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**Homework Statement**

Expansion of the universe is described by the scale factor R(t), where t is the time after the Big Bang. For a flat universe the scale factor is today

[itex]R(t)=C_{1}\cdot t^{\frac{2}{3}}[/itex]

When the Universe was radiation dominated, for t <200,000 years, the scale factor was

[itex]R(t)=C_{2}\cdot \sqrt{t}[/itex]

Today, about 12 billion years after the Big Bang, we measure the distance to another superhop to 1 billion light years. How big would the distance between these sites one year after the Big Bang, if we assume that we live in a flat universe.

**The attempt at a solution**

I express the distance in meters

[itex]R(t)=9.461\cdot 10^{24}\, m[/itex]

Then I express the time in seconds

[itex]t=3.787\cdot 10^{17}\, s[/itex]

I found that the constant is

[itex]C_{1}=1.8075\cdot 10^{13}[/itex]

I need to know the constant [itex]C_{2}[/itex], but I'm not sure how I'll be able to calculate it.

I have tested just to put [itex]C_{1}=C_{2}[/itex], but the result for the distance was wrong. I then realized that the two constants have different units.

Does anyone have a suggestion how I may proceed?