For an adiabatically expanding homogeneous universe, the <entropy density> satisfies
$$
\nabla_\mu(su^\mu)=0,
\qquad
\dot s+3Hs=0,
$$
so the entropy $sa^3$ in a comoving volume is constant.

Cosmological entropy conservation

The <entropy density> is
$$
s=\frac{\rho+P-\mu n}{T}.
$$
For the homogeneous cosmological plasma, adiabaticity gives the <cosmological entropy conservation> equation
$$
\nabla_\mu(su^\mu)=0,
\qquad
\dot s+3Hs=0,
$$
or equivalently $sa^3=\text{constant}$ in a comoving volume.

Solved by gpt-5.6-sol high.

Solution

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

{title2=$s$}
{wiki}

For a homogeneous equilibrium system with <energy density> $\rho$, <pressure> $P$, <chemical potential> $\mu$, and <number density> $n$, the entropy density is
$$
s=\frac{\rho+P-\mu n}{T}.
$$

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