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Approximating decoupling by a single time and retaining only a photon monopole and Doppler dipole omits the finite cosmological visibility function width, diffusion or Silk damping, quadrupole anisotropy and polarization generated as tight coupling fails. The photon-baryon acoustic oscillator also make the emission density and velocity strongly scale dependent. They can be included in the initial monopole and dipole, so their existence alone does not invalidate the subsequent collisionless line-of-sight solution for free-streaming photons. Smaller angles correspond to larger angular multipole $\ell$.

The CMB line-of-sight solution separates sharply localized last-scattering sources, weighted by the cosmological visibility function, from integrated gravitational sources weighted by $e^{-\tau }$ along the photon trajectory.

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.

The first mechanism is Silk damping: finite photon mean free paths allow radiation to diffuse out of compressions, while velocity slip and radiation shear dissipate acoustic motion. A Fourier diffusion equation has $X^{\prime }=-D(\tau )k^{2}X$, hence $X\propto \exp [-k^2\int Dd\tau ]$. For the polarization-inclusive tightly coupled plasma, a useful expression isThe positive scattering rate is $\dot{\kappa }=a{n}_e{\sigma }_T$, and $R$ is the baryon loading parameter. The two terms represent heat conduction from velocity slip and photon shear viscosity. The mode amplitude has an approximately Gaussian cutoff in $k$; power is suppressed by $e^{-2k^2/k_D^2}$. The formula assumes tight coupling and slowly varying coefficients before decoupling.

The second mechanism is finite-width last-scattering damping. Photons emerge from a range of times and depths, so rapidly varying sources are averaged rather than observed at one sharply defined phase. Approximate the normalized cosmological visibility function by a Gaussian of width ${\sigma }_{\tau }$. For a radial plane-wave phase,The associated power factor is $e^{-(k\mu {\sigma }_{\tau }{)}^2}$. An acoustic factor $e^{\pm ik{c}_s\tau }$ is likewise averaged, replacing $k\mu$ by $k(\mu \pm c_s)$ in the constant-sound-speed approximation. Thus fine radial structure and rapid acoustic time oscillations are suppressed once their phase changes substantially across the visibility width. This is an averaging effect after the perturbations exist; Silk damping physically erases them by diffusion. Angular projection and the evolving source determine the final isotropic damping envelope, so the radial $\mu$ dependence must not be silently discarded.

There is first an angular-scale typo in the PDF: smaller angles correspond to larger multipoles, approximately $\theta \sim \pi /\ell$. Small scales relative to the first acoustic peak therefore have $\ell \gtrsim 200$, not the printed inequality in the other direction.

The photon-baryon acoustic oscillator produces rapidly varying density and velocity sources around and below the sound horizon. Treating the emission density as a simple long-wavelength gravitational source misses these acoustic peaks and their phases. Silk damping suppresses short-scale perturbations through diffusion before decoupling. The cosmological visibility function has finite width, so different emission times and acoustic phases are averaged rather than sampled on one sharp surface. As tight coupling breaks down, the photon quadrupole and higher moments are generated; Thomson scattering of the quadrupole produces polarization. These require the full photon Boltzmann hierarchy, rather than exactly vanishing emission moments above the dipole. They are the angular-scale limits of instantaneous photon decoupling.

The distinction is important: acoustic evolution and diffusion can already be included in correctly calculated emission ${\delta }_{\gamma }$ and $\mathbf{v}$. They do not themselves invalidate collisionless propagation after decoupling. The two-moment, single-time initial condition and omission of scattering after that surface are the approximations used in the requested line-of-sight formula.

For the stated assumptions, multiply the synchronous photon brightness equation by $e^{ik\mu \tau }$ and integrate:Here ${\tau }_d$ is the decoupling time. Inverse Fourier transformation shifts each source to the unperturbed rayInsert the initial brightness monopole and Doppler dipole, and divide by four:This is the synchronous Sachs-Wolfe line-of-sight formula. The first term is intrinsic emission temperature, the second is the Doppler effect, and the last is the accumulated gravitational redshift in this gauge. The derivative $h_{ij}^{\prime }$ is the partial conformal-time derivative evaluated on the ray, not its total derivative along the ray. Its integral contains endpoint gravitational effects as well as evolving-potential effects when rewritten in Newtonian gauge; it should not be identified solely with the Integrated Sachs-Wolfe effect.

