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Write $T=\overline{T}(1+\Theta )$, with $\Theta =\Delta T/\overline{T}$ at linear order. Since $\overline{T}\propto a^{-1}$, the background Planck function depends only on comoving momentum. Expanding its temperature-shifted argument givesUsing $T$ rather than $\overline{T}$ in the perturbation denominator changes only higher-order terms.

The photon brightness perturbation is the energy-weighted angular perturbationIntegration by parts gives $\int q^4{f}_0^{\prime }dq=-4\int q^3{f}_{0}dq$; the boundary term vanishes at both endpoints for a Planck spectrum. Therefore $\Delta =4\Theta$. Multiplying the transport equation by $q^3$ and integrating yields the synchronous photon brightness equationThe PDF's definition is $4\Delta T/T$; the TeX's denominator $\hslash$ is a transcription error.

Expand $\Delta$ in Legendre moments in $\mu$. The monopole is the density/temperature perturbation, and the dipole is the bulk velocity. Before decoupling, rapid Thomson scattering isotropizes photons in the electron rest frame. In the tight-coupling approximation, the scattering rate is much larger than $k$ and $\mathcal{H}$, so the quadrupole and higher moments are suppressed by the mean-free-path parameter, with the quadrupole typically of order $k/\dot{\kappa }$ times the dipole. To leading order only monopole and dipole are needed. This suppression uses the collision term restored before decoupling; it does not follow from the collisionless equation by itself. Finite quadrupole corrections generate polarization and diffusion damping.

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.

The collision rate $\kappa =a{n}_e{\sigma }_T$ is the scattering probability per unit conformal time. Thomson scattering approximately conserves photon number and energy in the electron rest frame, but redistributes propagation directions. Consequently the collision term drives the photon angular distribution toward an isotropic monopole plus the dipole associated with the baryon/electron bulk velocity. It damps the photon-baryon relative velocity and the quadrupole and higher directional anisotropies. Photon and baryon momentum-transfer terms have equal and opposite forces when weighted by their background enthalpies.

The metric terms describe the change in photon energy along a geodesic in a time-dependent spatial metric. The trace perturbation supplies an isotropic redshift, while the scalar trace-free perturbation supplies a direction-dependent redshift proportional to $3{\mu }^2-1$. Brightness is proportional to the fourth power of temperature, accounting for the factor four relative to the temperature shift. These are gravitational sources, not scattering terms.

When $\kappa \gg k,\mathcal{H}$, angular anisotropy is scattered away faster than free streaming or background expansion can regenerate it. In the photon Boltzmann hierarchy, a moment of order $\ell \ge 2$ has damping of order $\kappa {\Delta }_{\ell }$ and is sourced by neighboring moments with coefficients of order $k$. Successive moments are therefore suppressed by powers of $k/\kappa =k{\tau }_c$. This is the tight-coupling approximation. At leading perfect-fluid order the quadrupole can be neglected; its first nonzero correction must be retained to calculate the diffusion damping in part (iii).

The printed equations do not all share a consistent velocity convention. To fix the intended one, take the Fourier velocity to be $v^i=i{k}^i\theta$. Ignoring the metric source, the synchronous perfect-fluid density equation then givesThe PDF's baryon continuity sign is opposite to this convention. Keeping it literally would make the adiabatic ratio and common velocity in part (ii) incompatible for any nonstationary perturbation. Likewise, the momentum-transfer terms for these potentials are $-\kappa ({\theta }_{\gamma }-{\theta }_b)$ and $-R\kappa ({\theta }_b-{\theta }_{\gamma })$, without the printed $k^{-2}$.

For isotropic scattering in this convention the brightness source is $\kappa [{\delta }_{\gamma }+4ik\mu {\theta }_b-\Delta ]$. Its dipole relaxes toward the baryon velocity. The printed source involving ${\theta }_{\gamma }-{\theta }_b$ is not this source, and cannot yield the stated Euler drag by taking its dipole. The following solutions use the consistent continuity and momentum equations. These repairs do not change the requested tight-coupling reduction because the opposite drag forces cancel from its total-momentum equation. The velocity-divergence convention in the standard photon and baryon evolution equations is converted to this potential convention by $\vartheta =-k^2\theta$.

In the tight-coupling approximation, the Thomson-scattering rate ${\Gamma }_T=-{\tau }^{\prime }$ is much faster than both spatial streaming and cosmological evolution: $k/{\Gamma }_T\ll 1$ and $\mathcal{H}/{\Gamma }_T\ll 1$. Frequent scattering locks the photon and baryon velocities and suppresses higher angular multipoles. The photon dipole is leading order, while the photon quadrupole is first order in the mean-free-time expansion.

