Cosmological Perturbation Theory, 26.4.2011 version

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15 OTHER GAUGES 3415.4 Comoving Curvature <strong>Perturbation</strong>The comoving curvature perturbationR ≡ −ψ C = −ψ − H(v − B) (15.24)turns out to be a useful quantity fro discussing superhorizon perturbations.From the second Einstein equation in the Newtonian gauge, Eq. (10.16), we have thatv N =23H 2 (1 + w) (Ψ′ + HΦ), (15.25)so that the relation of the comoving curvature perturbation and the Bardeen potentials isR = −Ψ −2 (H −1 Ψ ′ + Φ ) 2= −Ψ −3(1 + w)3(1 + w) Φ − 23(1 + w) H−1 Ψ ′ . (15.26)Derivating Eq. (15.26),R ′ = −Ψ ′ 2w ′ (+H −13(1 + w) 2 Ψ ′ + Φ ) )2−(− H′3(1 + w) H 2Ψ′ + H −1 Ψ ′′ + Φ ′= − 2H−13(1 + w) Ψ′′ − 4 + 6c2 s 23(1 + w) Ψ′ −3(1 + w) Φ′ + 2H w − c2 s1 + w Φ ,(where we used some background relations), and using the Einstein equations (10.18) and(10.18), we get an evolution equation for R,− 3 2 (1 + w)H−1 R ′= H −2 Ψ ′′ + H −1 ( Φ ′ + 2Ψ ′) + 3c 2 (s H −1 Ψ ′ + Φ ) − 3wΦ= H −2 Ψ ′′ + (2 + 3c 2 s)H −1 Ψ ′ + H −1 Φ ′ + 3(c 2 s − w)Φ) 2(Φ − Ψ) + 3 δp N2= 3c 2 (s H −1 Ψ ′ + Φ ) ( k+ 1 3H( ) k 2 ( k= −c 2 s Ψ + 1H3H¯ρ) 2 ( δp(Φ − Ψ) + 3 N)2¯ρ − c2 sδ N . (15.27)From Sec. 12 we haveS =13(1 + w)(δ − 1 c 2 s)δp¯ρ⇒ δp = c 2 s[δρ − 3(¯ρ + ¯p)S] (15.28)(valid in any gauge), so that Eq. (15.27) becomes the important result( ) k 2− 3 2 (1 + w)H−1 R ′ [= c2H s Ψ + 1 3 (Ψ − Φ)] + 9 2 c2 s (1 + w)S . (15.29)As an evolution equation, this does not appear very useful, since it mixes metric perturbationsfrom two different gauges. However, the importance of this equation comes from the two observationswe can now make immediately: 1) For adiabatic perturbations, S = 0, the second termon the rhs vanishes. 2) For superhorizon perturbations, i.e., for Fourier modes whose wavelengthis much larger than the Hubble distance, k ≪ H, we can ignore the first term on the rhs. Thus:For adiabatic perturbations, the comoving curvature perturbation stays constant outside thehorizon.Adiabatic (scalar) perturbations have only one degree of freedom, and thus their full evolutionis captured in the evolution of R.
