Cosmological Perturbation Theory, 26.4.2011 version

4 GAUGE TRANSFORMATIONS 7Thus we getand our final result for the gauge transformation of δs iss( ˜P) = s( ˆP) − ¯s ′ ξ 0 , (4.9)˜δs(x α ) = s( ˆP) − ¯s ′ ξ 0 − ¯s( ¯P) = ̂δs(x α ) − ¯s ′ ξ 0 . (4.10)In analogy with (4.8), the perturbations in vector and tensor fields in the two gauges aredefinedand̂δw α (x β ) ≡ wˆα ( ˆP) − ¯w α ( ¯P)˜δw α (x β ) ≡ w˜α ( ˜P) − ¯w α ( ¯P). (4.11)̂δA α β(x γ ) ≡˜δA α β(x γ ) ≡̂δB αβ (x γ ) ≡Aˆαˆβ( ˆP) − Āα β ( ¯P)A˜α˜β( ˜P) − Āα β ( ¯P)Bˆαˆβ( ˆP) − ¯B αβ ( ¯P)and˜δB αβ (x γ ) ≡ B˜α˜β( ˜P) − ¯B αβ ( ¯P). (4.12)Consider the case of a type (0,2) 4-tensor field. We haveBˆµˆν ( ˜P) = Bˆµˆν ( ˆP) + ∂Bˆµˆν [∂ˆx α ˆx α ( ˜P) − ˆx α ( ˆP)]= Bˆµˆν ( ˆP) − ∂ ¯B µν∂x α ( ¯P)ξ α (4.13)B˜µ˜ν ( ˜P) = X ˆρ˜µ Xˆσ˜ν Bˆρˆσ( ˜P) = (δ ρ µ − ξρ ,µ )(δσ ν − ξσ ,ν ) [= Bˆµˆν ( ˆP) − ξ ρ ,µ Bˆρˆν( ˆP) − ξ σ ,ν Bˆµˆσ( ˆP) − ∂ ¯B µν∂x α ( ¯P)ξ αBˆρˆσ ( ˆP) − ∂ ¯B ρσ∂x α ( ¯P)ξ α ]= Bˆµˆν ( ˆP) − ξ ρ ,µ ¯B ρν ( ¯P) − ξ σ ,ν ¯B µσ ( ¯P) − ∂ ¯B µν∂x α ( ¯P)ξ α , (4.14)where we can replace Bˆµˆσ ( ˆP) with ¯B µσ ( ¯P) in the two middle terms, since it is multiplied by afirst-order quantity ξ σ ,ν and we can thus ignore the perturbation part, which becomes secondorder.Subtracting the background value at ¯P we get the gauge transformation rule for the tensorperturbation δB µν ,˜δB µν ≡ B˜µ˜ν ( ˜P) − ¯B µν ( ¯P)= Bˆµˆν ( ˆP) − ¯B µν ( ¯P) − ξ ρ ,µ ¯B ρν ( ¯P) − ξ σ ,ν ¯B µσ ( ¯P) − ∂ ¯B µν∂x α ( ¯P)ξ α= ̂δB µν − ξ ρ ,µ ¯B ρν − ξ σ ,ν ¯B µσ − ¯B µν,α ξ α . (4.15)In a similar manner we obtain the gauge transformation rules for 4-vector perturbations,and perturbations of type (1,1) 4-tensors (exercise),˜δw α = ̂δw α + ξ α ,β ¯wβ − ¯w α ,β ξβ (4.16)˜δA µ ν = ̂δA µ ν + ξµ ,ρĀρ ν − ξσ ,νĀµ σ − Āµ ν,α ξα . (4.17)