Theory of Cosmological Perturbations Part III CMB anisotropy
1. Photon propagation equation Definitions Lorentz-invariant distribution function: fp µ, x µ ) Lorentz-invariant volume element on momentum space: dπp) 2 dπp) = 2θp 0 )δg µν p µ p ν ) gd 4 p d 4 p = dp 0 dp 1 dp 2 dp 3 ) = 2θˆp 0 )δη ab ˆp a ˆp b )d 4 ˆp ˆp a = e a µp µ tetrad components) f and dπ are invariant under local Lorentz transformation) number of photons within dπp) crossing 3-surface dσ µ : dn rate of change in dn due to collision: Boltzmann equation [ p µ ] x + dpµ f = C[f] ; µ dv p µ dp µ dv = Γµ αβp α p β, dn = fp µ, x µ )dπp) p α dσ α C[f]dπp) p α dσ α p α = dxα dv C[f] collision term v affine parameter)
Boltzmann equation in a perturbed FLRW universe 3 perturbed metric A, β i, H ij = Oɛ)) ds 2 = a 2 η) [ 1 + 2A)dη 2 + 2β i dx i dη + γ ij + H ij )dx i dx j] a 2 η)d s 2 g µν = a 2 ḡ µν ) perturbed distribution function fp µ, x µ ) = fp 0, η) + δfp µ, x µ ) Conformal transformation null geodesics are invariant under conformal transformation) q µ = a 2 p µ = a 2dxµ dv dxµ dλ, dq µ dλ = Γ µ αβq α q β [ q µ ] x + dqµ µ dλ q µ further transformation of the momentum variables: f = a 2 C[ f] q µ ˆq 1 + A)q 0, γ i = qi q 0 ˆq photon energy seen by observer normal to η=const. a γ i directional cosine: γ ij γ i γ j = 1 + Oɛ).
4 [ q µ x + dˆq ] µ dλ ˆq + dγi f = a 2 C[ dλ γ f] i [ η + γi x + 1 dˆq i q 0 dλ ˆq s Γ i jkγ j γ k where s Γ i kj is the connection of γ ij, and dˆq dλ = d [ ] 1 + A)q 0 = q 0 d dλ dλ A + dq0 dλ + Oɛ2 ) = q 0 d dλ A Γ 0 µνq µ q ν = q 0 ) 2 [ A i γ i 1 2 ] γ i [ ] d dλ = q0 η + γi x i ] ) H ij 2β i j γ i γ j f = a2 q0c[ f] ) + Oɛ) Here we have used the fact Γ 0 µν = δ Γ 0 µν = Oɛ) with δ Γ 0 00 = A, δ Γ 0 0i = A i, δ Γ 0 ij = 1 2 H ij β i j β j i )
5 Setting f = fˆq, η) + δfˆq, γ i, x µ ), 0th order: 0th order: 1st order: η f = 0 f = fˆq). [ η + γi x s Γ i jkγ j γ k ] δf i γ i [ = A i γ i + 1 ] ) H 2 ij 2β i j γ i γ j ˆq d f + C[ f] dˆq a2ˆq Assuming thermal equilibrium at sufficiently early stage η = η i ), fˆq) = 1 eˆq/a it i) 1 T i temperature at η = η i ) Define temperature T by T = a i T i /a irrespective of thermal equilibrium or not. Then 1 fˆq) = fa q) = e q/t 1 ; q ˆq physical energy a Planck distribution even if photons become out of thermal equilibrium.