15 OTHER GAUGES 35A general perturbation at a given time can be decomposed into an adiabatic component,which has S = 0, and an isocurvature component, which has R = 0. Because of the linearityof first order perturbation theory, these components evolve independently, and the evolution ofthe general perturbation is just the superposition of the evolution of these two components, andthus they can be studied separately. Beware, however, that the ”adiabatic” component doesnot necessarily remain adiabatic in its evolution, and the ”isocurvature” component does notnecessarily maintain zero comoving curvature. We later show that adiabatic perturbations stayadiabatic while they are well outside the horizon.Adiabatic perturbations are important, since the simplest theory for the origin of structureof the universe, single-field inflation, produces adiabatic perturbations. Using the constancyof R, it is easy to follow the evolution of these perturbations while they are well outside thehorizon.Using Eq. (15.21) we can write the second line of Eq. (15.27) as− 3 2(1 + w)HR′ =Ψ ′′ + (2 + 3c 2 s )HΨ′ + HΦ ′ + 3(c 2 s − w)H2 Φ= 3 2 H2δp C¯ρ − 1 3 ∇2 (Φ − Ψ) (15.30)or (using Eq. 15.19)H −1 R ′( )= − δpC¯ρ + ¯p − 2 wδ31 + w ∇2 Π = −c 2 Cs1 + w − 3S − 2 w31 + w ∇2 Π (15.31)which is completely in the comoving gauge.15.5 Perfect Fluid Scalar <strong>Perturbation</strong>s, AgainWith Π = 0 ⇒ Ψ = Φ, Eqs. (15.18,15.19,15.20,15.21,15.22,15.23) become∇ 2 Φ = 3 2 H2 δ C (15.32)Φ ′ + HΦ = 3 2 H2 (1 + w)v N (15.33)Φ ′′ + 3(1 + c 2 s )HΦ′ + 3(c 2 s − w)H2 Φ = 3 2 H2δp C¯ρ(15.34)δ ′ C − 3Hwδ C = (1 + w)∇ 2 v N (15.35)v ′ N + Hv N = δp C¯ρ + ¯p + Φ . (15.36)and Eqs. (15.26,15.31) becomeRH −1 R ′2 (= −Φ − Φ ′ + HΦ ) (15.37)3(1 + w)H( )= − δpC δC¯ρ + ¯p = −c2 s1 + w − 3S (15.38)and Eq. (15.34) asH −2 Φ ′′ + 3(1 + c 2 s )H−1 Φ ′ + 3(c 2 s − w)Φ = 3 2 c2 s [δC − 3(1 + w)S]. (15.39)which is Eq. (12.16), with just ∇ 2 Φ replaced by δ C using Eq. (15.32). (I seem to be repeatinghere some material from Sec. 12.) Our aim is to get a differential equation with preferably just
Page 2: About these lecture notes:I lecture
Page 6 and 7: 2 THE BACKGROUND UNIVERSE 2all clos
Page 8 and 9: 3 THE PERTURBED UNIVERSE 43 The Per
Page 10 and 11: 4 GAUGE TRANSFORMATIONS 6Let us now
Page 12 and 13: 4 GAUGE TRANSFORMATIONS 8Since the
Page 14 and 15: 5 SEPARATION INTO SCALAR, VECTOR, A
Page 16 and 17: 6 PERTURBATIONS IN FOURIER SPACE 12
Page 18 and 19: 6 PERTURBATIONS IN FOURIER SPACE 14
Page 20 and 21: 7 SCALAR PERTURBATIONS 16In Fourier
Page 22 and 23: 8 CONFORMAL-NEWTONIAN GAUGE 18andδ
Page 24 and 25: 9 PERTURBATION IN THE ENERGY TENSOR
Page 26 and 27: 10 FIELD EQUATIONS FOR SCALAR PERTU
Page 28 and 29: 11 ENERGY-MOMENTUM CONTINUITY EQUAT
Page 30 and 31: 13 SCALAR PERTURBATIONS IN THE MATT
Page 32 and 33: 13 SCALAR PERTURBATIONS IN THE MATT
Page 34 and 35: 15 OTHER GAUGES 30This discussion w
Page 36 and 37: 15 OTHER GAUGES 32We get to comovin
Page 40 and 41: 15 OTHER GAUGES 36one perturbation
Page 42 and 43: 16 SYNCHRONOUS GAUGE 38so thath ij
Page 44 and 45: 17 FLUID COMPONENTS 4017 Fluid Comp
Page 46 and 47: 17 FLUID COMPONENTS 42and Eq. (17.1
Page 48 and 49: 18 ADIABATIC AND ISOCURVATURE PERTU
Page 56 and 57: 20 THE REAL UNIVERSE 52interested i
Page 58 and 59: 21 EARLY RADIATION-DOMINATED ERA 54
Page 60 and 61: 21 EARLY RADIATION-DOMINATED ERA 56
Page 62 and 63: 22 SUPERHORIZON EVOLUTION AND RELAT
Page 64 and 65: 25 SACHS-WOLFE EFFECT 60andso thatH
Page 66 and 67: A GENERAL PERTURBATION 62A General
Page 68: REFERENCES 64The general perturbed

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