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All dots in this solution denote differentiation with respect to conformal time $\eta$. This is required by comoving $k$, the streaming term $k{\Theta }_{\ell }$, and $\dot{\tau }=-a{\bar{n}}_e{\sigma }_T$. The PDF's reference to proper-time derivatives is inconsistent with these coefficients. Proper-time equations follow by dividing the conformal-time equations by $a$, replacing streaming $k$ by $k/a$ and the collision rate by $-{\bar{n}}_e{\sigma }_T$.

The monopole equation of the photon Boltzmann hierarchy is ${\dot{\Theta }}_0+k{\Theta }_1/3=\dot{\varphi }$; its collision term vanishes. Consequently the photon continuity equation isFor the dipole the hierarchy givesUsing the relations from the preceding part, the photon Euler equation becomesBecause $\dot{\tau }<0$, the collision term drives the photon velocity towards the baryon velocity, as Thomson scattering should.

For pressureless baryons, stress-energy conservation without energy exchange gives $\boxed{{\dot{\delta }}_b-k{v}_b-3\dot{\varphi }=0}$. At this order Thomson scattering changes momentum but not the local photon energy: energy transfer by the relative bulk motion is second order, and recoil/thermal energy exchange is neglected. Total momentum conservation fixes the baryon force to be the negative of the photon force. The momentum densities are proportional to their enthalpy densities, $4{\bar{\rho }}_{\gamma }/3$ and ${\bar{\rho }}_b$. ThusIncluding the pressureless expansion and gravitational terms yields the baryon Euler equation with Thomson dragThe expansion coefficient for photons cancels against the changing background pressure, while pressureless baryons retain the Hubble drag term. The opposite drag signs and the enthalpy ratio ensure that the combined photon-baryon fluid conserves momentum.

In tight coupling, the scattering time $|\dot{\tau }{|}^{-1}$ is short compared with an acoustic period and an expansion time: $k/|\dot{\tau }|\ll 1$ and $\mathcal{H}/|\dot{\tau }|\ll 1$. Photons and baryons have nearly the same velocity; higher photon multipoles and the photon-baryon velocity slip are suppressed by powers of this small ratio. Here retain the collision operator as printed, which ignores CMB polarization. Its diffusion coefficient differs from the polarized result.

Neglect gravity and expansion, so $R$ is constant on the timescale under consideration. The quadrupole equation of the photon Boltzmann hierarchy isTo first order in $k/|\dot{\tau }|$, ${\dot{\Theta }}_2$ and $k{\Theta }_3$ are subleading relative to the dipole source. HenceThis is the temperature-only tight-coupling quadrupole. The negative collision rate is essential to its sign.

Put $\Delta =v_{\gamma }-v_b$. Subtracting the baryon Euler equation with Thomson drag from the photon Euler equation givesThe zeroth-order common velocity obeys $(1+R)\dot{v}=-k{\delta }_{\gamma }/4$. Solving the slip equation to its first nonzero order therefore givesThe second expression assumes that $R$ and $\dot{\tau }$ vary only on the neglected background timescale. Terms from their variation would need retaining if cosmic expansion were restored. Multiplying the baryon Euler equation with Thomson drag by $R$ and adding it to the photon equation eliminates drag. Since $v_b=v_{\gamma }-\Delta$,Using $v_{\gamma }=3{\dot{\delta }}_{\gamma }/(4k)$ from the photon continuity equation, the quadrupole term contributes $(2/9)k{\dot{\delta }}_{\gamma }/\dot{\tau }$, and the slip term contributes $R^{2}k{\dot{\delta }}_{\gamma }/[4(1+R)\dot{\tau }]$. HenceDifferentiating the photon continuity equation gives the photon-baryon diffusion damping equationThe sound speed reflects baryon inertia; the $R^2$ term is heat conduction through velocity slip, and the $8/9$ term is photon shear viscosity for the unpolarized hierarchy.

For constant coefficients the characteristic roots are $-D/2\pm \sqrt{D^2/4-c_s^2{k}^2}$. On the acoustic branch where $D<2c_{s}k$,Thus the solutions are damped acoustic oscillations, with positive diffusion damping growing as $k^2$. This is Silk damping. Slowly varying coefficients give an approximate envelope $\exp [-\frac{1}{2}\int Dd\eta ]$ and acoustic phase $k\int c_{s}d\eta$. The mathematical critically damped and overdamped cases follow from the roots, but extrapolation beyond the tight-coupling range is not reliable. For very large $R$, the acoustic and damping scales must be compared explicitly rather than inferring underdamping from $k/|\dot{\tau }|\ll 1$ alone.

