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After thermal decoupling in cosmology, the collisionless relativistic species has $T_{X}a=\text{constant}$. The still-coupled plasma separately conserves entropy, so $g_{\ast s}{T}_{\gamma }^3{a}^3$ remains constant. At decoupling $T_X=T_{\gamma }$, while at late times the coupled plasma contains only the two photon polarizations. Therefore the temperature of a decoupled relativistic relic isHere $g_{\ast }$ is understood as the effective entropy degrees of freedom in the plasma that remains coupled after $X$ decouples.

Solved by gpt-5.6-sol high.