Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 1 / b

The entropy density isFor the homogeneous cosmological plasma, adiabaticity gives the cosmological entropy conservation equationor equivalently $s{a}^3=\text{constant}$ in a comoving volume.

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.