 Solution

Write ${\psi }_0=\psi (0,\mathbf{k})$. For regular adiabatic initial conditions during radiation domination, use ${\Theta }_0=-{\psi }_0/2+O(k\eta )$, ${\Theta }_2=O((k\eta {)}^2)$, and constant potentials. The dipole equation in the neutrino Boltzmann hierarchy givesRegularity removes a constant dipole, and integration yieldsThen the quadrupole equation givesIts regular initial value is zero, soFor the usual regular growing solution, the monopole has no linear correction and parity improves the dipole and quadrupole errors to $O((k\eta {)}^3)$ and $O((k\eta {)}^4)$, respectively. The weaker bounds above already follow from the assumptions explicitly given. Although the quadrupole is small on superhorizon scales, the trace-free Einstein field equations divide its stress by $k^2$, allowing a finite leading potential difference.