Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 312 / 3 / ii / Solution

Let $Q^{ij}={\hat{k}}^i{\hat{k}}^j-{\delta }^{ij}/3$. Rotations around $\hat{\mathbf{k}}$ and the vanishing trace imply that the angular integral in the scalar neutrino anisotropic stress has the form $A_{\ell }{Q}^{ij}$. Contract with ${\hat{k}}_i{\hat{k}}_j$ and use ${\mu }^2-1/3=(2/3)P_2(\mu )$. By the Orthogonality of Legendre polynomials,Only the quadrupole in the neutrino Boltzmann hierarchy contributes. Its phase factor is $(-i{)}^2=-1$, so the supplied stress definition becomesComparing with the chosen normalization $-(4/3){\bar{\rho }}_{\nu }\Pi Q^{ij}$ givesThe sign and factor here follow jointly from the stress convention and the unweighted Legendre polynomial coefficients; they should not be transferred unchanged to a differently normalized hierarchy.