + R i i= 6a −2 (H ′ + H 2 )+ a −2 [ −6Ψ ′′ + 2∇ 2 (2Ψ − Φ) − 6H(Φ ′ + 3Ψ ′ ) − 12(H ′ + H 2 )Φ ] . (8.15)And, finally, the Einstein tensorG 0 0G 0 i= R 0 0 − 1 2 R= −3a −2 H 2 + a −2 [ −2∇ 2 Ψ + 6HΨ ′ + 6H 2 Φ ]= R 0 iG i 0 = R0 i = −Ri 0 = −G 0 i= Rj i − 1 2 δi j RG i j= a −2 (−2H ′ − H 2 )δ i j+ a −2 [ 2Ψ ′′ + ∇ 2 (Φ − Ψ) + H(2Φ ′ + 4Ψ ′ ) + (4H ′ + 2H 2 )Φ ] δ i j+ a −2 (Ψ − Φ) ,ij . (8.16)