Mass cancellation in sharp-freeze-out number dilution ($n_0=6P^{3/2}[k_B{T}_0/(\hslash c){]}^3{e}^{-P}$)

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.