= 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 $\int c\,dt/a(t)$. Since $a(t_0)=1$, the present proper radius of the <particle horizon> is
$$
\boxed{R_0=c\int_0^{t_0}\frac{dt}{a(t)}.}
$$
For the power-law scale factor, $a(0)=0$ requires $\alpha>0$. The integral converges only when $\alpha<1$, in which case
$$
\boxed{R_0=\frac{ct_0}{1-\alpha},\qquad
H_0=\frac\alpha{t_0},\qquad t_0=\alpha H_0^{-1}.}
$$
At $\alpha=1$ 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 \b[less than the present <Hubble time>]. These power laws describe idealized single-component epochs rather than the full late-time expansion history.
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