Physical Foundations of Cosmology

360 Cosmic microwave background anisotropies with the relationpαpα=0weget ω ̃=ω ( 1 + ∂ξi ∂η li ) , (9.10) where we have kept on ...

9.2 Sachs–Wolfe effect 361 whereλis an affine parameter along the geodesic. Since the photons have zero mass, the first integral ...

362 Cosmic microwave background anisotropies and we obtain the result that ( δT T + ) =const, (9.21) along null geodesics. The i ...

9.3 Initial conditions 363 Unlike Silk damping, free-streaming causes the spatial variation (the x- dependence) of the photon di ...

364 Cosmic microwave background anisotropies wherey≡ω/T andp 0 andphave been expressed in terms ofωusing (9.14) and (9.18). The ...

9.4 Correlation function and multipoles 365 energy density itself and the Sachs–Wolfe effect, and the second term is related to ...

366 Cosmic microwave background anisotropies inhomogeneities. The root-mean-square difference between a local measurement and th ...

9.5 Anisotropies on large angular scales 367 IfδT/Tis expanded in terms of spherical harmonics, δT(θ,φ) T 0 = ∑ l,m almYlm(θ, φ) ...

368 Cosmic microwave background anisotropies represents pristine information about the primordial inhomogeneities. In this secti ...

9.6 Delayed recombination and the finite thickness effect 369 forlup to 20 or so. Forl> 20 ,the neglected effects become esse ...

370 Cosmic microwave background anisotropies average over a scalex∼ηr,whereηris roughly the duration of recombina- tion. Clea ...

9.6 Delayed recombination and the finite thickness effect 371 ηrdetermined by the condition μ′′=μ′^2. (9.50) Since recombination ...

372 Cosmic microwave background anisotropies ηL=ηr.Then, substituting (9.55) in (9.49) and performing an explicit integration ov ...

9.6 Delayed recombination and the finite thickness effect 373 smaller than the horizon scale. Therefore, one can try to use the ...

374 Cosmic microwave background anisotropies scale of radiation inhomogeneities one has to use (9.62) and the finite thickness l ...

9.7 Anisotropies on small angular scales 375 the source function at recombination is ( (^) k+ δk 4 ) r  ⎡ ⎣Tp ( 1 − 1 3 c^2 s ) ...

376 Cosmic microwave background anisotropies 10 100 1 10 100 1000 0 0.5 1.0 0.1 1 0 0.5 1.0 1.5 2.0 0.1 kηeq Tp To kηeq Fig. 9.1 ...

9.7 Anisotropies on small angular scales 377 9.7.2 Multipole moments To calculate the multipole momentsCl, we have to substitute ...

378 Cosmic microwave background anisotropies with “oscillating”(O)and “nonoscillating” functions(N)in the integrands, so that l( ...

9.7 Anisotropies on small angular scales 379 is proportional to the baryon density and vanishes in the absence of baryons where ...