Cosmological Perturbation Theory, 26.4.2011 version

13 SCALAR PERTURBATIONS IN THE MATTER-DOMINATED UNIVERSE 28whose solution isΦ(η,⃗x) = C 1 (⃗x) + C 2 (⃗x)η −5 . (13.12)The second term, ∝ η −5 is the decaying part. We get C 1 (⃗x) and C 2 (⃗x) from the initial valuesΦ in (⃗x), Φ ′ in (⃗x) at some initial time η = η in,asΦ in (⃗x) = C 1 (⃗x) + C 2 (⃗x)ηin −5(13.13)Φ ′ in (⃗x) = −5C 2(⃗x)ηin −6(13.14)C 1 (⃗x) = Φ in (⃗x) + 1 5 η inΦ ′ in(⃗x) (13.15)C 2 (⃗x) = − 1 5 η6 inΦ ′ in(⃗x) (13.16)Unless we have very special initial conditions, conspiring to make C 1 (⃗x) vanishingly small, thedecaying part soon becomes ≪ C 1 (⃗x) and can be ignored. Thus we have the important resultΦ(η,⃗x) = Φ(⃗x), (13.17)i.e., the Bardeen potential Φ is constant in time for perturbations in the flat (k = 0) matterdominateduniverse.Ignoring the decaying part, we have Φ ′ = 0 and we get for the velocity perturbation fromEq. (13.9)v N HΦ 2Φ= =4πGa2¯ρ 3H = 1 3 Φη ∝ η ∝ a1/2 ∝ t 1/3 . (13.18)and Eq. (13.8) becomes∇ 2 Φ = 4πGa 2¯ρ [ δ N + 2Φ ] = 3 2 H2 [ δ N + 2Φ ] (13.19)orIn Fourier space this readsδ N = −2Φ + 23H 2 ∇2 Φ . (13.20)δ N ⃗ k(η) = −[2 + 2 3( kH) 2]Φ ⃗k . (13.21)Thus we see that for superhorizon scales, k ≪ H, or k phys ≪ H, the density perturbation staysconstant,δ N ⃗ k= −2Φ ⃗k = const. (13.22)whereas for subhorizon scales, k ≫ H, or k phys ≫ H, they grow proportional to the scale factor,δ ⃗k = − 2 3( kH) 2Φ ⃗k ∝ η 2 ∝ a ∝ t 2/3 . (13.23)Since the comoving Hubble scale H −1 grows with time, various scales k are superhorizonto begin with, but later become subhorizon as H −1 grows past k −1 . (In physical terms, in theexpanding universe λ phys /2π = k −1phys ∝ a grows slower than the Hubble scale H−1 ∝ a 3/2 .) Wesay that the scale in question “enters the horizon”. (The word “horizon” in this context refersjust to the Hubble scale 1/H, and not to other definitions of “horizon”.) We see that densityperturbations begin to grow when they enter the horizon, and after that they grow proportional