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

At neutrino decoupling, neutrinos, electrons, positrons, and photons share one temperature. The neutrinos subsequently free stream, so $T_{\nu }\propto a^{-1}$. In the still-coupled electromagnetic plasma, the effective entropy degrees of freedom change during electron-positron annihilation in cosmology fromto $g_{\ast s,\mathrm{after}}=2$. Separate cosmological entropy conservation in that plasma gives $g_{\ast s}{T}_{\gamma }^3{a}^3=\mathrm{constant}$, while $T_{\nu }a$ remains constant. Consequently the Cosmic neutrino background temperature obeys

For one species with two internal states, the relic-neutrino energy density obtained from the frozen Fermi-Dirac distribution isIn the relativistic limit, set $x=ap/T_{\nu }$ and use the standard Riemann zeta function integralThis gives

At $T_{\nu }/a\ll m_{\nu }$, the Taylor expansion of the relativistic energy is $\sqrt{p^2+m_{\nu }^2}=m_{\nu }+p^2/(2m_{\nu })+O(p^4/m_{\nu }^3)$. The number density isThe ratio of the next momentum moment to this one isThereforeso $\boxed{F=15\zeta (5)/(2\zeta (3))\simeq 6.47}$.

Comparing the correction in part c with the nonrelativistic kinetic energy $m_{\nu }⟨v^2⟩/2$ gives the characteristic late-time speedThus the late-time relic-neutrino speed redshifts as $a^{-1}$.

The Cosmic neutrino background began free streaming at neutrino decoupling, long before cosmological recombination, so its directional flux can retain information about density fluctuations from epochs inaccessible to the Cosmic microwave background. Earlier decoupling does not by itself guarantee a larger comoving radial distance. A sufficiently massive neutrino eventually becomes nonrelativistic, and its travelled distance iswhich may be smaller than the photon distance because $v(a)<c$. Hence the cosmic neutrino background last-scattering surface need not lie beyond the Cosmic microwave background last-scattering surface; it generally does for neutrinos that remain sufficiently relativistic for sufficiently long.