A Summing Up — Question 2

Question 2

A relic neutrino changes from radiation-like to matter-like behavior

For a decoupled relic use $T_{\nu0}=1.95$ K, typical momentum $pc=3.15k_BT_\nu$ and $T_\nu=T_{\nu0}(1+z)$. Let mass energy be .050 eV and $k_B=8.62\times10^{-5}$ eV/K.

1. Estimate the redshift where typical pc equals the rest energy.

2. For total particle plus antiparticle density 112 cm$^{-3}$ for this mass eigenstate, estimate present rest-energy density and state its later scale-factor dependence.

Question 2 - Solution

A relic neutrino changes from radiation-like to matter-like behavior

(a) Method and calculation

$$1+z_{\mathrm{nr}}=\frac{.050}{3.15(8.62\times10^{-5})(1.95)}.$$$$\boxed{1+z_{\mathrm{nr}}\simeq94.43,\quad z_{\mathrm{nr}}\simeq93.43}.$$

(b) Interpretation and further calculation

$$\boxed{u_{\nu,0}\simeq n_\nu mc^2=112(.050)=5.60\ {\mathrm{eV/cm^3}}}.$$$$\boxed{u_\nu\propto a^{-3}\ \text{when nonrelativistic},\quad u_\nu\propto a^{-4}\ \text{when relativistic}}.$$

Physical check

The transition is gradual across a momentum distribution. The specified density refers to one eigenstate including its antiparticle, avoiding an extra factor of two.