Cosmological Perturbation Theory, 26.4.2011 version

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25 SACHS–WOLFE EFFECT 60andso thatHere the total derivative along the photon geodesic is⃗qqso that Eq. (25.10) becomesddη = ∂ ∂η + dxk ∂dη ∂x k (25.11)(1 − Ψ)pk=(1 + Φ)p 0 ≈ pkp 0 = dxkdηto 0th order (25.12)(⃗q · ∇Φ−2 = −2 dxk ∂Φ dΦq dη ∂x k = −2 dη − ∂Φ ), (25.13)∂η1 dqq dη = −dΦ dη + Φ′ + Ψ ′ (25.14)The relative perturbation in the photon energy, δE/Ē, that the photon accumulates whentraveling from x ∗ to x obs in the perturbed universe is thusδEĒ = δq¯q ∫ ∫ ∫ dq(Φ = q = − dΦ + ′ + Ψ ′) dη∫ ηobs( ∂Φ= Φ(x ∗ ) − Φ(x obs ) +∂η + ∂Ψ )dη , (25.15)∂ηwhere the integrals are along the photon path.For a thermal distribution of photons, a uniform relative photon energy perturbation correspondsto a temperature perturbation of the same amount:( ) δT= δĒ Tjour E . (25.16)Here “jour” refers to the temperature perturbation the photon distribution accumulates on thejourney between x ∗ and x obs .The other contributions to the observed CMB temperature anisotropy are due to the localphoton energy density perturbation and photon velocity perturbation at the origin of the photon,on the last scattering surface:( ) δT= 1T4 δ γ(x ∗ ) − ⃗v(x ∗ ) · ˆn , (25.17)intrwhere ˆn is the direction the observer is looking at.For a given observer, the Φ(x obs ) part is common to photons from all directions, and theobserver interprets it as part of the mean (background) photon temperature. Thus the observedCMB temperature anisotropy isδTT = 1 4 δN γ (x ∗) − ⃗v N (x ∗ ) · ˆn + Φ(x ∗ ) +η ∗∫ ηobsη ∗( ∂Φ∂η + ∂Ψ )dη . (25.18)∂η(There is also a contribution from the motion of the observer that causes a dipole pattern inthe observed anisotropy. To get rid of that, the dipole of the observed anisotropy is subtractedaway from the observations before any cosmological analysis.)For large scales, much larger than the horizon size at photon decoupling, the Doppler effect−⃗v N (x ∗ ) · ˆn is small compared to the other terms. The contribution( ) δT= 1T4 δN γ (x ∗ ) + Φ(x ∗ ) (25.19)SW
26 MATTER POWER SPECTRUM 61is called the ordinary Sachs–Wolfe effect, and the contribution( ) ∫ δT ηobs( ∂Φ=T∂η + ∂Ψ )dη (25.20)∂ηISWis called the integrated Sachs–Wolfe effect.For more, see Section I5.3 of CMB Physics 2004.26 Matter Power Spectrumη ∗See Chapter M of CMB Physics 2004, which is based on Chapter 7 of Dodelson[3]. Note that themultipole moments of the photon brightness function Θ l are related to the Θ m lof CMB Physics2007 and to perturbations of the photon energy tensor byΘ 0 ≡ Θ 0 0 ≡ 1 4 δ γ Θ 1 ≡ 1 3 Θ0 1 ≡ 1 3 v γ . (26.1)
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About these lecture notes:I lecture
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2 THE BACKGROUND UNIVERSE 2all clos
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3 THE PERTURBED UNIVERSE 43 The Per
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4 GAUGE TRANSFORMATIONS 6Let us now
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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 38 and 39: 15 OTHER GAUGES 3415.4 Comoving Cur
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 66 and 67: A GENERAL PERTURBATION 62A General
Page 68: REFERENCES 64The general perturbed
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