Cosmological Perturbation Theory, 26.4.2011 version

7 SCALAR PERTURBATIONS 16In Fourier space this readsThe components of h µν areh µν =ψ ⃗k = D ⃗k − 1 3 E ⃗ k. (7.3)[ −2A B,iB ,i −2ψδ ij + 2E ,ij]. (7.4)Exercise: Curvature of the spatial hypersurface. The hypersurface η = const. is a 3-dimensional curved manifold. Calculate the connection coefficients (3) Γ i jkand the scalar curvature(3) R ≡ g ij(3) R ij of this 3-space for a scalar perturbation in terms of ψ and E.Now if we start from a pure scalar perturbation and do an arbitrary gauge transformation,represented by the field ξ µ = (ξ 0 ,ξ i ), we may introduce also a vector perturbation. This vectorperturbation is however, pure gauge, and thus of no interest. Just like we did for the shift vectorB i earlier, we can divide ξ i into a part with zero divergence (a transverse part) and a part withzero curl, expressible as a gradient of some function ξ,ξ i = ξ i tr − δij ξ ,j = ⃗ ξ tr − ∇ξ where ξ i tr,i = ∇ · ⃗ξ tr = 0. (7.5)The part ξtr i is responsible for the spurious vector perturbation, whereas ξ0 and ξ ,j change thescalar perturbation. For our discussion of scalar perturbations we thus lose nothing, if we decidethat we only consider gauge transformations, where the ξtr i part is absent. These “scalar gaugetransformations” are fully specified by two functions, ξ 0 and ξ,˜η = η + ξ 0 (η,⃗x)˜x i = x i − δ ij ξ ,j (η,⃗x) (7.6)and they preserve the scalar nature of the perturbation.Applied to scalar perturbations and gauge transformations, our transformation equations(4.27,4.28,4.31) becomeà = A − ξ 0 ′ a ′−a ξ0˜B = B + ξ ′ + ξ 0˜D = D − 1 3 ∇2 ξ + a′a ξ0Ẽ = E + ξ , (7.7)where we use the notation ′ ≡ ∂/∂η for quantities which depend on both η and ⃗x. The quantityψ defined in Eq. (7.2) is often used as the fourth scalar variable instead of D. For it, we get7.1 Bardeen Potentials˜ψ = ψ + a′a ξ0 = ψ + Hξ 0 . (7.8)We now define the following two quantities, called the Bardeen potentials,Φ ≡ A + H(B − E ′ ) + (B − E ′ ) ′Ψ ≡ D + 1 3 ∇2 E − H(B − E ′ ) = ψ − H(B − E ′ ). (7.9)These quantities are invariant under gauge transformations (exercise).