Strong energy condition in a Friedmann universe

With zero cosmological constant, the condition $\rho +3P\ge 0$ implies $\ddot{a}\le 0$ andwherever $H\ne 0$.

If $H>0$ at one time, the inequality $d(H^{-1})/dt\ge 1$ forces $H\to +\infty$ and $a\to 0$ within finite time to the past. If $H<0$, it forces $H\to -\infty$ and $a\to 0$ within finite time to the future.