 Solution

The Hubble time is the instantaneous inverse expansion rate $H_0^{-1}$, where $H=\dot{a}/a$. A radial light ray covers comoving distance $\int cdt/a(t)$. Since $a(t_0)=1$, the present proper radius of the particle horizon isFor the power-law scale factor, $a(0)=0$ requires $\alpha >0$. The integral converges only when $\alpha <1$, in which caseAt $\alpha =1$ the divergence is logarithmic, and at $\alpha >1$ it is a power divergence. The familiar flat-universe epochs have $\alpha =1/2$ for radiation and $\alpha =2/3$ for pressureless matter, obtained from $\rho \propto a^{-4}$ or $a^{-3}$ and the Friedmann equation. Both have $0<\alpha <1$, so their ages are less than the present Hubble time. These power laws describe idealized single-component epochs rather than the full late-time expansion history.