Recollapse of a closed Friedmann universe with nonnegative pressure
= Recollapse of a closed Friedmann universe with nonnegative pressure
For positive spatial curvature, zero cosmological constant, and $\rho,P\geq0$, continuity makes $\rho a^3$ nonincreasing during expansion. The negative curvature term then prevents unbounded growth of $a$, while the acceleration equation forces $\dot a$ to reach zero in finite time.