Cosmological Perturbation Theory, 26.4.2011 version

10 FIELD EQUATIONS FOR SCALAR PERTURBATIONS IN THE NEWTONIAN GAUGE22Thus the anisotropic stress is gauge-invariant (being the traceless part of δTj i ). Note that theδρ and δp equations are those of a perturbation of a 4-scalar, as they should be, as ρ and p are,indeed, 4-scalars.9.2.2 Scalar PerturbationsFor scalar perturbations, v i = −v ,i and ξ i = −ξ ,i , so that we haveṽ = v + ξ ′˜Π = Π. (9.29)These hold both in coordinate space and Fourier space (we use the same Fourier convention forξ as for v and B).9.2.3 Conformal-Newtonian GaugeWe get to the conformal-Newtonian gauge by ξ 0 = −B + E ′ and ξ = −E. Thusδρ N = δρ + ¯ρ ′ (B − E ′ ) = δρ − 3H(1 + w)¯ρ(B − E ′ )δp N = δp + ¯p ′ (B − E ′ ) = δp − 3H(1 + w)c 2 s ¯ρ(B − E′ )v N = v − E ′Π = Π. (9.30)9.3 Scalar Perturbations in the Conformal-Newtonian GaugeFrom here on we shall (unless otherwise noted)1. consider scalar perturbations only, so that v i = −v ,i and B i = −B ,i2. use the conformal-Newtonian gauge, so that B = 0.Thus the energy tensor perturbation has the formδT µ ν = [ −δρN−(¯ρ + ¯p)v N ,i(¯ρ + ¯p)v N ,i δp N δ i j + ¯p(Π ,ij − 1 3 δ ij∇ 2 Π)]. (9.31)10 Field Equations for Scalar Perturbations in the NewtonianGaugeWe can now write the Einstein equationsδG µ ν = 8πGδT µ ν (10.1)for scalar perturbations in the conformal-Newtonian gauge. We have the left-hand side δG µ νfrom Sect. 8.1 and the right-hand side δT µ ν from Sect. 9.3:δG 0 0= a −2 [ −2∇ 2 Ψ + 6H(Ψ ′ + HΦ) ] = −8πGδρ NδG 0 i = −2a −2 ( Ψ ′ + HΦ ) ,i= −8πG(¯ρ + ¯p)v N ,iδG i 0 = 2a −2 ( Ψ ′ + HΦ ) ,i= 8πG(¯ρ + ¯p)v N ,iδG i j= a −2 [ 2Ψ ′′ + ∇ 2 (Φ − Ψ) + H(2Φ ′ + 4Ψ ′ ) + (4H ′ + 2H 2 )Φ ] δ i j+ a −2 (Ψ − Φ) ,ij = 8πG [ δp N δ i j + ¯p(Π ,ij − 1 3 δ ij∇ 2 Π) ] . (10.2)