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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-310/1/b/i/solution