= Mass cancellation in sharp-freeze-out number dilution
{title2=$n_0=6P^{3/2}[k_BT_0/(\hbar c)]^3e^{-P}$}
For an imposed equilibrium cutoff $mc^2=Pk_BT_f$ and density $n_f=6[mc^2k_BT_f/(\hbar c)^2]^{3/2}e^{-P}$, 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 $g_{*s,0}/g_{*s,f}$ when these assumptions are relaxed.
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