Density bound on a relativistically decoupled massive relic ($m_x\lesssim \frac{{\Omega }_{x,\max }}{{\Omega }_{\gamma }}\frac{⟨E_{\gamma }⟩}{n_x/n_{\gamma }}$)

A species freezing out while relativistic retains its thermal comoving number. If it is nonrelativistic today, ${\rho }_x\simeq m_x{n}_x$. Comparing with photons gives ${\Omega }_x/{\Omega }_{\gamma }=m_x(n_x/n_{\gamma })/⟨E_{\gamma }⟩$. An upper density fraction therefore bounds its mass inversely with its relic number density. The relation does not apply unchanged to a species whose abundance is depleted by nonrelativistic annihilation.