 Solution

A scalar Fourier mode selects one spatial direction, $\hat{\mathbf{k}}$, and has no transverse vector or tensor polarization. Rotations about $\mathbf{k}$ therefore leave its temperature and polarization patterns unchanged. Only azimuthal number $m=0$ occurs in the corresponding spherical harmonic expansion.

Let $\mu =\hat{\mathbf{k}}\cdot \mathbf{e}$ and use the paper's conventionwhich omits the $(2\ell +1)$ factor used in some other definitions of the photon temperature multipole. In Fourier space the transport term is $ik\mu \Theta$. The Legendre polynomial recurrence relation makes its coefficient at multipole $\ell$ equal toThe gravitational time derivative is purely monopolar, and the potential gradient and baryon velocity are dipolar. Thomson scattering removes the monopole from its relaxation term and adds the quadrupole polarization correction. Thus the photon Boltzmann hierarchy isThe $\ell =0$ term containing ${\Theta }_{-1}$ is understood as zero.