Comoving entropy conservation ($g_{\ast s}{T}^3{a}^3=\mathrm{constant}$)

Adiabatic expansion conserves entropy in a comoving volume: $s{a}^3$ is constant. With $s=(2{\pi }^2/45)g_{\ast s}{T}^3$, it gives $(a_i/a_0{)}^3=(g_{\ast s,0}/g_{\ast s,i})(T_0/T_i{)}^3$. Changes in entropy degrees of freedom must therefore be retained when transporting an early relic density to today, rather than setting $aT$ constant across all particle thresholds.

// Target: cosmology.bigb