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.

Throughout this question dots denote conformal time derivatives, and $\mathcal{H}=\dot{a}/a$ is the conformal Hubble parameter. The monopole of the photon Boltzmann hierarchy gives ${\dot{\Theta }}_0+k{\Theta }_1/3=\dot{\varphi }$. With the specified velocity convention, the photon continuity equation is thereforeThe dipole equation is ${\dot{\Theta }}_1+k(2{\Theta }_2/5-{\Theta }_0)=\dot{\tau }({\Theta }_1+v_b)+k\psi$. Thus the photon Euler equation isIn particular, $\dot{\tau }<0$ makes the scattering term damp the photon-baryon velocity slip; it does not amplify it.

The linear momentum density of the photon-baryon fluid is weighted by enthalpy, not merely energy density:Elastic Thomson scattering exchanges momentum internally. Its photon force density is $(4{\bar{\rho }}_{\gamma }/3)\dot{\tau }(v_{\gamma }-v_b)$, so the baryon force density is its negative. Dividing by ${\bar{\rho }}_b$ gives the baryon Euler equation with Thomson dragThis $R$ is the baryon loading parameter, the baryon-to-photon inertia ratio. The two collision terms cancel exactly when the equations are weighted by their enthalpies. Background conservation gives ${\bar{\rho }}_b\propto a^{-3}$ and ${\bar{\rho }}_{\gamma }\propto a^{-4}$, so $\dot{R}=\mathcal{H}R$.

The collisionless Boltzmann equation is conservation of $f$ along the photon trajectory. To linear order, use the unperturbed trajectory $d\mathbf{x}/d\eta =\mathbf{e}$ when differentiating $\Theta$. Deflection of $\mathbf{e}$ is first order and multiplies the first-order angular dependence of $\delta f$, while the background has no angular dependence; it therefore contributes only at second order. Similarly, the energy derivative of $\delta f$ times the perturbed energy change is second order. ThusFor a nontrivial spectrum this identifies the linear Free-streaming photon Boltzmann equation:In Newtonian gauge in cosmology, substitute the specified scalar gravitational-redshift source to writeMultiply its angular average by four. Since $⟨e^j{⟩}_{\Omega }=0$, the photon angular temperature moments imply the photon continuity equationFor the photon Euler equation, multiply by $3e^i$ and average. The source is $-3⟨e^i{e}^j{⟩}_{\Omega }{\partial }_j\psi =-{\partial }^i\psi$, while the second angular moment isIt follows thatThe background energy density is spatially homogeneous, so no spatial derivative acts on $\bar{\rho }$. The minus sign of its anisotropic stress term follows from the PDF's $\Pi =-\pi$ convention. Switching to the conventional trace-free stress $\pi$ changes that term to a plus sign.

With $v_b=v_{\gamma }$ at leading order and $c_s^2=1/[3(1+R)]$, the photon continuity equation and total Euler equation give ${\ddot{\delta }}_{\gamma }+\dot{R}{\dot{\delta }}_{\gamma }/(1+R)+k^2{\delta }_{\gamma }/[3(1+R)]=4\ddot{\varphi }+4\dot{R}\dot{\varphi }/(1+R)-4k^2\psi /3$. The baryon loading parameter changes inertia and shifts the gravitational equilibrium. The leading equations omit diffusion damping.

For a constant growing adiabatic mode with curvature convention $\mathcal{R}=-\varphi -\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )/[4\pi G{a}^2(\bar{\rho }+\bar{P})]$, ${\varphi }_{\mathrm{rad}}=-2\mathcal{R}/3$ and ${\varphi }_{\mathrm{mat}}=-3\mathcal{R}/5$. The superhorizon photon continuity equation conserves ${\Theta }_0-\varphi$. With radiation initial value ${\Theta }_0=\mathcal{R}/3$, it gives ${\Theta }_{0,\mathrm{mat}}=2\mathcal{R}/5$ and $({\Theta }_0+\psi {)}_{\mathrm{mat}}=-\mathcal{R}/5$ when $\psi =\varphi$.