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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.

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.

For $\mathbf{k}=k\hat{\mathbf{z}}$, scalar axisymmetry leaves only $m=0$ in the photon quadrupole: ${\Theta }_{20}=-\sqrt{4\pi /5}{\Theta }_2$ with the normalization used above. The two spin-weighted spherical harmonics satisfy ${}_{+2}{Y}_{20}={}_{-2}{Y}_{20}\propto {\sin }^2\theta$. Translation from last scattering supplies $e^{-ik{\chi }_{\ast }\mu }$. Thus the two helicity combinations have the same scalar source:where $\mathcal{C}$ is a common normalization constant independent of $\mu$. In this meridian polarization basis the scalar contribution has $U=0$; this does not mean that every rotated polarization basis has zero $U$.

Apply the spin-raising and lowering operators to an axisymmetric scalar potential $f$. The first action is $-{\partial }_{\theta }f$. In the second action, the intermediate spin is respectively $+1$ or $-1$; both giveIt follows thatSubtracting the two equations gives $P_B^{\prime \prime }=0$. The possible affine $P_B=A+B\mu$ consists solely of the unobservable $\ell =0,1$ kernel of the twice-applied spin operators. Defining polarization potentials to have no such components sets the physical scalar B mode to zero, $\boxed{P_B=0}$. This is the low-multipole kernel of polarization potentials, not a physical dipole polarization.

For $x\ne 0$ the remaining equation integrates to $P_E=-\mathcal{C}{\Theta }_2{e}^{-ix\mu }/x^2+A_E+B_E\mu$. Remove the same monopole/dipole kernel and use the plane-wave expansion. Thus the scalar E-mode radial projection isThe overall minus sign is absorbed in the stated proportionality constant. Although the unprojected antiderivative has $x^{-2}$, its divergent monopole and dipole pieces are discarded; the physical $\ell \ge 2$ limit is finite because $j_{\ell }(x)=O(x^{\ell })$.

The temperature-only tight-coupling quadrupole is suppressed by $k/{\Gamma }_T$. For regular superhorizon scalar modes, the photon dipole is itself proportional to $k$, so the recombination quadrupole and the resulting polarization are especially small at large angular scales. Last-scattering polarization is generated by the small departure from perfect tight coupling, not by an isotropic monopole. Reionization can generate a separate large-angle signal after recombination and is absent from the instantaneous-last-scattering approximation here. The no-B-mode conclusion applies to linear scalar sources; it does not include tensor sources or conversion by lensing.