The direction $\mathbf{n}$ is photon propagation toward the observer. The observer-to-source sky direction is its negative, so changing that convention also changes the displayed Doppler sign and the ray parametrization. Correct emitter positions and this direction convention are needed to interpret the formula.

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.

To first order, propagate along the unperturbed ray $\mathbf{x}(\eta )={\mathbf{x}}_0-({\eta }_0-\eta )\mathbf{e}$ and keep its orthonormal tetrad direction fixed. Corrections to direction or trajectory multiply an already first-order temperature source and are second order. Define the angular scattering sourceUsing the photon energy redshift from a tensor metric perturbation, the Photon Boltzmann equation with Thomson scattering along this ray becomesMultiply by the survival probability $e^{-\tau }$. The cosmological visibility function is $g(\eta )=-\dot{\tau }{e}^{-\tau }$, so integration givesEvery source here is evaluated on the same ray, and $\tau ({\eta }_0)=0$.

The instantaneous-visibility approximation makes $e^{-\tau }$ a step from zero to one at ${\eta }_{\ast }$. It suppresses the early boundary term and localizes the scattering source to $S_{\ast }$. To obtain only the requested tensor CMB line-of-sight source, additionally neglect this last-scattering angular source: in leading tight coupling, the tensor-induced incident quadrupole is small, and a tensor mode has no scalar monopole or dipole. Ignore later reionization and its scattering as well. ThenInstantaneous visibility alone would leave the term $S_{\ast }$; neglect of that term is an additional approximation. The time derivative on $h$ is essential and is present in the PDF, although it was lost in the extracted TeX.

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.

Put $\Gamma (\eta )=a{\bar{n}}_e{\sigma }_T\ge 0$, the Thomson scattering rate per unit conformal time. The cosmological optical depth is the integrated scattering rate from emission time to observation, so a Poisson scattering process has survival probabilitySince $\dot{\tau }=-\Gamma$, the cosmological visibility function isA photon scatters in an interval $d\eta$ with rate factor $\Gamma d\eta$ and then reaches us without another scattering with probability $E$, so $gd\eta$ is its probability of last scattering in that interval. For an optically thick initial epoch,If the initial optical depth is finite, the missing weight represents photons that have already stopped scattering before ${\eta }_i$.

Before cosmological recombination, the optical depth to the present is enormous: $E\simeq 0$, and $g$ is also small because almost every scattering is followed by another. Through photon decoupling, $E$ rises rapidly and $g$ has its dominant positive peak. After decoupling, without reionization, $E$ is close to one and reaches exactly one at ${\eta }_0$, while $g$ is small because the remaining electron density and scattering rate are small. Residual ionization can give a weak tail; there is no second reionization peak. The peak marks the Cosmic microwave background last-scattering surface.

The cosmological visibility function is the probability density for last scattering, so the stipulated two instantaneous populations giveSince $g={\partial }_{\eta }{e}^{-\tau }$ and $e^{-\tau }$ vanishes before the first transparent epoch,Its plot is a nondecreasing step function: it jumps from zero to $p$ at recombination and from $p$ to one at ${\eta }_1$. The cosmological optical depth interpretation is that $e^{-\tau (\eta )}$ is the probability that a photon present at time $\eta$ reaches the observer without any later scattering.

Reionization returns some neutral cosmic gas to an ionized state after cosmological recombination. The resulting free electrons increase the cosmological optical depth through Thomson scattering and create a second contribution to the cosmological visibility function.

Ignoring the last-scattering angular source and later scattering, the tensor-induced temperature anisotropy is $\Theta =-\frac{1}{2}{\int }_{{\eta }_{\ast }}^{{\eta }_0}{\dot{h}}_{ij}(\eta ,{\mathbf{x}}_0-({\eta }_0-\eta )\mathbf{e})e^i{e}^{j}d\eta$. An instantaneous cosmological visibility function alone leaves a scattering source at last scattering; leading tensor tight coupling supplies the additional reason to neglect it.