First integral of the Friedmann acceleration equation
= First integral of the Friedmann acceleration equation
{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.