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Normalize the present scale factor to $a_0=1$. Conservation of the number of electrons associated with the baryons makes their physical number density scale as $a^{-3}$. At this early epoch they are free rather than bound in neutral atoms, soHere the specified present density is interpreted as the conserved total electron abundance, including electrons that might be bound today. A present free-electron density would require an ionization-fraction correction.

Density alone does not determine whether a gas is relativistic; its random kinetic energy must be compared with the rest energy. In the usual thermal Hot Big Bang model, use the measured cosmic microwave background temperature $T_0\simeq 2.7\mathrm{K}$ and the temperature scaling of thermal photon number and energy densities, $T_{\gamma }\propto a^{-1}$. Tight coupling then givesSince $m_e{c}^2\simeq 5.1\times 10^5\mathrm{eV}$,Their thermal speed is of order $\sqrt{3k_B{T}_e/m_e}\simeq 0.04c$. The gas is fully ionized because its temperature is far above the recombination temperature, but ionization does not imply relativistic electrons. Thermal electron-positron pairs are exponentially suppressed at this temperature.^[ref]

The given age and scale-factor ratio should be treated as independent supplied estimates in the following rate comparison. They do not describe the same epoch in standard radiation-era expansion: $a/a_0=10^{-6}$ normally corresponds to an age of order a year, not $10^4$ years. The density and temperature estimates above depend on the scale-factor ratio, not on assigning it the stated age.

The given formula is the integral of the Planck photon distribution. SetThenwhereEach photon of frequency $\nu$ has energy $h\nu$, so the energy density isThis is the temperature scaling of thermal photon number and energy densities.

After decoupling, cosmological redshift givesThe occupation number is conserved along the freely propagating photon trajectory. Its exponential argument at decoupling becomesThus the spectrum still has the Planck form ifFrequency and temperature redshift by the same factor, which is the redshift preservation of a thermal photon spectrum.