Cosmological Perturbation Theory, 26.4.2011 version

4 GAUGE TRANSFORMATIONS 8Since the background is isotropic and homogeneous, and our background coordinate systemfully respects these properties, the background 4-vectors and tensors must be of the form[ ]¯w α = ( ¯w 0 ,⃗0) Ā α β = Ā0 0 01 , (4.18)03 δi jĀk and they depend only on the (conformal) time coordinate η. Using these properties we can writethe gauge transformation rules for the individual components of 4-scalar, 4-vector and type (1,1)4-tensor perturbations (we now drop the hats from the first gauge),˜δs = δs − ¯s ′ ξ 0˜δw 0 = δw 0 + ξ 0 ,0 ¯w 0 − ¯w 0 ,0ξ 0˜δw i = δw i + ξ i ,0 ¯w 0˜δA 0 0 = δA 0 0 − Ā0 0,0 ξ0 (4.19)˜δA 0 i = δA 0 i + 1 3 ξ0 ,iĀk k − ξ0 ,iĀ0 0˜δA i 0= δA i 0 + ξ i ,0Ā0 0 − 1 3 ξi ,0Āk k˜δA i j = δA i j − 1 3 δi jĀk k,0 ξ0 . (4.20)The following combinations (the trace and the traceless part of ˜δA i j) are also useful:˜δA k k= δA k k − Āk k,0 ξ0˜δA i j − 1 3 δi j ˜δA k k = δA i j − 1 3 δi jδA k k . (4.21)Thus the traceless part of δA i jis gauge-invariant!4.1 Gauge Transformation of the Metric PerturbationsApplying the gauge transformation equation (4.15) to the metric perturbation, we have˜δg µν = δg µν − ξ ρ ,µḡ ρν − ξ σ ,νḡ µσ − ḡ µν,0 ξ 0 , (4.22)where we have replaced the sum ḡ µν,α ξ α with ḡ µν,0 ξ 0 , since the background metric dependsonly on the time coordinate x 0 = η, and dropped the hats from the first gauge. Rememberingḡ µν = a 2 (η)η µν from Eq. (2.7), we haveandFrom Eqs. (3.1) and (3.4) we haveḡ µν,0 = 2a ′ aη µν (4.23)˜δg µν = δg µν + a 2 [−ξ ρ ,µ η ρν − ξ σ ,ν η µσ − 2 a′a η µνξ 0 ]. (4.24)[δg µν ] = a 2 [ −2A −Bi−B i −2Dδ ij + 2E ij](4.25)