 Cosmological entropy conservation

Let $s$ be proper entropy density in a homogeneous FLRW metric and $u^{\mu }$ the comoving fluid four-velocity. Reversible adiabatic expansion has a conserved entropy current,Since a fixed comoving volume has physical volume proportional to $a^3$, the covariant divergence givesThus the conserved quantity is comoving entropy; the proper entropy density falls as $a^{-3}$. This dilution does not require a decrease of the total entropy in the selected material volume.

For equilibrium relativistic matter at negligible chemical potentials, $s$ is proportional to $g_{\ast s}{T}^3$, where $g_{\ast s}$ is the effective number of relativistic entropy degrees of freedom. Conservation gives $aT{g}_{\ast s}^{1/3}=\text{constant}$, so $T\propto a^{-1}$ only while $g_{\ast s}$ is constant. Annihilation into a still-coupled bath can change its temperature relative to a decoupled species while conserving the bath's comoving entropy.^[ref]

Irreversible entropy production replaces the zero divergence by a nonnegative source. Separately decoupled sectors need their own entropy balances; imposing one common temperature after decoupling would add an assumption.