Cosmological Perturbation Theory, 26.4.2011 version

8 CONFORMAL–NEWTONIAN GAUGE 178 Conformal–Newtonian GaugeWe can use the gauge freedom to set the scalar perturbations B and E equal to zero. FromEq. (7.7) we see that this is accomplished by choosingξ = −Eξ 0 = −B + E ′ . (8.1)Doing this gauge transformation we arrive at a commonly used gauge, which has many names:the conformal-Newtonian gauge (or sometimes, for short, just the Newtonian gauge), the longitudinalgauge, and the zero-shear gauge. We shall denote quantities in this gauge with the suborsuperscript N. Thus B N = E N = 0, whereas you immediately see thatA N= ΦD N = ψ N = Ψ . (8.2)Thus the Bardeen potentials are equal to the two nonzero metric perturbations in the conformal-Newtonian gauge.From here on (until otherwise noted) we shall calculate in the conformal-Newtonian gauge.The metric is thus justds 2 = a(η) 2 [ −(1 + 2Φ)dη 2 + (1 − 2Ψ)δ ij dx i dx j] , (8.3)org µν = a 2 [ −1 − 2Φ(1 − 2Ψ)δ ij][ ]and g µν = a −2 −1 + 2Φ, (8.4)(1 + 2Ψ)δ ijor[ ]−2Φh µν =−2Ψδ ij[ ]and h µν −2Φ=. (8.5)−2Ψδ ij8.1 Perturbation in the Curvature TensorsFrom the conformal-Newtonian metric (8.3) we get the connection coefficients[Γ 0 00 = a′a + Φ′ Γ 0 0k = Φ ,k Γ 0 ij = a′a δ ij − 2 a′a]δ (Φ + Ψ) + Ψ′ ijand the sumsΓ i 00 = Φ ,i Γ i 0j = a′a δi j − Ψ′ δ i jΓ i kl = − ( Ψ ,l δk i + Ψ ,kδli ) (8.6)+ Ψ,i δ klΓ α 0α = 4a′ a + Φ′ − 3Ψ ′Γ α iα = Φ ,i − 3Ψ ,i (8.7)where we have dropped all terms higher than first order in the small quantities Φ and Ψ. Thusthese expressions contain only 0 th and 1 st order terms, and separate into the background andperturbation, accordingly:Γ α βγ = ¯Γ α βγ + δΓα βγ , (8.8)where¯Γ 0 00 = H ¯Γ0 0k= 0 ¯Γ0 ij = Hδ ij¯Γ i 00 = 0¯Γi 0j = Hδ i j¯Γ i kl = 0 (8.9)