1st order: Evaluation of collision term For T m e = 0.5 MeV 5 10 9 K, electrons are non-relativistic. Thomson scattering of photons by electrons σ T = 8πre/3 2 6.7 10 25 cm 2 ): 1 q C[ f] = 3 8π n eσ T d Ω 1 + cos 2 θ 2 [ fq, γ i ) fq, γ i )] 6 q, γ ) i photon momentum after scattering d Ω solid angle sustained by γ i in electron rest frame e.r.f.) θ scattering angle in e.r.f. Let v i e) be the electron 3-velocity. Then q = q [ 1 βi + v e)i ) γ i ] Because the photon energy is unchanged under Thomson scattering, q = q = q [ 1 β i + v e)i ) γ i ]
Hence 7 fq, γ i ) = fq ) + δfq, γ i ) = fq) + q q) q f + δfq, γi ) = fq) + q q fq) ) βi + v e)i ) γ i γ i ) + δfq, γ i ) and From these, d Ω = q q 1 q C[ f] = 3 ) 2 dω = 1 + 2β i + v e)i)γ i ) dω = dω + Oɛ) dω 1 + γ k γ k ) 2 2 8π n eσ T [ q f ] q β i + v e)i )γ i γ i ) + δfq, γ ) i δfq, γ i ) = n e σ T [ δf Ω δf q f ) β i + v e)i )γ i + 34 ] q Q ijγ i γ j δf Ω = 1 δfdω 4π Q ij = 1 γ i γ j 1 4π 3 γ ij)δfdω
8 Since f = fˆq) = faq) Define q f q = ˆq df dˆq [ η + γi x s Γ i jkγ j γ k ] δf i γ i = [A i γ i + 12 ] H ij β i j β j i )γ i γ j ˆq df dˆq +an e σ T [ δf Ω δf ˆq df ) β i + v e)i )γ i + 34 ] dˆq Q ijγ i γ j and rescale the affine parameter λ as 1 ˆq d dλ d dλ Θ δf ˆq df ), dˆq [ η + γi x s Γ i jkγ j γ k i ] γ i
Then d [A dλ Θ = i γ i + 12 ] H ij β i j β j i )γ i γ j [ +an e σ T Θ Ω Θ β i + v e)i )γ i + 3 ] 4 Π ijγ i γ j 9 Π ij γ i γ j 1 3 γ ij)θ Ω Equation for Θ is independent of the photon energy q. If we consider a temperature fluctuation in f = e q/t 1) 1 by setting T T + δt δ T f = ft + δt ) ft ) = ˆq df ) δt dˆq T δt T = δ T f ˆq df ) dˆq Θ = δf ˆq df ). dˆq We may regard Θ as the temperature fluctuation.
10 2. Gauge-invariant formulation Gauge transformation induced by x µ = x µ + ξ µ, ξ µ = T, L i ): This gives p µ p µ = xµ x ν pν = p µ + ξ µ,νp ν ˆq ˆq = 1 + Ā x)) q = a2 η)1 + Ā x)) p0 = a 2 η) Hence, from f + δf = f + δf, ) ) 1 + 2 1 a a T + Ax) a p a T T 0 + T ν),νp = ˆq 1 + HT + T i γ i) ; H a a δf = δf ˆq df dˆq [ HT + T i γ i] Θ = Θ + H T + T i γ i.
For scalar-type metric perturbations, in addition to A, 1 2 HS ij = Rγ ij + H T ij, β i = B i, v x)i = U x) i ; x =electron, photon, etc.. These transform under a gauge transformation ξ µ = T, L i ), 11 Ā = A T H T, B = B + T + L R = R H T, HT = H T + L, Ū x = U x) + L. A convenient choice of gauge-invariant variables Ψ = A σ H σ ; σ = H T B, Φ = R H σ, V x) = U x) H T, Θ s = Θ + H σ + σ i γ i These correspond to perturbations variables defined in Newton logitudinal) slicing for which σ = 0. In terms of these gauge-invariant variables, d dλ Θ s = Φ + Ψ i γ i + 1 2 HT ij γ i γ j ) + an e σ T [ Θ s Ω Θ s + V e) i γ i + 34 ] Π ijγ i γ j Hij T tensor GW) perturbation, Π ij = γ i γ j 13 ) γij Θ is gauge-invariant. Ω
Qualitative derivation of CMB anisotropy Before recombination an e σ T H = n eσ T H 6 104n e T Ω b h 2 n b 1 ev Θ s = 1 ) H 4 s,r + V b i γ i + O n e σ T ) [ 1 + 10Ω b h 2 T 1 ev )] 1/2 1 where s,r = δρ/ρ = 4δT/T on Newton slices. V b = V e) ) is the baryon velocity potential. Thus, Θ s at the last scattering surface is approximately Θ s = 1 4 s,r + V b i γ i s,r is related to the radiation density perturbation on the matter comoving slices, r, as s,r = r 4H V where V is the matter shear velocity potential. Neglecting the difference between V and V b, Θ s = 1 4 r H V + V i γ i. 12
