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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.

Before decoupling, rapid Thomson scattering makes the photon distribution nearly isotropic in the local photon-fluid rest frame. A local blackbody temperature perturbation changes its energy density by ${\delta }_{\gamma }=4\delta T/T$. A first-order boost by the fluid velocity adds a temperature photon dipole $\mathbf{n}\cdot \mathbf{v}$; it does not generate a photon quadrupole until second order in that boost. Thus the photon brightness perturbation at instantaneous decoupling isHigher angular moments are suppressed by the photon mean-free-path to perturbation-length ratio in the tight-coupling approximation. The idealized equilibrium assumption neglects those finite-mean-free-path corrections, polarization effects and finite thickness of the last-scattering surface. It is not an assertion that collisionless photons remain isotropic after decoupling. The distinction between a fluid treatment and the subsequent hierarchy is discussed in the primary synchronous and Newtonian perturbation treatment.

Let $\mu =\hat{\mathbf{k}}\cdot \mathbf{n}$ and $W(\tau )=e^{ik\mu (\tau -{\tau }_0)}$. Multiplication of the synchronous photon brightness equation by its integrating factor givesIntegration from ${\tau }_{\ast }={\tau }_{\mathrm{dec}}$ to ${\tau }_0$ and division by four yieldFor the real-space observation point ${\mathbf{x}}_0$, the corresponding background ray is $\mathbf{x}(\tau )={\mathbf{x}}_0-\mathbf{n}({\tau }_0-\tau )$. Fourier inversion gives the synchronous Sachs-Wolfe line-of-sight formulaEvery source is evaluated at its appropriate point on the ray, rather than at the observation position. Here $\mathbf{n}$ is photon propagation direction, opposite to the outward observer-to-source sky direction; changing that convention changes the written sign of the Doppler projection consistently.

The first term is the intrinsic photon temperature at last scattering. On wavelengths larger than the sound horizon, it retains primordial adiabatic information; inside that horizon the photon-baryon fluid has acoustic energy density oscillations, producing the acoustic temperature peaks. The second term is the Doppler effect from the fluid's bulk motion at emission. A regular long-wavelength velocity is gradient-suppressed; the Doppler contribution is important near and below the acoustic horizon, with its acoustic phase displaced from that of the energy density oscillation.

The integral is the gravitational energy shift produced by the changing spatial metric tensor along the photon path. In synchronous gauge it includes endpoint gravitational redshifts as well as evolving-potential effects. It must not be identified solely with the Integrated Sachs-Wolfe effect: in Newtonian gauge the same total separates into the ordinary last-scattering Sachs-Wolfe effect, local observer terms, and an integral of time-varying potentials. The ordinary contribution is important on scales outside the sound horizon at decoupling. A late integrated contribution is strongest on large angular scales when dark energy or curvature changes the potentials; evolution around matter-radiation equality also contributes near the first acoustic scales.

The relevant comoving sound scale is $r_s({\tau }_{\ast })={\int }^{{\tau }_{\ast }}{c}_s(\tau )d\tau$, while the projection distance is $D_{\ast }\simeq {\tau }_0-{\tau }_{\ast }$. Modes project roughly to photon temperature multipole $\ell \sim k{D}_{\ast }$, and the acoustic scale is $\ell \sim D_{\ast }/r_s$, with successive peak spacing of order $\pi D_{\ast }/r_s$. At much shorter wavelengths Silk damping and the finite last-scattering thickness suppress energy density and velocity anisotropies; the instantaneous-decoupling approximation alone does not model that cutoff. These scale distinctions explain why the three terms cannot be assigned one common range of important wavelengths.

In the tight-coupling approximation, repeated Thomson scattering drives the photon distribution toward isotropy in the local baryon rest frame. At linear order a temperature perturbation gives a monopole and a small boost gives only a dipole. A photon propagating along $\mathbf{n}$ has rest-frame energy $p(1-\mathbf{n}\cdot \mathbf{v})$, soSince the rest-frame density contrast is ${\delta }_{\gamma }=4{\Theta }_{\mathrm{rest}}$, the initial photon brightness perturbation isThe quadrupole and higher photon temperature multipoles are small when the conformal collision rate $\kappa =a{n}_e{\sigma }_T$ greatly exceeds both $k$ and $\mathcal{H}$. For example the quadrupole is suppressed by a factor of order $k/\kappa$ relative to the dipole. Instantaneous decoupling idealizes this strongly coupled state as the initial data for collisionless propagation; finite mean free path and polarization corrections are omitted from that initial truncation.

Multiply the synchronous photon brightness equation by $e^{ik\mu \tau }$ and integrate from ${\tau }_{\ast }$ to ${\tau }_0$. The line-of-sight solution for free-streaming photons isThe phase is essential: it specifies the emission point and the points traversed by the ray. In real space define $\mathbf{x}(\tau )={\mathbf{x}}_0-\mathbf{n}({\tau }_0-\tau )$ and ${\mathbf{x}}_{\ast }=\mathbf{x}({\tau }_{\ast })$. Dividing by four gives the synchronous Sachs-Wolfe line-of-sight formula,The spatial arguments suppressed in a shorthand version must be understood in this retarded sense; evaluating the emission terms at ${\mathbf{x}}_0$ instead would discard free streaming. Here $\mathbf{n}$ is propagation direction. The observer-to-source direction on the sky is $-\mathbf{n}$, explaining the opposite Doppler sign often used with that convention. The formula describes the synchronous-frame observer; an additional observer peculiar velocity contributes a dipole.

The first term is the intrinsic temperature perturbation at emission, since ${\delta }_{\gamma }=4\Theta$. The second is the Doppler CMB anisotropy from the plasma bulk velocity. The third is gravitational frequency shifting by the time-dependent spatial metric along the ray, and can include scalar and tensor metric contributions. In scalar Newtonian gauge, rearranging that metric contribution supplies the emission gravitational potential as well as the Integrated Sachs-Wolfe effect, up to the observer terms. It should not be identified solely with the integrated Sachs-Wolfe term: a constant Newtonian potential need not give a vanishing synchronous metric integral.

On large angles, modes are outside or comparable to the sound horizon at emission. For the growing adiabatic mode during matter domination, the intrinsic and emission-potential contributions give the ordinary Sachs-Wolfe effect, ${\Theta }_0+\Psi =\Psi /3$ in Newtonian variables. Velocity is a gradient effect, of order $k{\tau }_{\ast }\Psi$ on superhorizon scales, so its Doppler contribution is suppressed there. A scale-invariant primordial curvature spectrum produces approximately the familiar large-angle plateau in $\ell (\ell +1)C_{\ell }$. Evolving potentials during radiation-to-matter transition or late acceleration can add integrated gravitational anisotropy; constant matter-era scalar potentials do not generate a genuine scalar integrated Sachs-Wolfe source.

On smaller angles, the photon-baryon fluid has undergone acoustic oscillations. Its density and velocity are approximately in cosine and sine phases, producing acoustic structure in the intrinsic and Doppler sources and their projection onto angular multipoles. Silk damping suppresses sufficiently short wavelengths before decoupling, while the finite width of last scattering also smooths the real small-angle signal. These corrections refine the instantaneous-decoupling model rather than follow from a collisionless equation alone. The separate intrinsic, velocity and metric pieces depend on gauge convention; the complete observed anisotropy, including the appropriate observer terms and removal of the unobservable mean, is the physical quantity.