= Solution
The physical <particle horizon> is the greatest physical distance at time $t$ from which a light signal could have reached the observer since the beginning of the cosmological model. If that beginning is $t=0$, radial null propagation has comoving distance $\int_0^tc\,dt'/a(t')$, so
$$
d_H(t)=ca(t)\int_0^t\frac{dt'}{a(t')}.
$$
For $a(t)=t^\alpha$ and $0<\alpha<1$, the integral converges at zero and equals $t^{1-\alpha}/(1-\alpha)$. Consequently
$$
\boxed{d_H(t)=\frac{ct}{1-\alpha}.}
$$
The <horizon problem> is that widely separated regions of the observed cosmic microwave background have nearly equal temperatures although their past causal regions did not overlap before recombination in the usual decelerating hot-big-bang history. That history offers no causal equilibration mechanism for their observed uniformity. The <particle horizon> used here is distinct from the <Hubble radius> $c/H$.
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