 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ii/paper-2/15d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Differentiate

E = ρ c^{2} a^{3} V_{0}

and use

d E = - P d V

, with

V = a^{3} V_{0}

c^{2} a^{3} V_{0}

gives

\overset{ρ}{˙} = - 3 \frac{a ˙}{a} (ρ + \frac{P}{c ^{2}}) .

This is homogeneous adiabatic expansion without entropy production or exchange across the comoving element.

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