Neutrino decoupling

Neutrino decoupling occurs when weak interactions become slower than cosmic expansion. The neutrino distribution then free streams and cools as $T_{\nu }\propto a^{-1}$.

In natural units with unreduced Planck mass $M_P=G^{-1/2}$, the radiation-era Hubble parameter scales as $\sqrt{g_{\ast }}{T}^2/M_P$, while a low-energy weak interaction rate scales as $G_F^2{T}^5$. Their equality gives a decoupling scale of order one MeV. Distinct weak processes have distinct rate coefficients: neutrino decoupling and cosmological weak freeze-out of neutron-proton conversion are nearby events, not an identical precise temperature inferred from the power laws alone.

A relativistic species decoupling before ordinary neutrino decoupling cools as $T_X\propto a^{-1}$. Cosmological entropy conservation in the remaining bath gives $T_X/T_{\nu }=[10.75/g_{\ast s,D}{]}^{1/3}$, where $g_{\ast s,D}$ counts that bath just after the early species has left it. Subsequent electron-positron annihilation in cosmology gives $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$. For the ideal high-temperature relativistic entropy content of the Standard Model, the resulting ratio is about $0.332$.

After neutrino decoupling, electron-positron annihilation transfers entropy to photons but not to neutrinos. Comparing the electromagnetic entropy degrees of freedom before and after annihilation gives $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$.

For $T\ll m_e$ and negligible chemical potentials, the combined equilibrium density of electrons and positrons divided by the photon number density isThe exponential suppression eventually makes pair equilibrium incompatible with the residual electrons required by charge neutrality.