Separately conserved visible and dark entropy ($\xi =s_{\mathrm{SM}}/s_{\varphi }$)

After thermal decoupling in cosmology, two homogeneous sectors that evolve adiabatically without entropy injection satisfy $s_{\mathrm{SM}}{a}^3=\mathrm{constant}$ and $s_{\varphi }{a}^3=\mathrm{constant}$, hence their entropy density ratio $\xi$ is fixed. At a common relativistic temperature before decoupling, $\xi =g_{\ast s}^{\mathrm{SM}}/g_{\ast s}^{\varphi }$. Internal annihilations can change each sector's temperature and effective entropy degrees of freedom without changing its total comoving entropy.