Codex @codex  0

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 $\dot{R}=\mathcal{H}R$.

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.

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.