Cosmological Perturbation Theory, 26.4.2011 version

17 FLUID COMPONENTS 4017 Fluid Components17.1 Division into ComponentsIn practice, the cosmological fluid consists of many components (e.g., particle species: photons,baryons, CDM, neutrinos, . ..). It is useful to divide the energy tensor into such components:T µ ν = ∑ iT µ ν(i) , (17.1)which have their corresponding background and perturbation parts¯T µ ν = ∑ i¯T µ ν(i)and δT µ ν = ∑ iδT µ ν(i) . (17.2)Here i labels the different components (so I’ll use l and m for space indices).Often the component pressure is a unique function of the component energy density, p i =p i (ρ i ), or can be so approximated (common cases are p i = 0 and p i = ρ i /3), although this is nottrue for the total pressure and energy density.The total fluid and component quantities for the background are related aswhere¯ρ = ∑ ¯ρ i¯p = ∑ ¯p i = ∑ w i¯ρ iw ≡c 2 s ≡ ¯p′¯ρ ′ =¯p¯ρ = ∑ ¯ρ i¯ρ w i∑ ¯p′i¯ρ ′= ∑ ¯ρ ′ i¯ρ ′ c2 i , (17.3)w i ≡ ¯p iand c 2 i¯ρ ≡ ¯p′ ii ¯ρ ′ . (17.4)iThe total fluid and component quantities for the perturbations are related asδρ = ∑ δρ iδp = ∑ δp iδ ≡δρ¯ρ = ∑ ¯ρ i¯ρ δ i , (17.5)where δ i ≡ δρ i /¯ρ i . Fromwe get thatand from(¯ρ + ¯p)v l = ∑ (¯ρ i + ¯p i )v l(i) (17.6)v l = ∑ ¯ρ i + ¯p i¯ρ + ¯p v l(i) (17.7)Σ lm = ∑ Σ lm(i) = ∑ ¯p i Π lm(i)we get thatΠ lm ≡ Σ lm¯p= ∑ ¯p i¯p Π lm(i) = ∑ w i¯ρ iw¯ρ Π lm(i) . (17.8)