Jakob K. Mogensen, Steen Hannestad, Thomas Tram

Cosmological constraints on self-interacting neutrinos require a Boltzmann hierarchy in which the collision term is projected onto momentum-averaged multipoles. We revisit the collision kernel for neutrino-neutrino scattering mediated by a light scalar and derive an exact analytic expression for the multipole integral that determines the coefficients $α_\ell$. The key idea is to express the integration kernel as angular derivatives of the Yukawa-potential $\frac{\mathrm{e}^{-P/2}}{P}$, move the derivatives onto Legendre polynomials, and reduce the remaining momentum integrals to a single base family obeying a first-order recurrence. This gives an exact rational-plus-$π^2$ representation for every multipole, together with a compact implementation based on exact rational arithmetic. We provide the recurrence relations, a closed form for the base integral, and an asymptotically constrained approximation suitable for Boltzmann codes such as CLASS. Our numerical implementation is publicly available in the Jupyter notebook IntegralComputation.ipynb.