Cosmological Perturbation Theory, 26.4.2011 version

12 PERFECT FLUID SCALAR PERTURBATIONS IN THE NEWTONIAN GAUGE 25With manipulations involving background relations, these can be worked (exercise) into theform(δ N ) ′ = (1 + w) ( )∇ 2 v N + 3Ψ ′) + 3H(wδ N − δpN(11.6)¯ρ(v N ) ′ = −H(1 − 3w)v N − w′1 + w vN + δpN¯ρ + ¯p + 2 3w1 + w ∇2 Π + Φ . (11.7)12 Perfect Fluid Scalar Perturbations in the Newtonian Gauge12.1 Field EquationsFor a perfect fluid, things simplify a lot, since now Π = 0 and thus for a perfect fluidΨ = Φ , (12.1)and we have only one degree of freedom in the scalar metric perturbation. We can now replaceΨ with Φ in the field equations. The original set becomesand the reworked set becomes∇ 2 Φ − 3H(Φ ′ + HΦ) = 4πGa 2 δρ N (12.2)(Φ ′ + HΦ ) ,i= 4πGa 2 (¯ρ + ¯p)v N ,i (12.3)Φ ′′ + 3HΦ ′ + (2H ′ + H 2 )Φ = 4πGa 2 δp N , (12.4)∇ 2 Φ = 4πGa 2¯ρ [ δ N + 3H(1 + w)v N]= 3 2 H2 [ δ N + 3H(1 + w)v N] (12.5)Φ ′ + HΦ = 4πGa 2 (¯ρ + ¯p)v N = 3 2 H2 (1 + w)v N (12.6)Φ ′′ + 3HΦ ′ + (2H ′ + H 2 )Φ = 4πGa 2 δp N = 3 2 H2 δp N /¯ρ , (12.7)where we have used Eq. (2.8).If we change the time variable from conformal time η to cosmic time t, they read∇ 2 Φ = 4πGa 2¯ρ [ δ N + 3aH(1 + w)v N] (12.8)˙Φ + HΦ = 4πGa(¯ρ + ¯p)v N (12.9)¨Φ + 4H ˙Φ(+ 2Ḣ + 3H2) Φ = 4πGδp N . (12.10)We define the total entropy perturbation as( δpS ≡ H¯p ′ − δρ )¯ρ ′(≡ H δṗ¯p − δρ˙¯ρ ). (12.11)From the gauge transformation equations (9.25,9.26) we see that it is gauge invariant.Using the background relations ¯ρ ′ = −3H(1 + w)¯ρ and ¯p ′ = c 2 s ¯ρ ′ we can also write(1S =3(1 + w)δρ¯ρ − 1 )δpc 2 , (12.12)s ¯ρfrom which we getδp = c 2 s [δρ − 3(¯ρ + ¯p)S] , (12.13)