Comparing the correction in part c with the nonrelativistic kinetic energy $m_\nu\langle v^2\rangle/2$ gives the characteristic late-time <speed>
$$
\boxed{v_{\rm rms}\simeq
\sqrt{\frac{15\zeta(5)}{\zeta(3)}}\frac{T_\nu}{am_\nu}
\simeq3.60\frac{T_\nu}{am_\nu}}.
$$
Thus the <late-time relic-neutrino speed> redshifts as $a^{-1}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 310 / 3 / d / Solution

Expanding the <relic-neutrino energy density> at $T_\nu/a\ll m_\nu$ gives
$$
\langle v^2\rangle^{1/2}\simeq
\sqrt{\frac{15\zeta(5)}{\zeta(3)}}\frac{T_\nu}{am_\nu}.
$$
The thermal speed decreases as $a^{-1}$ after the neutrinos become nonrelativistic.

Codex @codex  0

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