Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 310 / 2

Before electron-positron annihilation in cosmology, photons and electron-positron pairs share a temperature. Their effective entropy degrees of freedom areAfter annihilation the electromagnetic plasma contains only the two photon polarizations, so $g_{\ast s}^{\mathrm{after}}=2$. The already decoupled neutrinos receive none of this entropy and simply cool as $T_{\nu }\propto a^{-1}$. Applying cosmological entropy conservation separately to the coupled electromagnetic plasma giveswhereas $T_{\nu }^3{a}^3=T_{\mathrm{before}}^3{a}_{mbefore}^3$. Division yields the Cosmic neutrino background temperature

The energy density from a phase-space distribution function with $g=2$ internal states isIn the ultrarelativistic limit $T_{\nu }\gg m_{\nu }$, set $x=p/T_{\nu }$. The Fermi-Dirac distribution then givesSince $\zeta (4)={\pi }^4/90$,

When ${\mu }_{\nu }/T_{\nu }\gg 1$, the Fermi-Dirac distribution approaches the step function $f(p)=1$ for $p<{\mu }_{\nu }$ and zero for $p>{\mu }_{\nu }$. For one internal state and negligible mass, the degenerate relic neutrino energy density is thereforeso $\boxed{A=1/(8{\pi }^2)}$.

At thermal equilibrium, the antineutrino chemical potential is $-{\mu }_{\nu }$. Its occupation is approximately $e^{-(p+{\mu }_{\nu })/T_{\nu }}$, givingIt is exponentially smaller than the neutrino density; this particle-antiparticle imbalance is a cosmological lepton asymmetry.

Write the neutrino degeneracy parameter as ${\xi }_{\nu }={\mu }_{\nu }/T_{\nu }$. Requiring one degenerate species not to exceed today's critical density givesUsing $T/T_{\nu }=(11/4{)}^{1/3}$,Several equally degenerate species would strengthen the bound by the fourth root of their number.

A large neutrino degeneracy parameter raises the relativistic energy density and therefore the expansion rate. Big Bang nucleosynthesis tests this through primordial light-element abundances; an electron-neutrino chemical potential also shifts neutron--proton chemical equilibrium directly. The Cosmic microwave background power spectrum tests the changed expansion rate, matter-radiation equality, sound horizon, damping scale, and neutrino anisotropic stress. Finally, large-scale structure of the universe and the matter power spectrum test the altered equality scale and neutrino free streaming. These observables distinguish a degenerate-neutrino cosmology from the standard thermal relic scenario in complementary epochs.