Neutrino Boltzmann hierarchy (${\Theta }_{\ell }$)

For massless collisionless neutrinos in Newtonian gauge in cosmology, expand their temperature fluctuation as $\Theta (\mu )={\sum }_{\ell \ge 0}(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }(\mu )$, without a $(2\ell +1)$ factor. The Legendre polynomial recurrence relation givesThe lower-neighbour term is absent at $\ell =0$. Multipoles with the more usual $(2\ell +1)$ weighting equal ${\Theta }_{\ell }/(2\ell +1)$.