In a homogeneous and isotropic universe $(\Lambda=0)$, the acceleration equation for the scale factor $a(t)$ is given by

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

where $\rho(t)$ is the mass density and $P(t)$ is the pressure.

If the matter content of the universe obeys the strong energy condition $\rho+3 P / c^{2} \geqslant 0$, show that the acceleration equation can be rewritten as $\dot{H}+H^{2} \leqslant 0$, with Hubble parameter $H(t)=\dot{a} / a$. Show that

$$H \geqslant \frac{1}{H_{0}^{-1}+t-t_{0}}$$

where $H_{0}=H\left(t_{0}\right)$ is the measured value today at $t=t_{0}$. Hence, or otherwise, show that

$$a(t) \leqslant 1+H_{0}\left(t-t_{0}\right)$$

Use this inequality to find an upper bound on the age of the universe.