After recombination For dust-dominated universe, Φ = Ψ. Then, ignoring H T ij tensor) for the moment, d dλ Θ s = Φ Ψ i γ i = 2Ψ d dλ Ψ d dλ Θ s + Ψ) = 2Ψ Integrating this along a photon path to the present λ = λ 0 ) gives 13 Θ s + Ψ)λ 0 ) = Θ s + Ψ)λ d ) + λ0 λ d 2Φ dλ x η dec ) γ o x µ λ d ) = η d, x i η d )) x µ λ 0 ) = η 0, 0) x i η d ) = γ i η 0 η d ) Last Scattering Surface
Temperature anisotropy measured on the matter comoving frame: 14 Θ c = Θ s + H V + V i γ i Therefore, recovering tensor contribution, we obtain Θ c λ 0 ) = Ψ H V V i γ i) 0 + Ψ H V + V i γ i + 1 ) λ0 4 r + 2Ψ 12 ) HTij γ i γ j dλ d λ d Ψ H V ) 0 V i γ i ) 0 Monopole: unobservable. Dipole: our peculiar motion relative to CMB rest frame. Ψ H V ) d Sachs-Wolfe effect. V i γ i) Doppler effect. d ) 1 4 r Intrinsic temperature fluctuation at LSS. d λ0 λ0 2 Ψ dλ Integrated SW effect. 12 ) HTij γ i γ j dλ Tensor effect. λ d λ d
Large angular scale anisotropy 15 Large angle small wavenumber k: θ 10 2 1 ) k 1 superhorizon scale at LSS ah For adiabatic perturbations, at LSS assuming dust-dominance), HV = 2 3 Ψ, r 3 4 tot 8 ) 9 Ψ k2 k 2 a 2 H Ψ for 1 2 a 2 H 2 Θ ad = 1 3 Ψ d + ISW d For isocurvature perturbations, ) δρ S 3 ) δρ ρ dust 4 ρ Θ iso = 2Ψ d + ISW rad at η 0, 1 4 r 1 3 S = 5 3 Ψ at η = η d. SW part of Θ iso is enhanced by factor 6 relative to Θ ad Stringent constraint on isocurvature perturbations
3. Cosmological parameter dependence 16 Observed temperature anisotropy spectrum by WMAP. CMB anisotropy depends on Nature of primordial perturbation spectrum spectral index, ad or iso, etc...) Physics at decoupling time baryon density, CDM density, etc...) Evolution of the universe after decoupling cosmological constant, dark energy, etc...) Spatial geometry of the universe flat, closed or open)
Physics at decoupling is determined by the values of ρ b and ρ CDM. 17 Peaks are caused by accoustic oscillations of the baryon-photon fluid: Ψ HV F ρ b, ρ m ) cosc s kη d ) c s sound velocity) 1) Baryon and CDM density in terms of cosmological parameters: ρ x ) d = 1 + z d ) 3 ρ x ) 0 = 1 + z d ) 3 ρ cr ) 0 Ω x Ω b h 2 x = bbaryon) or CDM Thus, physics at decoupling is the same for the same values of Ω b h 2, Ω CDM h 2 ). degeneracy in CMB anisotropy. Evolution after decoupling is affected by ρ Λ cosmological constant) ρ Λ is important at z O1): Late time ISW effect. Larger Ω Λ h 2 causes larger power on large angular scales small l). ρ Λ causes increase in the conformal distance to LSS: η 0 η d increases. The same physical length on LSS is sustained by a smaller angle. Dependence on spatial geometry For open space, the same physical length on LSS is sustained by a smaller angle.
Increase in Ω Λ decrease in Ω tot slight degeneracy 18 Geometrical degeneracy
CMB degeneracy of cosmological parameters Agular diameter distance to LSS: d A z) = 1 H 0 sinh k 0 Rz) 1 + z) k 0 ; Rz) = 1+z 1 dy Ωm y 3 + Ω Λ k 0 y 2 19 Ω m = Ω b + Ω CDM, k 0 curvature parameter : 1 + k 0 = Ω m + Ω Λ θ = r phys d A h fω d, Ω vac ) 4+1) cosmological parameters: h, Ω b, Ω CDM, Ω Λ, n s n s spectral index) h Position of the first peak: θ peak 1 fω m, Ω Λ ) Height of the first peak: height h 1 Ω b h 2, Ω CDM h 2 ) Height of the second peak: height h 2 Ω b h 2, Ω CDM h 2 ) Global temperature anisotropy spectral shape: determination of n s. WMAP determined Ω b h 2, Ω CDM h 2 and n s. Ω b h 2 = 0.024 ± 0.001, Ω CDM h 2 = 0.012 ± 0.02, n s = 0.99 ± 0.04 5 parameters 4 observational constraints = 1 degree of degeneracy
Degeneracy line projected on Ω m, Ω Λ ) plane 20