Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 9B / b / Solution

{c}

Combining the perfect-fluid continuity equation with
$$
\frac{\ddot a}{a}=-\frac{4\pi G}{3c^2}(\rho+3P)
$$
shows that
$$
\frac{8\pi G}{3c^2}\rho a^2-\dot a^2
$$
is constant in time. This constant is the spatial-curvature integration constant in the first Friedmann equation.


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 Incoming links: First integral of the Friedmann acceleration equation