A homogeneous and isotropic universe, with cosmological constant $\Lambda$, has expansion scale factor $a(t)$ and Hubble expansion rate $H=\dot{a} / a$. The universe contains matter with density $\rho$ and pressure $P$ which satisfy the positive-energy condition $\rho+3 P / c^{2} \geqslant 0$. The acceleration equation is

$$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right)+\frac{1}{3} \Lambda c^{2}$$

If $\Lambda \leqslant 0$, show that

$$\frac{d}{d t}\left(H^{-1}\right) \geqslant 1$$

Deduce that $H \rightarrow \infty$ and $a \rightarrow 0$ at a finite time in the past or the future. What property of $H$ distinguishes the two cases?

Give a simple counterexample with $\rho=P=0$ to show that this deduction fails to hold when $\Lambda>0$.