Sharp-freeze-out relic dilution ($n_0=n_f(a_f/a_0{)}^3$)

In a stable-particle approximation with negligible later annihilation, comoving particle number is conserved and physical number density falls as $a^{-3}$. Replacing scale-factor ratios by photon-temperature ratios additionally assumes no change in the entropy degrees of freedom.

For an imposed equilibrium cutoff $m{c}^2=P{k}_B{T}_f$ and density $n_f=6[m{c}^2{k}_B{T}_f/(\hslash c{)}^2{]}^{3/2}{e}^{-P}$, dilution by $(T_0/T_f{)}^3$ cancels the explicit mass in the final number density. The rest-energy density still scales with particle mass. This result assumes constant entropy degrees of freedom and no post-cutoff annihilation; entropy conservation adds a factor $g_{\ast s,0}/g_{\ast s,f}$ when these assumptions are relaxed.