= 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 convention
$$
\Theta(\eta,\mathbf k,\mathbf e)
=\sum_{\ell\geq0}(-i)^\ell\Theta_\ell(\eta,\mathbf k)P_\ell(\mu),
$$
which 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 to
$$
k\left(\frac{\ell+1}{2\ell+3}\Theta_{\ell+1}
-\frac{\ell}{2\ell-1}\Theta_{\ell-1}\right).
$$
The 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> is
$$
\boxed{
\begin{aligned}
\dot\Theta_\ell+
k\left(\frac{\ell+1}{2\ell+3}\Theta_{\ell+1}
-\frac{\ell}{2\ell-1}\Theta_{\ell-1}\right)
={}&-\dot\tau\left[
(\delta_{\ell0}-1)\Theta_\ell-\delta_{\ell1}v_b
+\frac{\delta_{\ell2}}{10}(\Theta_2-\sqrt6E_2)\right]\\
&+\delta_{\ell0}\dot\phi+\delta_{\ell1}k\psi.
\end{aligned}}
$$
The $\ell=0$ term containing $\Theta_{-1}$ is understood as zero.
Back to article page
