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Use Newtonian gauge with scalar metric perturbationsLet $\hat{n}$ point from the observer toward the emitter, $\Theta =\delta T/T$, and evaluate the unperturbed ray at $\mathbf{x}(\tau )=({\tau }_0-\tau )\hat{n}$. In the sudden-decoupling approximation the Sachs-Wolfe effect and its linear companions giveHere ${\Theta }_0={\delta }_{\gamma }/4$ is the intrinsic photon monopole; ${\Psi }_{\ast }$ is the emission gravitational redshift; the velocity term is the Doppler CMB anisotropy of the baryon-photon plasma; and the integral is the Integrated Sachs-Wolfe effect from evolving potentials during travel. The last two local terms are an observer monopole and dipole, removed when discussing the higher measured multipoles. Changing the direction convention reverses both velocity signs. A finite cosmological visibility function replaces the sharp emission surface by a weighted source integral.

For scale-invariant primordial potentials, ${\mathcal{P}}_{\Psi }=k^3{P}_{\Psi }/(2{\pi }^2)$ is constant before transfer effects. In matter domination, the adiabatic superhorizon relation is ${\Theta }_0=-2\Psi /3$, so the observable Sachs-Wolfe combination is ${\Theta }_0+\Psi =\Psi /3$. Its amplitude per logarithmic scale is therefore approximately independent of angle. In a flat geometry $\ell \sim k{D}_{\ast }$ and $\theta \sim {\ell }^{-1}$, and projection gives the Sachs-Wolfe plateauBoth the intrinsic and gravitational emission terms have this leading large-angle scaling; they must be combined, not treated as unrelated random sources.

The Euler equation gives $v_b\sim k{\tau }_{\ast }\Psi$ for the growing large-scale matter mode. Thus the Doppler amplitude grows roughly as ${\theta }^{-1}$ relative to the large-scale potential amplitude, while its logarithmic power grows as ${\theta }^{-2}$, until acoustic dynamics and damping invalidate that simple estimate. On subhorizon pressureless scales the Poisson equation gives ${\delta }_m\sim (k\tau {)}^2\Psi$: the corresponding density amplitude has the often-used ${\theta }^{-2}$ scaling. This is a matter-density estimate, not a universal small-angle photon-temperature law; the actual intrinsic photon source undergoes the oscillations in part(b). On superhorizon scales it instead has the constant-potential behavior just described.

The integrated source vanishes for constant linear potentials in pure matter domination. Potential decay during radiation domination or late accelerated expansion makes it nonzero. Its broad, slowly evolving sources favor large angular scales and give a potential-sized contribution there; crossings of many structures cancel progressively on small scales. Scale invariance alone fixes no universal power of angle for the integrated term: its time evolution and transfer functions are also required. Consequently the large-angle plateau, velocity-gradient enhancement and subhorizon density-gradient estimate describe distinct physical regimes, rather than one extrapolated power law across the whole spectrum.

Thomson scattering generates linear Cosmic microwave background polarization when the incident radiation has a photon quadrupole. The cross-section prefers electric fields perpendicular to the scattering direction. Contributions from an isotropic field cancel, and a dipole alone does not supply the required spin-two angular pattern. Schematically the outgoing Stokes combination is proportional towhich selects the incident quadrupolar angular dependence. Tight coupling nearly isotropizes the radiation, making the quadrupole small; it develops when the photon mean free path becomes finite near decoupling. At leading tight-coupling order its size scales as $(k/\dot{\kappa })v_{\gamma }$.

The polarization is therefore weaker than the temperature anisotropy: a characteristic amplitude is of order a few percent to roughly ten percent of the temperature signal, or several microkelvin compared with temperature fluctuations of several tens of microkelvin on acoustic scales. This is an order-of-magnitude comparison, not a scale-independent ratio of every power-spectrum multipole.

The photon monopole follows the acoustic displacement, approximately $\cos (k{r}_s)$; the velocity sourcing the quadrupole follows $\sin (k{r}_s)$. Hence polarization maxima occur near displacement zeros, between the main temperature peaks. In the leading coherent-oscillator approximation the shift is half the peak spacing, $\Delta \ell /2$, although projection and gravitational driving modify exact locations. Scalar perturbations generate E-mode polarization at linear order, not B-mode polarization. Tensor perturbations and lensing can give B modes, and reionization generates an additional large-angle polarization feature.

