{title2=$\Theta_\ell$}

For massless collisionless <neutrinos> in <Newtonian gauge in cosmology>, expand their temperature fluctuation as $\Theta(\mu)=\sum_{\ell\ge0}(-i)^\ell\Theta_\ell P_\ell(\mu)$, without a $(2\ell+1)$ factor. The <Legendre polynomial recurrence relation> gives
$$
\dot\Theta_\ell+k\left(\frac{\ell+1}{2\ell+3}\Theta_{\ell+1}-\frac{\ell}{2\ell-1}\Theta_{\ell-1}\right)=\delta_{\ell0}\dot\phi+\delta_{\ell1}k\psi.
$$
The lower-neighbour term is absent at $\ell=0$. Multipoles with the more usual $(2\ell+1)$ weighting equal $\Theta_\ell/(2\ell+1)$.


 Neutrino Boltzmann hierarchy

ID: neutrino-boltzmann-hierarchy