# Horizon distance for $a(t) \propto t^q$

Suppose a universe with scale factor $a(t)\propto t^q$, where $q>0$, described by RW metric. In such universe, at time $t = 0$, $a(0) = 0$, which implies that the proper distance between everything is $0$. The horizon distance is defined as the distance of the furthest objects one can see, that is the distance which light emitted at $t=0$ has reached us at $t=t_0$. My question is: since at time $t=0$ everything was effectively a single dot, light emitted at $t=0$ instantaneously reached everywhere. So how can light, emitted at $t=0$, can just now, at $t=t_0$ reach us (in a universe with scale factor $a(t)\propto t^q$)?

• So if I want to calculate the horizon distance using the RW metric, then $$d_{hor}(t_0) = c\int\limits_0^{t_0}\frac{dt}{a(t)}$$, the lower limit $0$ is just a mathematical idealization to make the integral nicer? That is, since it is just a limit (since $a^{-1}(t)$ is singular at $0$), we look at times where the universe had dimensions, just extremely small ones? Jul 30, 2016 at 7:59