Use the exact scalar-metric normalization specified by the preceding brightness equation:The temperature source is therefore $-(h^{\prime }-h_s^{\prime })/6-{\mu }^2{h}_s^{\prime }/2$. Let $W=e^{ik\mu (\tau -{\tau }_0)}$, so $W^{\prime }=ik\mu W$ and $W^{\prime \prime }=-k^2{\mu }^{2}W$. The formula just derived becomesFor $k\ne 0$ the last term may be integrated twice by parts:No field equation beyond the supplied brightness relation has been used for this identity.

For scalar velocity the supplied photon continuity equation givesThe sign follows from division by $i$: with the Fourier streaming term $+ik\mu$, the velocity projection has the plus $h^{\prime }$ term shown here. Combining it with the lower endpoint from the integrations by parts gives the complete endpoint terms in the synchronous Sachs-Wolfe formula:The local observer bracket is an angular photon monopole plus photon dipole. Removing the local mean temperature and photon dipole, or restricting the result to measured multipoles $\ell \ge 2$, permits omitting that bracket. It must be retained for the literal full brightness amplitude. The $k=0$ limit should be taken in the original line-of-sight integral, rather than treating the separate inverse-$k$ endpoint terms as independently defined.

The final target formula printed in the PDF does not follow from its preceding equations. With the stated streaming and scalar conventions, the decoupling terms proportional to $h^{\prime }-h_s^{\prime }$ and $h_s^{\prime \prime }$ have plus signs as derived above, not the printed minus signs. Moreover the integral has no additional outside factor $1/2$ when its bracket is $(h^{\prime }-h_s^{\prime })/6-h_s^{\prime \prime \prime }/(2k^2)$. Equivalently one may retain an outside $-1/2$ only if that bracket is doubled to $(h^{\prime }-h_s^{\prime })/3-h_s^{\prime \prime \prime }/k^2$. Omitting the observer photon monopole and photon dipole does not fix these discrepancies.

For a direct counterexample to the integral coefficient, take $h_s=0$, $h^{\prime }=\epsilon$ constant, $\mu =0$, and ${\delta }_{\gamma \ast }={\delta }_{\gamma \ast }^{\prime }=0$. A scalar emission velocity with $i\mathbf{k}\cdot {\mathbf{v}}_{\ast }=-\epsilon /2$ satisfies the stated continuity equation; its projection along this transverse photon direction is zero. Keep $\epsilon ({\tau }_0-{\tau }_{\ast })\ll 1$ so perturbation theory is valid. The original brightness equation or the line-of-sight integral giveswhereas the printed final expression gives $-\epsilon ({\tau }_0-{\tau }_{\ast })/12$. All $h_s$ observer terms vanish in this example, so an observer-term convention cannot remove the contradiction. The boxed corrected identity supplies the requested integration-by-parts result with every boundary contribution and its observable higher-multipole version specified.

Along the zeroth-order photon path, set $Y=\Theta +\psi$ and $\mathcal{S}=(3/4)\int d\hat{\mathbf{m}}\Theta (\hat{\mathbf{m}})[1+(\mathbf{e}\cdot \hat{\mathbf{m}}{)}^2]/(4\pi )$. The scalar photon energy redshift and the Photon Boltzmann equation with Thomson scattering giveThe integrating factor is $e^{-\tau }$, because ${\tau }^{\prime }<0$ in this convention. HenceTight coupling before decoupling makes the incident distribution nearly isotropic in the baryon frame. Neglecting its quadrupole, the angular kernel integrates to $\mathcal{S}={\Theta }_0$, since $(3/4)⟨1+(\mathbf{e}\cdot \hat{\mathbf{m}}{)}^2⟩=1$. The primordial term at the optically thick lower endpoint is erased. With instantaneous last scattering, no later scattering and $g=\delta (\eta -{\eta }_{\ast })$, the line-of-sight solution isThe approximation is linear, uses an unperturbed ray, neglects the scattering quadrupole and polarization feedback, and idealizes the cosmological visibility function as a delta function. It also excludes reionization or any other post-recombination opacity.

