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For a nonzero Fourier mode $\mathbf{k}=k\hat{z}$, the divergence-free condition implies $B_z=0$. The complex vectors $m^{(\pm )}=(\hat{x}\pm i\hat{y})/\sqrt{2}$ form a Hermitian orthonormal basis of this transverse plane. Their opposite rotation phases give the helicity decomposition of a transverse Fourier mode; the factor $i/\sqrt{2}$ in the amplitudes is a convention. The baryon velocity in this vector sector has the analogous decomposition.

For helicity $\sigma =\pm 1$, the angular source isThe same formula holds for ${\mathbf{v}}_b^{(\sigma )}$. Thus these sources have only angular order $m=\sigma$. Streaming multiplies by $ik\cos \theta$ and the stated Thomson scattering operator is diagonal in $m$, so neither mixes this sector with other angular orders. Assuming the initial anisotropy is in the same vector-helicity sector, or considering the anisotropy generated from zero initial data, the solution therefore contains only $Y_{\ell \sigma }$ with $\ell \ge 1$. Arbitrary independently imposed anisotropies of other $m$ would instead evolve in their own homogeneous sectors.

There is a genuine normalization inconsistency in the PDF. To retain its printed temperature expansion, writewhere $q_{\ell }$ denotes exactly the multipoles called ${\Theta }_{\ell }$ in that expansion. The spherical-harmonic streaming recurrence gives the coefficient of $Y_{\ell \sigma }$ in $ik\cos \theta \Theta$, divided by $C_{\ell }$, asHere $C_{\ell +1}/C_{\ell }=-i\sqrt{(2\ell +3)/(2\ell +1)}$ and $C_{\ell -1}/C_{\ell }=i\sqrt{(2\ell -1)/(2\ell +1)}$. Both streaming denominators are therefore $2\ell +1$, rather than the PDF's $2\ell +3$ and $2\ell -1$. The dipole source also changes: $C_1=-i\sqrt{3/(8\pi )}$, so $-(\mathbf{e}\cdot \dot{\mathbf{B}}+\dot{\tau }\mathbf{e}\cdot {\mathbf{v}}_b)/C_1$ is $-\sigma (4\pi /3)({\dot{B}}^{(\sigma )}+\dot{\tau }{v}_b^{(\sigma )})Y_{1\sigma }$. The quadrupole collision correction is still $-\dot{\tau }{q}_2{\delta }_{\ell 2}/10$.

Thus the vector photon Boltzmann hierarchy consistent with the literal printed expansion isA concrete counterexample to the printed hierarchy with this expansion is a collisionless instant with $q_2=1$ and every other multipole zero. Direct streaming gives ${\dot{q}}_1=-k\sqrt{3}/3$, whereas the printed hierarchy would give $-k\sqrt{3}/5$.

Alternatively, preserve the PDF's desired hierarchy by making the photon multipole normalization changeThis is the reciprocal angular normalization, not the printed expansion. Multiplying the corrected hierarchy by $(2\ell +1)/(4\pi )$ gives preciselyThis supplies both consistent conventions explicitly. The lower coupling vanishes at $\ell =1$, so no vector monopole is introduced. Cosmological optical depth to the observer has $\dot{\tau }<0$, giving damping of the dipole and the factor $9/10$ quadrupole damping in this temperature-only collision model; no polarization collision term has been added to the supplied equation.

Put $\Gamma (\eta )=a{\bar{n}}_e{\sigma }_T\ge 0$, the Thomson scattering rate per unit conformal time. The cosmological optical depth is the integrated scattering rate from emission time to observation, so a Poisson scattering process has survival probabilitySince $\dot{\tau }=-\Gamma$, the cosmological visibility function isA photon scatters in an interval $d\eta$ with rate factor $\Gamma d\eta$ and then reaches us without another scattering with probability $E$, so $gd\eta$ is its probability of last scattering in that interval. For an optically thick initial epoch,If the initial optical depth is finite, the missing weight represents photons that have already stopped scattering before ${\eta }_i$.

Before cosmological recombination, the optical depth to the present is enormous: $E\simeq 0$, and $g$ is also small because almost every scattering is followed by another. Through photon decoupling, $E$ rises rapidly and $g$ has its dominant positive peak. After decoupling, without reionization, $E$ is close to one and reaches exactly one at ${\eta }_0$, while $g$ is small because the remaining electron density and scattering rate are small. Residual ionization can give a weak tail; there is no second reionization peak. The peak marks the Cosmic microwave background last-scattering surface.

Dropping gravity and expansion from the photon Boltzmann hierarchy givesThe first equation is photon energy continuity: the photon dipole transports the photon monopole. The second is the photon Euler equation: a pressure gradient accelerates the radiation, its quadrupole contributes shear stress, and Thomson scattering drags the velocity toward the baryon velocity.

With $v_{\gamma }=-{\Theta }_1$, the scattering contribution to ${\dot{v}}_{\gamma }$ is $\dot{\tau }(v_{\gamma }-v_b)$. Radiation momentum density is $(4{\bar{\rho }}_{\gamma }/3){\mathbf{v}}_{\gamma }$, whereas baryon momentum density is ${\bar{\rho }}_b{\mathbf{v}}_b$. Conservation of total momentum therefore requiresorSince the cosmological optical depth is measured to the observation time, $\dot{\tau }<0$; the two drag terms reduce their relative velocity.

The cosmological visibility function is the probability density for last scattering, so the stipulated two instantaneous populations giveSince $g={\partial }_{\eta }{e}^{-\tau }$ and $e^{-\tau }$ vanishes before the first transparent epoch,Its plot is a nondecreasing step function: it jumps from zero to $p$ at recombination and from $p$ to one at ${\eta }_1$. The cosmological optical depth interpretation is that $e^{-\tau (\eta )}$ is the probability that a photon present at time $\eta$ reaches the observer without any later scattering.

Reionization returns some neutral cosmic gas to an ionized state after cosmological recombination. The resulting free electrons increase the cosmological optical depth through Thomson scattering and create a second contribution to the cosmological visibility function.