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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 $a(t)\propto t^q$)?

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It can't. Physics (as we know it) doesn't work at the singularity. Trying to use equations that apply at other times there doesn't make any sense.

Now you could ask how light emitted when the universe was tiny is just now reaching us and the answer would be due to a fantasically large expansion rate, that stretched the space between us (the observer) and the source prior to it reaching us.

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  • $\begingroup$ 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? $\endgroup$
    – JonTrav1
    Jul 30, 2016 at 7:59
  • $\begingroup$ Also, shouldn't such an extreme expansion rate be extremely rapid? That is, even faster than the speed of light? $\endgroup$
    – JonTrav1
    Jul 30, 2016 at 8:01
  • $\begingroup$ What you're doing is more complex than that though. There is a critical temperature in the early universe when space was opaque to the propagation of light. It's similar to why the sun's light takes soo long from being made to escape it: gizmodo.com/… $\endgroup$
    – R. Rankin
    Jul 30, 2016 at 8:03
  • $\begingroup$ @JonTrav1 For any hubble rate there is a distance at which one can say the space between the observer and the emitter is moving faster than light. This in no way violates relativity. It makes for a great read if you get a book on it. (: $\endgroup$
    – R. Rankin
    Jul 30, 2016 at 8:06

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