The photon monopole ${\Theta }_0$ is the intrinsic emission-temperature perturbation. The emission potential ${\psi }_{\ast }$ gives gravitational redshift on leaving the potential well; together they form the Sachs-Wolfe combination. The baryon peculiar velocity supplies the Doppler CMB anisotropy. The integral is the Integrated Sachs-Wolfe effect, from changes in gravitational potentials during travel. The observer potential is a direction-independent endpoint shift absorbed into the observed monopole. An observer peculiar velocity, if included, contributes an additional kinematic dipole; the stated observer is coordinate-comoving.

Write $S_{\ast }(\mathbf{x})=({\Theta }_0+\psi )({\eta }_{\ast },\mathbf{x})$ and $E=e^{-{\tau }_{\mathrm{re}}}$. Under the stated matter-era growing-mode approximation, $\dot{\varphi }+\dot{\psi }=0$, and ignoring velocities removes the Doppler CMB anisotropy. The two peaks of the cosmological visibility function give the Cosmic microwave background line-of-sight solutionApply the same solution with observation time ${\eta }_{\mathrm{re}}$, just before the thin reionization screen. Between recombination and that screen there is no further scattering, so free propagation givesA direction-independent potential commutes with the angular average. Thus the incoming photon monopole plus the potential at the screen isSubstitution givesThis derivation inherits the neglected scattering quadrupole and polarization assumptions of part (ii); the screen is an idealized isotropizing approximation.

For a plane Fourier mode $S_{\ast }(\mathbf{x})=S_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{x}}$, the angular average is explicit:where $j_0$ is the zeroth Spherical Bessel function. The mode's observed transfer isIf $k\Delta \chi \gg 1$, the rescattered monopole is suppressed by at least $1/(k\Delta \chi )$, because the incoming directions sample incoherent phases. If $k\Delta \chi \ll 1$, $j_0=1+O((k\Delta \chi {)}^2)$ and the two propagation phases agree to leading order. The two weights then sum to one. ConsequentlyThese are the small-scale damping and large-scale coherence limits of reionization damping of cosmic microwave background temperature anisotropy.

Each small-scale primary temperature amplitude is multiplied by $e^{-{\tau }_{\mathrm{re}}}$, so its Cosmic microwave background power spectrum is multiplied by $e^{-2{\tau }_{\mathrm{re}}}$. Holding transfer parameters fixed,Many measured high-$\ell$ modes determine this combination accurately. Their derivatives $\partial \log C_{\ell }/\partial \log A_s=1$ and $\partial \log C_{\ell }/\partial {\tau }_{\mathrm{re}}=-2$ are degenerate: a change $d\log A_s=2d{\tau }_{\mathrm{re}}$ leaves the leading temperature spectrum unchanged. The undamped large-scale modes can distinguish the parameters, but there are few of them and cosmic variance gives fractional full-sky uncertainty $\sqrt{2/(2\ell +1)}$ per multipole. This explains the primordial-amplitude–optical-depth degeneracy. Polarization from reionization, lensing, and more complete late-time effects partly break it. The TeX's final ${\tau }_e$ is a transcription mismatch; the PDF uses ${\tau }_{\mathrm{re}}$ consistently.

Dropping gravity and expansion from the photon Boltzmann hierarchy givesThe first equation is photon energy continuity: the photon dipole transports the photon monopole. The second is the photon Euler equation: a pressure gradient accelerates the radiation, its quadrupole contributes shear stress, and Thomson scattering drags the velocity toward the baryon velocity.

With $v_{\gamma }=-{\Theta }_1$, the scattering contribution to ${\dot{v}}_{\gamma }$ is $\dot{\tau }(v_{\gamma }-v_b)$. Radiation momentum density is $(4{\bar{\rho }}_{\gamma }/3){\mathbf{v}}_{\gamma }$, whereas baryon momentum density is ${\bar{\rho }}_b{\mathbf{v}}_b$. Conservation of total momentum therefore requiresorSince the cosmological optical depth is measured to the observation time, $\dot{\tau }<0$; the two drag terms reduce their relative velocity.

When gravity and expansion can be neglected and $R$ is constant, the photon monopole satisfiesThe first damping contribution comes from photon-baryon velocity slip; the second comes from the photon quadrupole. This is a local form of Silk damping.