Use the quadrupole equation. Its time derivative is smaller than its collision term by an additional expansion parameter, and ${\Theta }_3$ starts one order beyond ${\Theta }_2$. The leading balance is thereforeThus for the temperature-only collision operator supplied here,The temperature-only tight-coupling quadrupole is not numerically interchangeable with a hierarchy that includes polarization feedback. The latter changes the leading collision coefficient and gives the existing photon quadrupole in tight coupling with polarization instead. Here the stated temperature-only approximation is the one being expanded.

The tight-coupling approximation requires the positive scattering rate $\kappa =-\dot{\tau }$ to exceed both $k$ and the conformal Hubble parameter. The photon-baryon velocity slip and photon quadrupole are then small, and the leading fluid motion has $v_b=v_{\gamma }=v$. One must combine the equations before setting the slip to zero: a small slip times a large collision rate can exert a finite force.

Add the photon Euler equation to $R$ times the baryon Euler equation with Thomson drag. The collisions cancel and, neglecting ${\Theta }_2$ at this order, the result isSince $\dot{R}=\mathcal{H}R$, the common velocity obeysThe inertia of the baryons reduces the photon-baryon sound speed to $c_s^2=1/[3(1+R)]$.

Differentiate the photon continuity equation and substitute this velocity equation:Thus the photon-baryon acoustic oscillator isThe terms on the right drive the oscillation gravitationally; the first-derivative term comes from changing baryon inertia. Diffusion damping enters only beyond this leading tight-coupling approximation.

Write $\Gamma =-\dot{\tau }>0$. The tight-coupling approximation requires $\Gamma \gg k,\mathcal{H}$: frequent Thomson scattering makes the photon-baryon velocity slip small and suppresses higher multipoles. Work with constant $R$ on the short scales of the question.

Set $s=v_{\gamma }-v_b$ and $A=k{\Theta }_0-2k{\Theta }_2/5$. The dipole and baryon equations implyTo leading order, $s=RA/[(1+R)\dot{\tau }]$. Differentiating this and keeping the first correction in $k/\Gamma$ yieldsTerms involving derivatives of ${\Theta }_2$ contribute at the next order here.

The leading $\ell =2$ polarization balance isThe $\ell =2$ temperature equation then becomesso the photon quadrupole in tight coupling with polarization isFinally substitute ${\Theta }_1=-3{\dot{\Theta }}_0/k$ into the corrected dipole equation. This gives the photon-baryon diffusion damping equationThe oscillation frequency agrees with the photon-baryon sound speed. The positive damping coefficient comes from velocity slip and shear viscosity. Its growth as $k^2$ suppresses small-scale oscillations, producing the Silk damping tail and reducing the high-multipole Cosmic microwave background acoustic peaks.

Insert the photon temperature multipole expansion into the Fourier-space transport equation. The Legendre polynomial recurrence relation turns multiplication by $\mu$ into nearest-neighbour multipole couplings, while Orthogonality of Legendre polynomials projects onto a fixed $\ell$. The monopole projection has no collision term because Thomson scattering conserves photon number:The dipole projection receives the electron-velocity source,For every $\ell \ge 2$, the collision term damps the anisotropic multipole and the photon Boltzmann hierarchy isThus the Free-streaming photon Boltzmann equation moves angular structure between neighbouring multipoles, whereas Thomson scattering suppresses all multipoles above the dipole in the tight-coupling approximation.

In the tight-coupling approximation, finiteness of the photon Euler equation as $\Gamma \to \infty$ requiresLet $S=3{\Theta }_1+v_b$. Rearranging the baryon Euler equation and substituting the leading relation only on its right-hand side givesInsert this slip into the photon Euler equation:The photon continuity equation gives ${\Theta }_1=-({\Theta }_0^{\prime }-{\Phi }^{\prime })/k$. Eliminating the dipole yieldswhere the photon-baryon sound speed is

The velocity slip is the difference between the bulk velocities of photons and baryons. Thomson scattering suppresses it at a rate $\Gamma (1+R^{-1})$, where $R=3{\rho }_b/(4{\rho }_{\gamma })$ and $\Gamma =-\dot{\tau }>0$. Its first correction in the tight-coupling approximation produces heat-conduction damping of acoustic waves.