Cosmological Perturbation Theory, 26.4.2011 version

16 SYNCHRONOUS GAUGE 39We get the Einstein tensor perturbations from Eq. (A.7), setting A = B = 0,δG 0 0 = a −2 [ 2k 2 ψ + 6HD ′] (16.17)δG 0 i = a −2 [ −2ik i ψ ′]δG i 0 = a −2 [ 2ik i ψ ′]δG i j= a −2 [ 2D ′′ + k 2 D + 4HD ′ − 1 3 k2 E + 1 3 E′′ + 2 3 HE′] δji−k i k j a[D −2 − 1 3 E + 1 ]k 2(E′′ + 2HE ′ )In the MB η, h notation,δG i i = a −2 [ 6D ′′ + 2k 2 ψ + 12HD ′] .δG 0 0 = a −2 [ 2k 2 η − Hh ′] (16.18)δG 0 i = a −2 [ −2ik i η ′]δG i 0 = a −2 [ 2ik i η ′]δG i j= a −2 [ − 1 2 h′′ − η ′′ − H(h ′ + 2η ′ ) + k 2 η ] δji−k i k j a[η −2 − 1 ]k 2(1 2 h′′ + 3η ′′ + Hh ′ + 6Hη ′ )The Einstein equations are thusδG i i = a −2 [ −h ′′ + 2k 2 η − 2Hh ′] .k 2 η − 1 2 Hh′ = −4πGa 2 δρ Z = − 3 2 H2 δ Z (16.19)k 2 η ′= 4πGa 2 (¯ρ + ¯p)kv Z = 3 2 H2 (1 + w)kv Zh ′′ + 2Hh ′ − 2k 2 η = −24πGa 2 δp Z = −9H 2δpZ¯ρh ′′ + 6η ′′ + 2Hh ′ + 12Hη ′ − 2k 2 η = −16πGa 2¯pΠ = −6H 2 wΠ,which is MB Eq. (21). Note that MB uses the notationθ ≡ ∇ · ⃗v = −∇ 2 v = kv and (¯ρ + ¯p)σ ≡ 2 3 ¯pΠ. (16.20)We get the continuity equations from Eq. (A.15), setting A = B = 0 and D = − 1 6 h,δρ Z ′ = −3H(δρ Z + δp Z ) − (¯ρ + ¯p)( 1 2 h′ + kv Z ) (16.21)(¯ρ + ¯p)v Z ′= −(¯ρ + ¯p) ′ v Z − 4H(¯ρ + ¯p)v Z + kδp Z − 2 3 k¯pΠ)δ Z ′= −(1 + w)(kv Z + 1 2 h′ ) + 3H(wδ Z − δpZ¯ρv Z ′ = −H(1 − 3w)v Z − w′1 + w vZ + kδpZ¯ρ + ¯p − 2 3w1 + w kΠ.The two last equations are MB Eq. (30).Exercise: Derive the synchronous gauge Einstein equations and continuity equations from the correspondingNewtonian gauge equations by gauge transformation .