= 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,
$$
\nabla_\mu(su^\mu)=0.
$$
Since a fixed <comoving volume> has physical <volume> proportional to $a^3$, the covariant divergence gives
$$
\dot s+3Hs=0,\qquad \frac d{dt}(sa^3)=0.
$$
Thus 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 $g_{*s}T^3$, where $g_{*s}$ is the <effective number of relativistic entropy degrees of freedom>. Conservation gives $aTg_{*s}^{1/3}=\text{constant}$, so $T\propto a^{-1}$ only while $g_{*s}$ is constant. Annihilation into a still-coupled bath can change its <temperature> relative to a decoupled species while conserving the bath's <comoving entropy>.<Cosmology by David Tong>[Section 2.5.4, Electron-Positron Annihilation and Neutrino Decoupling: separate thermal baths and entropy conservation]{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.
Back to article page
