By Cox R.T., Hubbard J.C.

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In a first step we have 1 1 δΓi i0 = g ij (δgij,0 + δgj0,i − δgi0,j ) + δg iν (gνi,0 + gν0,i − δgi0,ν ), 2 2 62 so δΓi i0 = 1 2 1 ij γ δgij,0 + δg ij (a2 ),0 γij . 233) becomes δΓi i0 = (3D + △E)′ . 236) ˙ + 3D] ˙ = 0. 56), (ρδ)′ + 3Hρδ + 3HpπL + (ρ + p)[△V + 3D ′ ] = 0. 238) Momentum equation For ν = i eq. 227) gives δ(T µ i;µ ) = δT µ i,µ + δΓµ µλ T λ i + Γµ µλ δT λ i − δΓλ µi T µ λ − Γλ µi δT µ λ . 229), δT µ i,µ δΓµ µj T λ i Γµ µλ δT λ i −δΓλ µi T µ λ −Γλ µi δT µ λ = = = = = δT j i,j + δT 0 i,0 , δΓµ µj T j i = δΓ0 0j T j i + δΓk kj T j i , Γµ µ0 δT 0 i + Γµ µj δT j i = 4HδT 0i + Γk kj δT j i , −δΓ0 µi T µ 0 − δΓj µi T µ j = −δΓ0 0i T 0 0 − δΓj kiT k j , −Γ0 µi δT µ 0 − Γj µi δT µ j = −Hγij δT j 0 − HδT 0 i − Γj ki δT k j .

42) 3 39 Sometimes we shall also use the quantity Q := a(ρ + p)(v − B), in terms of which the energy flux density can be written as 1 δT 0 i = Q,i , (⇒ T t i = Q,i ). 44) where cs is the sound velocity c2s = p/ ˙ ρ. 45) Γ measures the deviation between δp/δρ and p/ ˙ ρ. ˙ As for the metric we have four perturbation amplitudes: δ := δρ/ρ , v , Γ , Π . 46) Let us see how they change under gauge transformations: δT µ ν → δT µ ν + (Lξ T (0) )µ ν , (Lξ T (0) )µ ν = ξ λT (0)µ ν,λ − T (0)λ ν ξ µ ,λ + T (0)µ λ ξ λ,ν .

The following relations between the three gauge invariant density perturbations are useful. 171) gives ∆α = ∆sα − 3H(1 + wα )(1 − qα )Vα . 173) implies ∆cs = ∆sα − 3H(1 + wα )(1 − qα )V. 56), Vα = vα + E ′ . 176) From now on we use similar notations for the total density perturbations: ∆ := δQ , ∆s := δχ (∆ ≡ ∆c ). 160). For instance, ρα ∆cα = α αρα δα + 3H(v − B) hα (1 − qα ) = ρδ + 3H(v − B)h = ρ∆. 181) pα Πα . 182) α hV = α pΠ = α We would like to write also pΓ in a manifestly gauge invariant form.

### A Statistical Quantum Theory of Regular Reflection and Refraction by Cox R.T., Hubbard J.C.

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