Cosmological Perturbation Theory, 26.4.2011 version

16 SYNCHRONOUS GAUGE 38so thath ij = 1 3 hδ ij + (∂ i ∂ j − 1 3 δ ij∇ 2 )µ= −2ηδ ij + µ ,ij . (16.9)Unfortunately, the MB notation fro the synchronous curvature perturbation is the same as ournotation for conformal time (MB use τ for the latter). Let’s hope this does not lead to confusion.MB discuss the synchronous metric perturbation in terms of the variables h and η, instead ofthe pair h,µ or η,µ. In coordinate space this appears impractical, since to solve µ from h andη requires integration (η = 1 6 (−h + ∇2 µ)). However, MB work entirely in Fourier space, whereη = − 1 6(h + µ) or µ = −h − 6η. Thus, in Fourier space, the metric ish ij= −2D Z δ ij − 2ˆk iˆkj E Z + 2 3 EZ δ ij= 1 3 hδ ij − ˆk iˆkj µ + 1 3 µδ ij= ˆk iˆkj h + (ˆk iˆkj − 1 3 δ ij)6η , (16.10)which is MB Eq. (4).From Eq. (8.1), we get from the synchronous gauge to the Newtonian gauge byFrom Eq. (7.9), the Bardeen potentials areξ Z→N = −E Z = − 1 2 µ (16.11)ξ 0 Z→N = EZ ′ = −ξ ′ Z→N .Φ = −HE Z ′ − E Z ′′ = − 1 2 Hµ′ − 1 2 µ′′Ψ = ψ Z + HE Z′ = η + 1 2 Hµ′ . (16.12)One has to be careful when Fourier transforming equations which employ different Fourierconventions for different terms. Thus, in Fourier space, Eq. (16.11) readsand Eq. (16.12) readsξ Z→N = − 12k µ = 1 (h + 6η) (16.13)2kξ 0 Z→N = − 1 k ξ′ Z→N (16.14)Φ = 12k 2 (−Hµ ′ − µ ′′) = 12k 2 [h ′′ + 6η ′′ + H(h ′ + 6η ′ ) ]Ψ = η − 12k 2 H(h′ + 6η ′ ), (16.15)which is MB Eq. (18).The opposite transformation, from Newtonian to synchronous gauge is, of course, justξ 0 N→Z = −ξ0 Z→N ξ N→Z = −ξ Z→N , (16.16)so we can easily express it in synchronous gauge quantities, which is actually what we typicallywant.