How Far Could Light Travel by the Time of Decoupling in an Expanding Universe?

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Homework Statement



At the time of decoupling the universe was 1/1000 of its present size. How for could light have traveled in the time up to decoupling? (assume that the universe was dominated by radiation until then)


Homework Equations





The Attempt at a Solution


I'm a little confused about how to include the fact that the universe was expanding while the light was traveling up to the time of decoupling.

I have the age of the universe at decoupling(calculated earlier) and I have calculated the scale factor as a function of time for a radiation dominated universe. I assume that the scale factor means that light travels less distance with time as distance is expanding but I'm not sure about this...and how would I calculate this distance?
 
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Here's a hint: light travels a distance of cdt in a Newtonian world, but there's a scale factor, so dr=cdt/a(t).
 
the equation given above is right but according to the book's answer the distance light has traveled is equal to simple [c x t]
 

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